Jane Street Interview Questions - Complete List

Quant Interview Questions

671 questions

In a group of 10 people standing in a circle, how many handshakes occur if each person shakes hands with every other person exactly once?

Quant Trader Intern Interview

A car has a uniformly distributed value between 0 and 1000. If you bid less than this value, you get nothing but keep your bid. If you bid more than this value, you can sell the car for 1.5 times the value. What should you bid? The surface of a 3x3 block is painted. The block is split into 27 cubes and one is dropped on the floor. What is the probability that no visible face is painted? If no visible face is painted, what is the probability that it is the center cube?

Trader Interview

You flip 4 fair coins and receive $1 for each coin that lands heads. What is the expected value of your total winnings?

Trader Interview

If you roll two dice, and you have the option to roll again and get paid the highest roll, when is it optimal to reroll?

Trader Interview

What is 2,000,000 minus 2,022?

Trader Interview

How many digits does 99 raised to the 99th power have?

Trader Interview

You and I play a game: we both pick a number before rolling two dice. The one whose chosen number is closer to the sum of the dice wins. The sum is not uniformly distributed—there is a 40% chance of getting a sum of 12, and the remaining probability is distributed uniformly among all other possible sums. What is your optimal strategy, and should you go first or second?

Trader Interview

There are four coins. For each heads you get, you receive $1. After flipping all four coins, you may re-flip one coin of your choice. What is the maximum amount you would pay to play this game?

Trader Interview

Given the cards that are already on the table, make a market on the probability that the next card drawn is a face card.

Trader Interview

In a best-of-seven World Series, what are the odds that the series goes to seven games if each team has an equal chance of winning each game?

Trader Interview

If you have an asset that increases or decreases by 20% each day with probability 1/2, what is the probability that after n days, the asset's price is unchanged?

Trader Interview

What is the probability of drawing two kings from a standard deck of cards?

Trader Interview

What is the expected value when rolling two fair dice? Also, if I offer you a game where I receive X dollars for outcome set A and you receive Y dollars for outcome set B, would you play? Explain your reasoning.

Trader Interview

Two teams, each with an equal probability of winning, are playing a best-of-7 tournament series. What is the probability that the series lasts until the 7th game?

Trader Interview

On a multiple-choice exam with three possible answers for each of the five questions, what is the probability that a student would get four or more correct answers just by guessing?

Trader Interview

I flip four coins and give you a dollar for each head. After all four coins are flipped, you may choose to re-flip one coin that landed tails. What is the expected value of this game?

Trader Interview

You roll a 30-sided die (numbered 1 to 30) repeatedly and sum all the outcomes. You stop rolling when the sum is greater than or equal to 300. What is the most likely final value of the sum?

Trader Interview

There are infinitely many cards numbered 1 to 5. You construct a deck by selecting any composition of these cards. A player guesses a value between 1 and 5 and draws a card at random from the deck. If they guessed the correct number, they receive a payout equal to that value; otherwise, they get nothing. What composition of the deck minimizes the player's expected payout?

Trader Interview

What day of the week will it be exactly 10 years from today?

Trader Interview

You roll a fair six-sided die and receive a payoff equal to the number shown. a) What is the expected payoff? b) If you are given the option to re-roll after seeing the result, under what conditions should you choose to re-roll? What is your expected payoff with this re-roll option?

Trader Interview

You have 4 coins. I flip them, and you receive one dollar for each heads. How much are you willing to pay to play this game? What if, after the initial flip, I give you the option to reflip any number of coins? What if I require you to reflip any number of coins of your choice?

Trader Interview

Five pirates of different ages have a treasure of 100 gold coins. On their ship, they decide to split the coins as follows: The oldest pirate proposes how to share the coins, and all pirates (including the oldest) vote for or against the proposal. If 50% or more of the pirates vote for it, the coins are shared accordingly. Otherwise, the proposing pirate is thrown overboard, and the process repeats with the remaining pirates. If a pirate would receive the same number of coins whether he votes for or against the proposal, he votes against so that the proposer is thrown overboard. Assuming all five pirates are intelligent, rational, greedy, do not wish to die, and are skilled in mathematics, what will happen?

Trader Interview

If I write down all the numbers from 1 to 1,000,000 on a page, how many times do I write the digit 2?

Trader Interview

If there are N points on a circle and you draw a chord between each pair of points, how many regions is the circle subdivided into?

Trader Interview

Part 1: We are playing a game where we start at 0 and flip a coin until someone wins. Every heads adds +1 to the number and every tails adds -1. If the number reaches -10, the interviewer wins; if it reaches 20, you win. What is the probability that the interviewer will win? Part 2: I am thinking of a 10-digit number. The first digit is the number of 0s in the number, the second is the number of 1s, and so on. What is the number?

Trader Interview

If you have four coins and I have four coins, we both toss our four coins. If the outcome of your four coins matches exactly with mine, I will give you $2; otherwise, you give me $1. Should you take this bet?

Trader Interview

What is 37 multiplied by 43?

Trader Interview

If two teams are equally talented, what is the probability that a 7-game series between them reaches the 7th game?

Trader Interview

First, throw a die with numbers 1 to 8. After seeing the result, you can either stop or throw a second die with numbers 1 to 12. You win the amount of money equal to the final number shown. What strategy should you use to maximize your expected winnings?

Trader Interview

What is the expected number of heads when you toss a fair coin 6 times?

Trader Interview

Given n unbiased coins, what is the probability that exactly half of them are heads? Answer the question for n = 2, 3, and 20,000.

Trader Interview

Given two fair dice, what is the probability that both show six if it is observed that at least one die shows a six?

Trader Interview

What is 15% of 115?

Trader Interview

A has 5 fair coins and B has 4 fair coins. A wins only if he flips more heads than B does. What is the probability that A wins?

Trader Interview

Given 5 pumpkins and the sum of the weights of every pair of two pumpkins, determine the weight of each individual pumpkin.

Trader Interview

Suppose there are n people on a bus. At each of three stops, 2/3 of the people on the bus get off and 7 people get on. What is the minimum number of people that were initially on the bus before the first stop, given that the number of people on the bus remains an integer after each stop?

Trader Interview

You flip 4 fair coins and receive x dollars, where x is the square of the number of heads. What are your expected winnings?

Trader Interview

If there are N points on a circle and a chord is drawn between each pair of points, how many regions does the circle get subdivided into?

Trader Interview

What is 38 squared?

Trader Interview

Consider the following game: You are given a 12-sided die and will be paid the number that the die lands on. If you are unhappy with the roll, you may instead choose to roll two 6-sided dice and will be paid the sum of those two dice. What is the expected value of this game, and how much would you be willing to pay to play?

Trader Interview

In Linux, how can you find the total number of files in the current folder, including all subfolders?

Trader Interview

If two equally talented teams are playing in the World Series, what is the probability that the series lasts seven games?

Trader Interview

A bus pulls up to a stop. Three-quarters of the people get off and seven people get on. The bus goes to two more stops with the same results. What is the smallest number of people that could have initially been on the bus?

Trader Interview

You and a friend are betting on individual games of the World Series. For each game, if your team wins, you win a certain positive amount of money, and if your team loses, you lose that same amount. You and your friend want to construct a betting scheme so that you will win $1000 if your team wins the series and lose $1000 if your team loses the series, regardless of the final game score (e.g., 4-3, 2-4, etc.). How much should you bet on the first game?

Trader Interview

On an infinite chessboard, to how many possible positions can a knight move after 10 moves? Provide a 90% confidence interval. Is the actual answer even or odd?

Trader Interview

Player A had a better batting average than Player B in both the first and second halves of the season. Is it possible for Player B to have a better average than Player A over the entire season?

Trader Interview

Apples cost 27 cents each. How many apples can you purchase with $10? (Solve this in under one minute.)

Trader Interview

If you roll a 10-sided die and a 20-sided die, what is the probability that the 10-sided die shows a higher number than the 20-sided die?

Trader Interview

A steel ball with an 8 cm diameter weighs 8 ounces. How much does a steel ball with a 12 cm diameter weigh, assuming both balls are made of the same material?

Trader Interview

You throw a die two times. What is the probability of getting at least one six?

Trader Interview

We play a game where we each guess the sum of two dice rolls. Whoever's guess is closest to the actual outcome wins. What would your optimal strategy be?

Trader Interview

You want to buy a car at an auction where the car's price is uniformly distributed between $0 and $1000. If you bid more than the car's price, you win it at the price you bid; if you bid less than the car's price, you do not win it but also do not lose anything. You can sell the car for 1.5 times the value for which you bought it. What should you bid to maximize your expected profit?

Trader Interview

You have two packs of poker cards: one with 52 cards, the other with 104 cards. You will pick two cards separately from the same pack. If both cards are red, you win. Which pack should you choose to maximize your chances of winning?

Trader Interview

I roll a die and will give you an amount in pounds equal to the number on the die. For example, if I get 5, I will give you 5 pounds. What is the price you are willing to pay for this game?

Trader Interview

What is the sum of the digits of all the numbers from 1 to 1,000,000? This is different from the sum of the numbers. For instance, the sum of the numbers from 1 to 10 is 55, whereas the sum of the digits is 46.

Trader Interview

What is 87 multiplied by 46?

Trader Interview

What is the expected value of a fair six-sided die?

Trader Interview

What is the expected value of the maximum of two fair six-sided dice?

Trader Interview

You have two ropes, each of which takes 1 hour to burn from end to end, but they burn at non-uniform rates along their lengths. How can you use them to measure exactly 45 minutes?

Trader Interview

Given a fair six-sided die, what is the expected value of the difference between two independent die rolls?

Trader Interview

There are three pancakes: one with both sides burned, one with only one side burned, and one with neither side burned. The pancakes are stacked in a plate, and the top visible side is burned. What is the probability that the other side of this pancake is also burned?

Trader Interview

What is 45 multiplied by 69?

Trader Interview

What is 34 multiplied by 82? Solve within ten seconds without using a pen or pencil.

Trader Interview

I have a square and place three dots randomly along the four edges. (1) What is the probability that the dots, when connected, do not form a triangle? (2) What is the probability that the dots lie on distinct edges?

Trader Interview

What is 91 squared (91^2)?

Trader Interview

Suppose you have n points on a circle arranged so that, after connecting every pair of points with straight chords, the number of regions inside the circle is maximized. How many regions are there?

Trader Interview

We are going to play a game. We flip a fair coin repeatedly until either the sequence HHT or HTT appears. Which sequence would you choose, HHT or HTT, and why?

Trader Interview

You have 3,000 apples in Edinburgh and want to transfer as many apples as possible to London. You have a truck with a maximum capacity of 1,000 apples. London is 1,000 miles away from Edinburgh, and when the truck is carrying apples, it consumes one apple per mile traveled. What is the maximum number of apples you can deliver to London?

Trader Interview

In Russian Roulette, a six-chamber gun contains two adjacent bullets. You pull the trigger once, and do not get shot. What is the probability that you will get shot on the next trigger pull, without spinning the cylinder?

Trader Interview

What is the expected value of rolling one die?

Trader Interview

A number between 0 and 1 is written on a piece of paper. You and Person X are playing a game. You must choose a number less than the number on the paper, but greater than X's guess. X picks a number at random. What is the lowest number you can choose to maximize your probability of winning?

Trader Interview

Is it possible to relabel the numbers on two six-sided dice with other positive numbers so that the probability distribution of their sum remains unchanged?

Trader Interview

You are offered a game where you flip a fair coin. Each time it comes up heads, you win $1 and can choose to continue playing or stop. Each time it comes up tails, you lose $1. When is the optimal time to stop?

Quantitative Trader Interview

You and another player each independently sample a number uniformly from [0,1] (each number is private information). Both numbers are put into a pot, and you take turns bidding for the pot. What is your optimal bidding strategy?

Quantitative Trader Interview

You are presented with three games and must decide which one to play: 1) Throw a standard die and take the square of the resulting value as your score. 2) Throw two dice and take the product of the results as your score. 3) Throw five dice and take the square of the mode of the results as your score. Which game should you choose to maximize your expected payoff?

Quantitative Trader Interview

You have 100 light bulbs labeled from 1 to 100, all initially off. You perform 100 rounds of toggling: on the nth round, you toggle every nth bulb. Toggling means turning a bulb on if it's off and off if it's on. After all 100 rounds, which bulbs are on?

Quantitative Trader Interview

Two players each roll a die. Each wins if their number is higher than the other's, but they are not obligated to bet. If the first player declines the bet, that round ends. The second player knows whether the first player made a bet or not. Which player has a higher expected payout?

Quantitative Trader Interview

Sum all odd numbers from 1 to 2n + 1.

Quantitative Trader Interview

What is the smallest number whose digits multiply to 10,000?

Quantitative Trader Interview

You play rock, paper, scissors with a friend, but your friend cannot play rock. What is the optimal strategy for each player (the winner gets $1 from the loser), and what is the expected pay-off?

Quantitative Trader Interview

A coin has a probability of 0.8 of landing heads. What is the expected number of coin tosses needed to observe two consecutive heads?

Quantitative Trader Interview

In a robotic long jump contest, each robot advances from position 0 to the takeoff point at 1 by repeatedly drawing a real number uniformly from [0, 1] and adding it to their position. After each advance, the robot must choose to either jump or continue. If the robot crosses 1 without jumping, it scores 0. If it jumps before crossing 1, it draws a final number from [0, 1] and adds it to its position as its score. In a head-to-head, each robot is programmed to maximize its probability of winning and knows the other's strategy. Without knowing the other’s result, what is the probability that a robot's first attempt scores 0?

Quantitative Trader Interview

You have a fair coin and continually toss it. What is the expected number of tosses needed to obtain the sequence TTT (three tails in a row)?

Quantitative Trader Interview

If you flip 9 fair coins, what is the probability of getting an even number of heads? What about for n fair coins?

Quantitative Trader Interview

How much should you be willing to pay to play a game where you roll two 6-sided dice (die A and die B), and can then choose to be paid out either (A/B) or (B/A), whichever is higher?

Quantitative Trader Interview

How many handshakes occur in a group of 25 people if each person shakes hands with every other person exactly once?

Quantitative Trader Interview

There are 52 cards: 26 red and 26 black. You draw cards one at a time, indefinitely, with replacement. For each red card drawn, you lose 1 point; for each black card, you gain 1 point. What strategy maximizes your total points?

Quantitative Trader Interview

If Worker A takes 1 hour to finish a job and Worker B takes 2 hours to finish the same job, how long would it take them to complete the job working together?

Quantitative Trader Interview

Imagine I flip 100 coins in a sequence, and you need to guess the sequence in one try. You are allowed to ask one yes/no question. What do you ask to maximize the probability of guessing the sequence?

Quantitative Trader Interview

You are given an array of k linked lists, each of which is sorted in ascending order. Merge all the linked lists into one sorted linked list and return it.

Quantitative Trader Interview

You roll a standard 6-sided die (d6) and a 10-sided die (d10). Before seeing the result, you must bid on the sum of the two dice. If your bid matches the actual sum, you win that amount. What is the optimal sum to bid in order to maximize your expected winnings?

Quantitative Trader Interview

Eight people walk into a room and each person shakes hands with every other person exactly once. How many handshakes occur?

Quantitative Trader Interview

Three random points are chosen on the circumference of a circle. What is the probability that a table with legs placed at those points would stand (i.e., that all three points lie on the same semicircle)?

Quantitative Trader Interview

Eight quants from different banks want to know the average salary of the group, but each prefers not to disclose their own salary to the others. Can you design a strategy for the group to calculate the average salary without revealing individual salaries?

Quantitative Trader Interview

What is the expected number of tries needed to get 3 consecutive heads in repeated fair coin flips?

Quantitative Trader Interview

Suppose in a market making context, you are asked to make a market (set buy and sell prices, as well as quantities) on the sum or the product of two rolls of fair 6-sided dice. How would you determine appropriate prices and quantities in this scenario?

Quantitative Trader Interview

100

Q1: What is the smallest number whose digits multiply to 216? What about 10,000? Q2: Calculate the probability of getting 3 heads in 4 coin flips. What is the probability of getting an odd number of heads in 4 flips? What about in 9 flips? In N flips? Q3: What is the next date whose digits are all unique? Q4: After three coin flips, heads-tails-heads and heads-heads-tails are equally probable. But if I keep flipping, one combination becomes more probable than the other. Why is that?

Quantitative Trader Interview

101

Two symmetric dice each have two sides painted red, two sides painted black, one side painted yellow, and one side painted white. When this pair of dice is rolled, what is the probability that both dice land with the same color face up?

Quantitative Trader Interview

102

If you and your opponent each roll a fair six-sided die, and you win $1 if your number is larger, what is your expected winning?

Quantitative Trader Interview

103

The probability of raining on Saturday is 30%, and the probability of raining on Sunday is 40%. (a) What is the probability it will rain on the weekend? (b) What assumption is made in this calculation? (c) If the events are not independent, what are the maximum and minimum possible probabilities that it will rain during the weekend? (Answer: 40%, 70%)

Quantitative Trader Interview

104

If you earn one dollar for every head you get when flipping a fair coin, what is the expected amount you will make after 4 flips?

Quantitative Trader Interview

105

If you have x coins, into how many stacks, and with how many coins per stack, should you divide them in order to maximize the product of the stack sizes?

Quantitative Trader Interview

106

What is the expected value of the number obtained when rolling a fair six-sided die?

Quantitative Trader Interview

107

We toss a coin repeatedly until either the sequence HHT appears or HTH appears. What is the probability that HHT appears first? How does this change if the coin is unfair, with probability p of heads?

Quantitative Trader Interview

108

If you have 5 digits, what is the largest number you can create such that the product of its digits is 120? Can you make the number arbitrarily large if the restriction on the number of digits is removed?

Quantitative Trader Interview

109

How many coin tosses are expected before getting 5 heads in a row?

Quantitative Trader Interview

110

Person A and B are going to play a coin toss game. There is an initial score of 0, and whenever a head or tail appears, the score increases by 1 or decreases by 1, respectively. The coin is tossed repeatedly until one wins, that is, when the score reaches +2 or -2, A or B wins the game. There is also an initial stake of $1 for the game and person A has the option to double the stake before each coin toss. When one person wins the game, the other player needs to pay the current stake to the winner. The question is: if you are person A, what is your optimal strategy and what is your highest expected payoff in the game?

Quantitative Trader Interview

111

You may throw a six-sided die and, if you are dissatisfied with the result, you may re-roll it once. How should you decide when to keep the initial roll and when to re-roll in order to maximize the expected value?

Quantitative Trader Interview

112

What is the probability of getting an even number of heads when flipping n coins?

Quantitative Trader Interview

113

We randomly select 3 numbers from the set of the first 9 prime numbers, without replacement. What is the probability that the sum of those numbers is even, and why?

Quantitative Trader Interview

114

What is the smallest positive integer whose digits multiply to 10,000?

Quantitative Trader Interview

115

What is the expected value of a roll of a standard six-sided die?

Quantitative Trader Interview

116

How many coin tosses are required to obtain at least 4 tails with a probability of at least 0.7?

Quantitative Trader Interview

117

You roll a 100-sided die and may either accept the face value in dollars, or pay an additional dollar to re-roll. You can play as many times as you like. What is the optimal strategy?

Quantitative Trader Interview

118

Solve the following mental math problems: 1) What is 14% of 42? 2) What is 36 squared? 3) What day of the week is April 15th, 2142?

Quantitative Trader Interview

119

What is the angle between the hour and minute hands on a clock at 9:40?

Quantitative Trader Interview

120

You throw 1,000 fair coins. What is the probability of getting an even number of heads?

Quantitative Trader Interview

121

You have a drawer with an infinite number of socks in two colors, with each color equally likely. What is the expected number of socks you must draw individually from the drawer before you obtain a matching pair?

Quantitative Trader Interview

122

If I flip 100 coins and then multiply the number of heads by the number of tails, what is the expected value of that product? Can you provide a confidence interval for this value?

Quantitative Trader Interview

123

I play a game where I start with a score of 100. I then flip 10 coins in a row. Every time I get a head, I add 1 to my score. When I get a tails, I take the reciprocal of my score. If you are running this game, and people are given their score in pounds at the end of the game, how much would you charge people to play?

Quantitative Trader Interview

124

In the game 'Shut the box', what is the average score a player achieves by the end if they play using the optimal strategy? How does this average score change if, instead, the player makes random moves?

Quantitative Trader Interview

125

The date is written as DD/MM/YYYY. What is the closest date that will use at most one digit throughout the date?

Quantitative Trader Interview

126

Decompose 300,000 into its prime factors. Then, arrange all the digits from these prime factors to form the smallest possible number.

Quantitative Trader Interview

127

a) The odds of having pizza on a Saturday are 60%, and on Sunday, 30%. What are the odds of having pizza at least once during the weekend? b) How do the odds change if you are only allowed pizza on one of the two days? c) What if you can only have pizza on Sunday if you had it on Saturday first?

Quantitative Trader Interview

128

Find the smallest positive integer x such that the product of all the digits of x is 10,000.

Quantitative Trader Interview

129

Two teams are playing a best-of-7 tournament. Each team has a 50% chance of winning each game. What is the probability that the series reaches the 7th game?

Quantitative Trader Interview

130

You have all the clubs from a standard deck (13 cards). You can choose 2 cards and receive a payout equal to their product, where all face cards are valued at 0. You may pay $1 to reveal the difference between any two cards of your choice. What is the expected value you should be willing to pay to play this game?

Quantitative Trader Interview

131

If you have balls weighing 1g, 2g, ..., up to 40g, and you have a fair balance, what is the minimum number of their weights you need to know in advance so that you can measure all the others?

Quantitative Trader Interview

132

How can you create a fair coin using an unfair coin? Is there any method that ensures you do not waste any of your coin flips during the process?

Quantitative Trader Interview

133

If we flip a coin 100 times, what is the probability of getting an even number of heads?

Quantitative Trader Interview

134

There are an unknown number of people on a bus. At the first stop, three-quarters of the passengers get off and 7 people get on. This process repeats for two more stops. After this sequence, what is the minimum possible number of people that could be on the bus?

Quantitative Trader Interview

135

What is the result of multiplying 17 by 3.3?

Quantitative Trader Interview

136

An asymmetrical 12-sided die has a 40% chance of rolling a 12, and the remaining faces are equally likely. Two people each choose a number between 1 and 12. The person whose chosen number is closer to the result of the die roll wins. Which number should you choose to maximize your chances of winning?

Quantitative Trader Interview

137

Two dice are rolled, and the result is hidden, but it is known that at least one die shows a 6. What is the expected value of the sum of the two dice?

Quantitative Trader Interview

138

You have a standard deck of 52 cards. I draw a card and tell you I drew a heart. Now you draw a card from the remaining cards. What is the probability that you will also draw a heart? Again, you have a standard deck of 52 cards. I draw 13 cards and tell you I drew exactly 5 hearts. Now you draw 13 cards from the cards that are remaining. What is the expected number of hearts you will draw?

Quantitative Trader Interview

139

1. What is the probability of getting a head when flipping a fair coin once? 2. What is the expected value of the sum obtained when flipping 8 fair coins, where a head counts as 1 and a tail as 0?

Quantitative Trader Interview

140

What is the minimum number of people needed in a room so that at least five of them share the same birth month?

Quantitative Trader Interview

141

Imagine I flip 100 coins in sequence, and you must guess the exact sequence. You can ask one yes/no question. What question should you ask to maximize your probability of correctly guessing the sequence?

Quantitative Trader Interview

142

What is the probability of getting at least 4 heads in 7 coin flips?

Quantitative Trader Interview

143

Devise a betting strategy for a seven-game series such that if team 1 wins the series, you win $1000 no matter what, and if team 1 loses the series, you lose $1000 no matter what.

Quantitative Trader Interview

144

It is 10:45 in London. What is the angle, in degrees, between the hour and minute hands of a clock at this time?

Quantitative Trader Interview

145

Find the sum of all odd numbers between 1 and 50. Then, how many 4-digit numbers can be made using the digits 1, 2, 3, and 4 (with or without repeating digits), and what is their average?

Quantitative Trader Interview

146

If 1.5 chickens lay 1.5 eggs in 1.5 days, how many eggs do 9 chickens lay in 9 days?

Quantitative Trader Interview

147

A fair coin is flipped repeatedly. If the sequence HHT appears before HTT, player A wins. Otherwise, player B wins. What is the probability that player A wins?

Quantitative Trader Interview

148

Given two bowling balls of the same density, if one weighs X kg and has a diameter of 10 inches, and the other has a diameter of 16 inches, how much does the second ball weigh?

Quantitative Trader Interview

149

You and your opponent play a coin-flipping game. If the first player flips heads, the second player pays him $30. If the first player flips tails, the coin passes to the second player, who then flips. If the second player flips heads, he wins $30 from the first player. This process continues. How much should you pay to go first?

Quantitative Trader Interview

150

Given a fair six-sided die (faces numbered 1 to 6), what is the expected value of a single roll? Suppose you have two chances: after the first roll, you can choose whether to keep the result or roll again. What is the maximum expected value you can achieve if you use the best possible strategy?

Quantitative Trader Interview

151

If January 1st, 2015 was a Monday, which day of the week would January 1st, 2025 fall on?

Quantitative Trader Interview

152

What is the sum of all odd integers from 1 to 50?

Quantitative Trader Interview

153

What is the probability of obtaining exactly 2 tails in 5 coin flips?

Quantitative Trader Interview

154

There are two painters: Mr. Fast can paint a room in 1 hour, and Mr. Slow needs 1 hour 15 minutes. If they work together, how long does it take them to paint one room?

Quantitative Trader Interview

155

How many numbers from 0 to 1000 contain at least one digit 2?

Quantitative Trader Interview

156

What is the weight of a bowling ball with a diameter of 12 inches, given that one with a diameter of 8 inches weighs 10 pounds?

Quantitative Trader Interview

157

What is the probability that in n coin tosses, the number of heads will be even?

Quantitative Trader Interview

158

In a best-of-7, first-to-4 competition, what is the probability that a game 7 is played?

Quantitative Trader Interview

159

What is the probability of getting a sum of 10 when you roll three dice?

Quantitative Trader Interview

160

What is the largest amount of postage that cannot be obtained using any combination of 5-cent and 11-cent stamps?

Quantitative Trader Interview

161

A bus has some initial number of passengers. At the first stop, three quarters of the passengers get off and 7 get on the bus. At the second stop, the same process occurs: three quarters of the passengers get off and 7 get on. At the third stop, the same thing happens again. What is the minimum number of passengers to start with?

Quantitative Trader Interview

162

You are waiting for a bus which takes a round trip of 10 minutes to complete (starting again when it finishes one tour). What is your expected waiting time if you arrive at a random time (uniformly from 0 to 10 minutes)? Next, consider that there is an ice cream shop somewhere on the bus route; every time the bus passes it, the driver flips a fair coin and, if heads, spends 10 minutes eating ice cream before continuing the tour. What is your expected waiting time now? How does the answer change if the coin is biased?

Quantitative Trader Interview

163

It is noon. The hour hand and minute hand overlap. When is the next time these two hands overlap again?

Quantitative Trader Interview

164

There are three coins: one with heads on both sides, one with tails on both sides, and one with heads on one side and tails on the other. You pick a coin at random and toss it, resulting in heads. What is the probability that the coin you picked has heads on both sides?

Quantitative Trader Interview

165

You have 25 horses and can race 5 horses at a time. How many races do you need to conduct to determine the top three fastest horses among all 25?

Quantitative Trader Interview

166

You are bidding on a coin with a price uniformly distributed between 0 and 100. If your bid is greater than the price, you acquire the coin and can sell it to your friend at 1.5 times the price. What bid should you make to maximize your expected profit?

Quantitative Trader Interview

167

You flip four coins. At least two coins show tails. What is the probability that exactly three of them are tails? Solve this mentally without writing anything down.

Quantitative Trader Interview

168

N points lie on a circle. You draw lines connecting all the points to each other. These lines divide the circle into a number of regions. How many regions are created? Assume the points are positioned so as to give the maximum number of regions for that N.

Quantitative Trader Interview

169

In the land of 10-digit numbers, what is the probability that the product of these numbers ends with five zeros?

Assistant Trader Interview

170

Create a four-wide market (i.e., offer four adjacent price ranges) on the possible values of the sum of four fair six-sided dice rolls.

Assistant Trader Interview

171

Calculate the sum of all odd numbers from 1 to 100.

Assistant Trader Interview

172

You have 27 cubes, each with all faces painted white. You assemble them into a 3x3x3 cube and paint the exterior blue. After breaking it apart, you pick a cube and place it on the table such that the five visible faces are all white. What is the probability that the non-visible face is blue?

Assistant Trader Interview

173

Given a fair 6-sided die, you roll once and observe the outcome n (where n is the number rolled). What is your expected value (EV) of the roll? If you are allowed one optional reroll (meaning you can choose to accept the first roll or reroll), what is your EV now? If you are allowed to reroll one additional time (meaning you can roll up to two times, always choosing whether to accept the current roll or reroll), what is your EV?

Assistant Trader Interview

174

What is 12% of 47?

Assistant Trader Interview

175

If you toss 4 coins and receive $1 for 1 head, $2 for 2 heads, $3 for 3 heads, and $4 for 4 heads, how much should you be willing to pay to play this game? If you have the option to re-toss all 4 coins after the first toss (keeping the best outcome), how much should you pay now?

Assistant Trader Interview

176

If I randomly choose 6 numbers from the first 12 prime numbers, what is the probability that the sum of those six numbers is odd?

Assistant Trader Interview

177

If I roll a die multiple times and can keep the value of each roll, what strategy should I use to maximize my total without exceeding 6, given that if my sum reaches 7 or more I go bust?

Assistant Trader Interview

178

Today is Monday. Ten years from now, on this same date, what day of the week will it be? Explain your reasoning and state your confidence level.

Assistant Trader Interview

179

Find the probability that the first two top cards you pick from a standard deck of cards are both tens.

Assistant Trader Interview

180

1. How many digits are in 99 raised to the power of 20? 2. What is the expected value of rolling 20 dice with 3 sides each versus 30 dice with 2 sides each? If you were playing a game, which option would you choose, or does it not matter? What is the probability that you will win?

Assistant Trader Interview

181

You have a 3x3x3 cube painted yellow on the outside, which is then split into 1x1x1 cubes. You randomly pick one of these cubes and observe that the side facing you is painted. Without knowing if any other sides are painted, what is the probability that this cube is a corner cube?

Assistant Trader Interview

182

If two dice are rolled, what is the probability that the sum is greater than 7?

Assistant Trader Interview

183

1. Four fair coins are tossed: a) What is the probability of getting more than 2 heads? b) What are the expected winnings if you receive $1 for each head? c) What are the expected winnings if you may re-toss one coin of your choice? 2. A 20-sided die is rolled. Player A and B each choose a number before the roll. Whichever player's number is closest to the value of the die wins the amount shown on the die. What is the optimal strategy? Is it better to choose first? What are the expected winnings?

Assistant Trader Interview

184

What is the smallest number whose digits have a product of 10,000?

Assistant Trader Interview

185

What is the probability of getting an even number of heads when flipping 39 fair coins?

Assistant Trader Interview

186

I have one fair coin and one biased two-headed coin, and I put both in my pocket. I randomly choose one coin and flip it. It shows heads. What is the probability that the coin has tails on the other side?

Assistant Trader Interview

187

If you have $10 and each mint costs 27 cents, how many mints can you buy?

Assistant Trader Interview

188

If you have $20 in $1 and $5 denominations, how many bills do you expect to have?

Assistant Trader Interview

189

You have two dice and two players. Each player picks a number. The winner is the player who picks the number closest to, or equal to, the sum of the rolls of the dice. What is the optimal strategy to maximize your chances of winning? Is it always better to pick the number that has the highest probability of being rolled? Discuss whether this strategy also applies if the two dice numbers are multiplied instead of summed.

Assistant Trader Interview

190

The value of a car is uniformly distributed between 0 and 1000. You can sell the car for 1.5 times its actual value, provided you win the car by bidding at least its value. What is the optimal bid amount?

Assistant Trader Interview

191

What is the sum of the odd numbers from 1 to 100?

Assistant Trader Interview

192

Suppose you own a car worth an unknown amount between $0 and $1000. You can make a single bid to buy the car. If your bid exceeds the car's value, you pay your bid and receive the car; if your bid is less than the value, you do not get the car. 1) What should your optimal bid be? 2) If you win, how much do you stand to lose? 3) If you have a mechanic who can increase the value of the car by a constant multiple of the car's current value, what should this constant be to make bidding worthwhile?

Assistant Trader Interview

193

How many digits are in 99 raised to the 10th power (99^10)?

Assistant Trader Interview

194

1. A larger cube is made up of 27 smaller cubes. If the exterior of the larger cube is painted red and then a small cube is chosen at random and rolled like a die, what is the probability that no red faces are showing? Given that no red faces are showing, what is the probability that the rolled cube is the center cube (which has no red faces)? 2. If I roll two ten-sided dice, which is more likely: rolling two odd numbers or rolling two even numbers?

Assistant Trader Interview

195

A company has a value V uniformly distributed between 0 and 1. You are planning to place a bid B for the company. If B is less than V, your bid loses and you get nothing. If B is greater than or equal to V, you purchase the company at price B, and the company ends up being worth 1.5 × V. What bid B should you place to maximize your expected profit?

Assistant Trader Interview

196

Person A has a 30-sided die and person B has a 20-sided die. Both players roll, and the person with the highest roll wins (on a draw, B wins). The loser pays the winner the value shown on the winner's die. 1. Calculate the expected value of this game for player A. 2. How does this value change if player B can re-roll, and when should B re-roll? 3. Now, how much is it worth for player A to get a re-roll option in this scenario? 4. If player A cannot re-roll, how many re-rolls would player B need in order to be favored in the game?

Assistant Trader Interview

197

We are playing a game with an unbiased 30-sided die. Before rolling, both players pick distinct numbers between 1 and 30. After rolling the die, if the number rolled is closer to your number, I pay you the number on the die in dollars. If it is closer to my number, you pay me the number on the die in dollars. If it is equidistant, neither pays. What is the optimal strategy: should you go first or second, and how should you pick your number? What is your expected value?

Assistant Trader Interview

198

If you could ask one true or false question to all Americans and wanted to minimize the number of people who answer correctly, what would you ask?

Assistant Trader Interview

199

Four fair coins are flipped. You receive one dollar each time a head appears. What is the expected payoff? If you are told that there are at least two heads among the four coins, what is the expected payoff under this condition?

Assistant Trader Interview

200

A seller is offering you a car whose value is uniformly distributed between 0 and 1000, but you do not know the real value and must bid for the car. If your bid is higher than the car's actual value, you buy the car at your bid price and can then resell it elsewhere for 1.5 times its real value. Otherwise, the transaction does not occur. You can only bid once. What is your optimal bid price?

Assistant Trader Interview

201

The value of a box is uniformly distributed between 0 and 100. You place a bet. If your bet is higher than the value, you get the box and can sell it at 1.5 times its value. What is the optimal bet amount?

Assistant Trader Interview

202

What is the largest positive integer that cannot be expressed as a sum of non-negative multiples of 7 and 11?

Assistant Trader Interview

203

You are given a die with 100 sides, numbered from 1 to 100. When you roll the die, you can choose to either (a) take the number of dollars equal to the number of dots that show up, or (b) pay $1 to reroll. You can continue to reroll as many times as you like, but you can only keep the result of your final roll. What is the optimal strategy, and what is the expected value of playing optimally?

Assistant Trader Interview

204

Given two dice—one 10-sided and one 6-sided—you must guess the roll of each die. If you guess under or exactly, there is no penalty. You receive the total amount shown on the dice if your guess matches the rolls exactly. What is the expected value of your winnings? What guessing strategy would maximize your expected payout?

Assistant Trader Interview

205

You have a dartboard that is split in half. If you hit the left half, you get 2 points; if you hit the right half, you get 3 points. You have an 80% chance of hitting the dartboard on any given throw and, if you hit the board, a 70% chance of hitting the side you are aiming for (and a 30% chance of hitting the other side). With an unlimited number of throws, what is the probability that you achieve a score of exactly 7?

Assistant Trader Interview

206

Given a black box that generates random numbers uniformly from 0 to 1, with independent draws and the probability of drawing any exact number being zero, design an algorithm to simulate a fair coin flip (with exactly 50% probability of heads or tails).

Assistant Trader Interview

207

If you toss 4 coins and receive 1 dollar for each heads, how much are you willing to pay to play the game? What is your answer if you have the option to toss one coin again, or two coins again?

Assistant Trader Interview

208

You have a die and are told that when you roll it, you receive the amount rolled. What is the expected value? What if you have the option to roll again and choose the higher result, what is the expected value? What if you have the option to roll a third time and choose the highest roll, what is the expected value?

Assistant Trader Interview

209

Suppose you have a sequence of 36 integers consisting only of 1's and 0's (e.g., 010101..., 10000100...). Can you construct a sequence where every subsequence of length 5 is unique? Assume the sequence starts with 00000. How would you construct such a sequence? For example, if you start with 00000100001..., it is not valid because it contains two identical subsequences (00001, 00001) of length 5.

Assistant Trader Interview

210

What is the probability of getting exactly 2 heads in 4 flips of a fair coin?

Assistant Trader Interview

211

How many digits are there in 7 raised to the 7th power?

Assistant Trader Interview

212

What is eleven million minus one thousand and eleven? Calculate the answer in your head without writing it down.

Assistant Trader Interview

213

What is the sum of all even numbers from 0 to 100?

Assistant Trader Interview

214

You have a tricycle (with three tyres) and two spare tyres, making five tyres in total. You plan to travel 1,000 miles. a) If you want each tyre to be worn equally by the end of the journey, how many miles will each tyre travel on the ground? b) What is the minimum number of stops needed to achieve this?

Assistant Trader Interview

215

You play a game with an urn containing 75 blue balls, 25 red balls, and 1 yellow ball. You earn a dollar for every red ball drawn. If you draw the yellow ball, you lose all accumulated winnings. After drawing each ball, you can choose to stop and keep your winnings or continue drawing. What should be your optimal strategy in this game?

Assistant Trader Interview

216

You play a game where you roll a 100-sided die labeled with the numbers 1 to 100. After each roll, you may either accept the payout equal to the number rolled in dollars or pay $1 to roll again. You can continue to pay $1 to reroll as many times as you like until you choose to accept your current roll. What is the optimal strategy for this game, and what are your expected winnings using this strategy?

Assistant Trader Interview

217

You have a lottery ticket with 10 slots. Behind each slot is an independently and uniformly distributed number between 0 and 1. Your payout is the maximum of these ten numbers. How much should you be willing to pay for the lottery ticket (i.e., what is the expected value of your payout)?

Assistant Trader Interview

218

What is the value of 49 × 51?

Assistant Trader Interview

219

If I flip three coins, what is the probability that at least two of them are heads?

Assistant Trader Interview

220

A tosses n+1 fair coins. B tosses n fair coins. B wins if he has at least as many heads as A. What is the probability that B wins?

Assistant Trader Interview

221

What is the probability of getting at least 2 heads when tossing a fair coin 4 times?

Assistant Trader Interview

222

In a best-of-7 game series between two teams, A and B, you get even odds on both teams for each game. Determine how much you should bet on each game so that you win $100 if Team A wins the series and lose $100 if Team B wins the series. Specifically, how much should you bet on the first game?

Assistant Trader Interview

223

You roll an n-sided die. You can accept the outcome and get paid that amount, or pay 1/n dollars to reroll. How much would you pay to play this game, and how much would you sell the game for? (Provide approximate numbers. You have five minutes to answer.)

Assistant Trader Interview

224

Consider the numbers from 1 to 1,000,000. How many times does the digit '2' appear in all of these numbers?

Assistant Trader Interview

225

If you extend the faces of a tetrahedron as planes infinitely in all directions, how many regions does this divide three-dimensional space into?

Assistant Trader Interview

226

Two people play a game with a 20-sided die. Player A chooses a number, and then Player B chooses a number. The die is rolled, and whichever player's number the result is closer to wins, and gets paid an amount. Who should be allowed to choose their number first to maximize fairness or advantage?

Assistant Trader Interview

227

An apple costs $0.33. How many apples can you buy with $1? What is the change if you buy apples with $5?

Assistant Trader Interview

228

If the probability of seeing at least one car in an hour is 544/625, what is the probability of seeing no cars in a 15-minute interval?

Assistant Trader Interview

229

You and your opponent each pick a number, then roll two dice. The person whose chosen number is closer to the sum of the dice wins. Would you prefer to pick your number first or second, and how would you choose your number?

Assistant Trader Interview

230

Calculate the sum of the digits of all the numbers from 1 to 1,000,000.

Assistant Trader Interview

231

There is a token you can purchase for $XX, which allows you to propose a bet. If you win the bet, you retain the token for future use. If you lose, you may use the token to recover half of your bet. Based on the chips both sides have on that day, how much is the token worth?

Assistant Trader Interview

232

You repeatedly flip a fair coin until either the sequence HHT or HTT appears. If you can choose which combination to bet on (HHT or HTT), which should you pick and why? Calculate the probability of winning for each choice.

Assistant Trader Interview

233

If you have a 6 by 6 matrix where each element is either 1 or -1, how many unique matrices are there such that the product of the elements in each row and each column is 1?

Assistant Trader Interview

234

You and a friend are playing a game where a random number X between 1 and 20 is chosen, and each of you picks a different number. The person whose chosen number is closer to X wins X dollars. Should you choose to pick first, and what number should you pick if you do?

Assistant Trader Interview

235

How many ways are there to partition 20 into prime numbers?

Assistant Trader Interview

236

Given an unlimited supply of coins of denominations 3, 8, and 20, what is the smallest positive integer that cannot be formed as a non-negative integer combination of these numbers?

Assistant Trader Interview

237

A truck at a depot starts with 2,000 apples and can carry a maximum of 1,000 apples at a time. For every mile traveled, the truck loses 1 apple. What is the maximum number of apples that can be transported across a 1,000 mile desert?

Assistant Trader Interview

238

Convert 16/32 to decimal.

Assistant Trader Interview

239

Given a 12-sided die, you roll the die repeatedly until the cumulative sum is odd. What value of the cumulative sum occurs with the highest probability?

Assistant Trader Interview

240

What is the nearest integer to the square root of 5500?

Assistant Trader Interview

241

If you roll a fair coin 10 times, what is the expected value of the product of the number of heads and the number of tails?

Assistant Trader Interview

242

What is the expected value of the product when rolling a six-sided die and an eight-sided die?

Assistant Trader Interview

243

Estimate the mass of the Earth.

Assistant Trader Interview

244

If it is 12 o'clock right now, when was the last time the hour and minute hands of the clock were aligned?

Assistant Trader Interview

245

At 12:00, the hands of a clock are perfectly aligned. Assuming continuous motion, at what time will they be aligned again?

Assistant Trader Interview

246

What quantitative characteristics would you use to qualitatively assess the difficulty of a maze?

Assistant Trader Interview

247

At 12:00, the hands of a clock are perfectly aligned. Assuming continuous motion, at what time will they be aligned again?

Assistant Trader Interview

248

What is 19% of 19?

Assistant Trader Interview

249

On a bus with three stops, at each stop, three-quarters of the passengers get off and seven new passengers board. What is the minimum integer number of passengers that must have been on the bus at the start so that the number of passengers after each stop is always an integer?

Assistant Trader Interview

250

Flip a fair coin repeatedly until either the pattern HHT or HTT appears. What is the probability that HHT appears before HTT? Please elaborate your reasoning.

Assistant Trader Interview

251

Today is 15/09/2011. What is the nearest future date such that all eight digits in the date (DDMMYYYY) are unique?

Assistant Trader Interview

252

What is the last digit in the value of 3 raised to the power of 33 (i.e., 3^33)?

Assistant Trader Interview

253

What is 29 multiplied by 33? Solve it without using pen and paper.

Assistant Trader Interview

254

What is 1,000,000 minus 112, without using pen and paper?

Assistant Trader Interview

255

If I have a 4x4x4 cube and I paint the outside of the cube, then cut it into 64 1x1x1 cubes: 1) How many of the smaller cubes have at least one side painted? 2) If I pick one of the 64 cubes at random and roll it, what is the probability that the top side is painted?

Assistant Trader Interview

256

If I roll two dice, what is the probability that the second die shows a higher number than the first?

Assistant Trader Interview

257

You dip a cube into paint and let it dry. Then you cut it into 27 smaller sub-cubes and put them in a bag. If you randomly pick one sub-cube and roll it on the table, what is the probability that no paint is visible?

Assistant Trader Interview

258

In an infinite sequence of coin flips, what is the probability that HHT appears before THH?

Assistant Trader Interview

259

In a World Series (baseball) between two teams, A and B, each with a 50% chance of winning any game, and the first team to win 4 games wins the series, what is the probability that the series reaches game 7 (i.e., each team wins 3 games before the final game)?

Assistant Trader Interview

260

There are 10 lightbulbs in a row, each of which can be either on or off. No two adjacent lightbulbs may be on at the same time. How many possible combinations are there?

Assistant Trader Interview

261

If you roll a fair coin 12 times, what is the expected product of the number of heads and the number of tails?

Assistant Trader Interview

262

You have 100 green marbles and 100 red marbles and can divide them however you like between two bags, with at least one marble in each bag. A player, who does not know the distribution, randomly chooses one of the bags and draws a single marble. If the marble is green, the player wins $100; if red, they win nothing. What is the fair price for this game?

Assistant Trader Interview

263

There are 4 coins. You earn a dollar for every head. All four coins are flipped. What is the expected payout? Additionally, if you have the option to either keep the result of the first flips or reflip all four coins, what is the optimal expected payout?

Assistant Trader Interview

264

Three coins are tossed. What is the probability of getting at least two heads?

Assistant Trader Interview

265

X is uniformly distributed on [0, 5]. What is E(|5 - X|) if X is continuous vs. discrete? Which value is larger, and why?

Assistant Trader Interview

266

How would you simulate the outcome of a six-sided die using only a fair coin?

Assistant Trader Interview

267

What is the expected number of coin flips required to simulate a fair 6-sided die?

Assistant Trader Interview

268

You toss 4 coins. For each head that appears, the player receives $1,000. The player is then given a second chance to play the game again. What is the optimal strategy for the game, and what is the expected monetary value of the game?

Assistant Trader Interview

269

You have a box filled with cash, where the cash value is uniformly distributed between 1 and 1000. You participate in an auction: you win the box if you bid at least the value of the cash inside; you win nothing and lose nothing otherwise. If you win, you can resell the box for 150% of its value. How much should you bid to maximize the expected value of your profit (resale value minus bid)?

Assistant Trader Interview

270

Calculate 15% of 155. You have 10 seconds.

Assistant Trader Interview

271

What is 76 multiplied by 84?

Assistant Trader Interview

272

There is a bus with n people. At the first stop, 2/3 of the people get off and 7 get on. At the second stop, again 2/3 of the people get off and 7 get on. At the third stop, the same happens. What is the minimum possible number of people on the bus after the third stop?

Assistant Trader Interview

273

What is the angle between the hour and minute hands of a clock at 2:30?

Assistant Trader Interview

274

I drop 4 coins on the table. Given that one is heads, what is the probability that the rest are tails?

Assistant Trader Interview

275

You have five coins, one of which is double-headed. You pick a coin at random without looking and flip it five times. If all five outcomes are heads, what is the probability that the coin you picked is the double-headed one?

Assistant Trader Interview

276

You toss four coins in the air with the goal of maximizing the number of heads. After the first toss, you have the option to toss any subset of the coins again. What is the expected number of heads in this game, assuming you play optimally?

Assistant Trader Interview

277

What is 56 squared? Calculate it using mental math.

Assistant Trader Interview

278

You are sitting in front of a roulette table with a 6-sided die and a standard deck of playing cards. What is the probability that you play all three games and receive the same number in all three?

Assistant Trader Interview

279

What is 37 multiplied by 43? (Solve without using pen and paper.)

Assistant Trader Interview

280

I roll a single die. If I am not satisfied with the number, I can choose to reroll once (for a maximum of two rolls). What is my expected value, assuming I play optimally?

Assistant Trader Interview

281

You roll a die repeatedly until the sum of the numbers rolled is greater than 13. On which number are you most likely to stop?

Trader Assistant Interview

282

What is 75 squared? You have ten seconds and cannot use pen or paper.

Assistant Trader Interview

283

What is the expected value of the absolute difference between the outcomes of two 30-sided dice?

Assistant Trader Interview

284

Flip a coin four times. If you flip a head on the first flip, you win $1. For each consecutive head after the first, you win double your previous winnings. What is the expected value of your total winnings?

Assistant Trader Interview

285

Construct a 2-point wide market for the probability that a randomly selected integer between 1 and 100 does not contain the digit 7.

Assistant Trader Interview

286

You are trying to get to Orlando, which is 800 miles away. You have 2,500 apples and a truck that can hold 1,000 apples at a time. You have unlimited gas, can take as many trips as you like, and can store apples anywhere along the road to pick up later. However, for every mile you drive, one apple falls out of the truck and is lost forever. What is the maximum number of apples you can deliver to Orlando? Solve this mentally, without paper or pencil.

Trader Assistant Interview

287

There exists a six-sided die. The die is rolled, and you are paid $x if the die shows x dots (e.g., if you roll a 3, then you are paid $3). What is a fair price for this game? Additional layer: After rolling the die once, you have the option of taking the rolled amount or rolling again. However, if you roll again, you must take the amount corresponding to the second roll. What is a fair price for this game?

Assistant Trader Interview

288

Flip three fair coins. What is the probability that all three land heads?

Assistant Trader Interview

289

Suppose you have two regular 6-sided dice. When you roll them, the sums of the faces range from 2 to 12, each with their characteristic probabilities. Now, imagine erasing the numbers on both dice and rewriting them with a new set of positive integers. Is it possible to assign numbers so that the sum distribution when rolling both dice remains unchanged? If so, how can this be done?

Assistant Trader Interview

290

Find the smallest positive integer x such that x raised to the power x (x^x) contains the digits '2016' consecutively. Determine a range [n, 2n] that contains this x. Additionally, specify how much you would bet on your choice and your level of confidence.

Quant Trader Intern Interview

291

How many throws of a fair 6-sided die are expected until each number appears at least twice?

Quant Trader Intern Interview

292

If copying, pasting, and typing a letter each take one second, what is the fastest way to reach at least 200 letters written?

Quant Trader Intern Interview

293

You roll a 4-sided die and sequentially accumulate the total score from each roll. What is the expected value of the first total that exceeds 100? Follow-up: What if you start at 96 instead of 0?

Quant Trader Intern Interview

294

You and an opponent are each given a number uniformly at random between 0 and 1. The player with the higher number wins 1 point from the other. After seeing your own number (but not your opponent's), you can choose to offer 'double' odds or just continue. If you offer double, your opponent, having seen their own number, can either reject the offer (losing 1) or accept it. If they accept, the winner takes 2 points from the loser according to who has the higher number. What is the optimal strategy for playing this game?

Quant Trader Intern Interview

295

You have a jar containing 3 balls labeled '+1$' and 2 balls labeled '-1$'. You draw balls one by one, without replacement, and you can choose to stop at any time. What is the fair price to pay to play this game?

Quant Trader Intern Interview

296

What is the probability that the sequence HHT appears before HTT in a sequence of fair coin tosses?

Quant Trader Intern Interview

297

Suppose there is a fair 4-sided die (numbered 1-4) and a fair 6-sided die (numbered 1-6). One die is chosen at random and rolled, resulting in a 2. What is the probability that the die rolled was the 4-sided die?

Quant Trader Intern Interview

298

You repeatedly roll a fair six-sided die. After each roll, you add the result to your cumulative score. At any point, you may choose to stop the game and keep your current score. However, if your total score ever becomes a perfect square, you immediately lose and receive zero points. What is the optimal strategy to maximize your expected score?

Quant Trader Intern Interview

299

How would you bet in a best-of-seven series so that you win 100 dollars if the team you support wins the series, and lose 100 dollars otherwise?

Quant Trader Intern Interview

300

Given a phone number with one omitted digit, what is the minimum number of phone calls needed to guarantee reaching the correct number?

Quant Trader Intern Interview

301

Two players take turns flipping a fair coin. The first player starts and has already obtained one head. They continue flipping until one player gets three heads in total. What is the probability that the first player wins?

Quant Trader Intern Interview

302

Suppose you are playing a dice game with a six-sided die. You may roll the die as many times as you like, and after each roll, you may choose to stop and take the sum of the numbers rolled as your score. However, if you roll a 6 at any time, you must stop and your total score becomes 0. What is the optimal strategy to maximize your expected score?

Quant Trader Intern Interview

303

What is the probability of getting an odd number of heads when flipping 200 fair coins?

Quant Trader Intern Interview

304

100 coins are flipped in a sequence. What six yes/no questions would you ask to maximize your chance of guessing the sequence?

Quant Trader Intern Interview

305

You are playing a game in which you roll a fair 6-sided die. If you roll n, you get n dollars. How much would you pay for this game? Suppose you may reroll if you're not satisfied with the first roll, but must accept the result of the second roll. Now, how much would you pay for the game? What if you can reroll up to two times? How much would you pay?

Quant Trader Intern Interview

306

What is the probability of getting exactly 4 heads in 9 coin tosses?

Quant Trader Intern Interview

307

If there's a 20% chance of rain on Saturday and a 30% chance of rain on Sunday, what is the probability it rains at least once this weekend?

Quant Trader Intern Interview

308

There are two balls in a bag, each either black or white. If you draw a white ball, put it back, then draw again and it's still a white ball, what is the probability of drawing a white ball the next time?

Quant Trader Intern Interview

309

There are 4 players in a game, each rolling a standard die. The player with the largest value and the player with the smallest value form one team; the remaining two players form the other team. The winning team is the one with the larger sum of their dice values, and their payoff is the difference between the team sums. Analyze the expected payoff of the winning team.

Quant Trader Intern Interview

310

Between rolling a single 20-sided die and rolling the sum of three 6-sided dice, which option gives a better chance of obtaining the highest number?

Trader Interview

311

Can you generate a number from 1 to 10 uniformly at random using two dice?

Trader Interview

312

You have 100 coins. How can you split them into piles such that the product of the number of coins in each pile is maximized? In the general setting, what is the optimal strategy?

Trader Interview

313

What is the probability of getting an odd number of heads when 100 fair coins are flipped?

Trader Interview

314

Develop a strategy for the Colonel Blotto game with 10 battlefields valued from 1 to 10 and 100 soldiers to allocate.

Trader Interview

315

What is the optimal way to bet on the outcomes of a six-sided die roll?

Trader Interview

316

What is the sum of the odd positive integers less than 60? What is the probability that n coins have an even number of heads, and how does this probability change if one of the coins is biased?

Trader Interview

317

In a game where a fair coin is tossed repeatedly, player A wins if the sequence HTT appears first, and player B wins if the sequence TTH appears first. Is this game fair?

Trader Interview

318

Starting from today's date in DD/MM/YYYY format, what is the next date in the future whose digits are all distinct?

Trader Interview

319

What is 5 to the power of 5?

Trader Interview

320

What is the expected value of a game where you roll a fair die and receive the number shown in dollars?

Trader Interview

321

Suppose you are offered the opportunity to bet on a coin flip resulting in heads, with the odds increasing from 1.5:1 up to 7:1. Explain your reasoning for whether you would bet or not bet at each odds level.

Trader Interview

322

Compute the expected value of the median when rolling a fair six-sided die three times.

Trader Interview

323

How many handshakes occur among 8 men in a room if every pair shakes hands?

Trader Interview

324

You are given a deck of cards numbered 1 to 100 and two boxes. For each card, flip a fair coin: place the card in the left box if heads, right box if tails. Find the expected value of the minimum card in the box that contains card 100.

Trader Interview

325

Player A rolls three 6-sided dice. Player B rolls one 20-sided die. The player with the highest score wins. Which player would you prefer to be, and why?

Trader Interview

326

What day of the week is 180 days from today? What is the probability that it will rain on the weekend if there is a 50% chance it will rain on Saturday and a 60% chance for Sunday?

Trader Interview

327

What is the probability of flipping four heads in a row with a fair coin?

Trader Interview

328

If 4 coins are flipped, what is the probability that at most 2 heads are flipped?

Trader Interview

329

A watch was broken into three equal parts, and the sum of the digits on each part was equal. What are the numbers on each of the parts?

Trader Interview

330

If 1.5 chickens lay 1.5 eggs in 1.5 days, how many eggs will 9 chickens lay in 9 days?

Trader Interview

331

You roll a die repeatedly until you get a 5. What is the expected value of the minimum value rolled during this process?

Trader Interview

332

If I flip four coins and give you a dollar for every head, what is the expected value of this game?

Trader Interview

333

Given 99 coins (not necessarily fair) and 1 fair coin, what is the probability that the total number of heads flipped is even?

Trader Interview

334

A fair coin is flipped repeatedly. What is the expected number of flips required to obtain three consecutive heads (HHH)? What is the expected value of the game?

Trader Interview

335

Two players each choose a different integer between 1 and 30. Then, a random integer x between 1 and 30 is generated. The player whose chosen number is closer to x receives x dollars. Analyze the optimal strategies for each player.

Trader Interview

336

What is the minimum number of people required to guarantee that at least 5 people share the same birthday month?

Trader Interview

337

(i) How many 7-digit numbers are there in the form abcdcba (symmetric about the middle digit)? (ii) What is the average value of all such numbers?

Trader Interview

338

What is the sum of all odd numbers between 1 and 80?

Trader Interview

339

You throw two eight-sided dice. What is the probability that the product of the two numbers shown is even?

Trader Interview

340

What is the last digit of 17 raised to the power of 17?

Trader Interview

341

What is the probability that a series of fair coin flips results in an even number of heads? What is the probability if the coins are not all fair, but a subset are?

Trader Interview

342

What is the fair price of a game in which you shuffle 9 cards labeled 1 through 9, and then repeatedly choose whether to open the next card or stop, receiving the sum of the last decreasing sequence of cards revealed?

Trader Interview

343

The chance of rain on Saturday is 50%, and the chance of rain on Sunday is 60%. What is the probability that it will rain on at least one day during the weekend?

Trader Interview

344

If I flip a fair coin five times, what is the probability of getting at least two heads?

Trader Interview

345

What is the angle between the hands of a clock at 9:30?

Trader Interview

346

What is the probability of flipping at most two tails when flipping five fair coins?

Trader Interview

347

What is the minimum number of people required to ensure that at least 7 of them share a birthday in the same month?

Trader Interview

348

Given two bowling balls of the same density, if one weighs X kg and has a diameter of 10 inches, and the other has a diameter of 16 inches, how much does the larger ball weigh?

Trader Interview

349

You roll a fair six-sided die. If you roll an even number, for every $2 you bet, you win $3. If you roll an odd number, for every $4 you bet, you win $6. What is your expected return, and how should you bet?

Trader Interview

350

Given 100 coin flips, what is the probability of getting an even number of heads?

Trader Interview

351

Two painters work at different rates. How long will it take them to paint a room if they work together, given their individual painting rates?

Trader Interview

352

If you throw a fair die with 6 faces repeatedly, what is the expected number of throws until you get two consecutive sixes?

Trader Interview

353

There are 10 people in a room. If each person shakes hands with every other person exactly once, how many total handshakes are there?

Trader Interview

354

If you flip 4 fair coins, what is the expected number of heads?

Trader Interview

355

You throw a fair die and are paid the number you roll. If you are not satisfied with the first result, you may choose to throw a second time and must accept the result of the second throw. What is the expected payoff?

Trader Interview

356

Is Simpson's paradox possible? If so, provide an example illustrating the paradox.

Trader Interview

357

What day of the week will it be exactly one year from today?

Trader Interview

358

What is the probability of drawing two queens from a standard deck of 52 playing cards?

Trader Interview

359

A cube painted with red paint is divided into 27 smaller cubes (3×3×3). If one small cube is randomly selected and tossed, what is the probability that at least one of the sides that are visible has paint on it?

Trader Interview

360

You are given a six-sided die. What is the expected value of the difference between two dice rolls?

Trader Interview

361

If I roll a pair of dice and know that one die shows a six, what is the probability that both dice show six?

Trader Interview

362

I am thinking of two positive integers, a and b. I can tell you that a/b belongs to the closed interval [0.48, 0.52]. Give all possible values for b, no matter what value a takes.

Trader Interview

363

What is the expected value of rolling a standard six-sided die, if you can reroll up to 3 times and take the highest number obtained?

Trader Interview

364

In a Rock-Paper-Scissors game where your opponent is not allowed to play rock, you win $1 for each win, lose $1 for each loss, and draw $0 for a tie. What should you play to maximize your expected profit?

Quantitative Researcher Interview

365

You are rolling a die. Each time you roll, you can either stop the game and receive payment equal to the face-up value, or choose to roll again. You may roll up to three times in total. What is the expected gain from playing this game optimally?

Quantitative Researcher Interview

366

Suppose you are playing Russian roulette. What is the probability that the person who fires first wins?

Quantitative Researcher Interview

367

Given the probabilities with which your opponent will play Rock, Paper, and Scissors, how should you respond to maximize your expected winnings?

Quantitative Researcher Interview

368

You have 3,000 apples in Edinburgh and want to transfer as many as possible to London. You have a truck with a maximum capacity of 1,000 apples. London is 1,000 miles away from Edinburgh. For every mile the truck drives while carrying apples, it consumes (or drops) one apple. What is the maximum number of apples you can deliver to London?

Quantitative Researcher Interview

369

You are given a biased coin with a known probability p of landing heads. How would you determine the expected number of flips until it shows heads twice in a row?

Quantitative Researcher Interview

370

You toss 4 fair coins. You earn one dollar for each head. Compute the expected payoff.

Quantitative Researcher Interview

371

Twenty-five people each shake hands with every other person exactly once. How many handshakes take place in total?

Quantitative Researcher Interview

372

Given the probability of rain on Saturday and Sunday, find the probability that it is sunny for the whole weekend.

Quantitative Researcher Interview

373

I toss 10 fair coins. What is the probability of getting an even number of heads?

Quantitative Researcher Interview

374

a) Flip 4 coins and you get paid $1 for every head. What is the expected value of your earnings? b) If you have a magic wand that allows you to tap on every pair of coins with opposite faces (one head and one tail) to flip both of them again, what is the expected value of your earnings after using the wand?

Quantitative Researcher Interview

375

Two people each make a bid from 1 to 100. The highest bid is reduced by 10. What should be your strategy to maximize your probability of winning the auction?

Quantitative Researcher Interview

376

Compare the expected value of (1) the square of the outcome when rolling one die, (2) the product of the outcomes when rolling two dice, and (3) the square of the median outcome when rolling five dice.

Quantitative Researcher Interview

377

Given 3 black and 2 white balls in a bag, you may take out several balls and stop whenever you choose. How should you determine when to stop in order to maximize the expected value of the number of black balls minus the number of white balls (b - w) you draw?

Quantitative Researcher Interview

378

What is the probability that a seventh game will be needed in a best-of-seven game series?

Quantitative Researcher Interview

379

What is the probability that the sum of two prime numbers between 1 and 20 is even?

Quantitative Researcher Interview

380

Person A walks up an escalator at 2 steps/second. Person B walks up the same escalator at 3 steps/second and gets on 5 seconds after Person A, arriving at the top X seconds after Person A. If the escalator moves at 1 step/second, how many steps are visible on the escalator at any given moment?

Quantitative Researcher Interview

381

If I throw two regular dice and tell you that one of them is a six, what is the probability that both dice are sixes?

Quantitative Researcher Interview

382

Given that the probability of rain on Saturday is p1 and the probability of rain on Sunday is p2, what is the probability that it rains on at least one day of the weekend?

Quantitative Researcher Interview

383

What is the expected value of the sum when rolling two dice, one with 11 sides and the other with 7 sides?

Quantitative Researcher Interview

384

You play a game where you roll two dice. If you get two sixes (6,6), you win $100. If you get one six and the other die is not a six, you lose $x. In all other cases, you can choose to roll both dice again. When is it optimal to play the game, in terms of the value of x?

Quantitative Research Interview

385

A deck of cards is shuffled and placed face down on a table. The cards are turned over one by one. If a black card is turned up, you win $1.00; if a red card is turned up, you lose $1.00. What is your strategy, and what are the expected winnings of your strategy?

Quantitative Researcher Interview

386

We have two integers A and B, and we know that A divided by B (A/B) is in the interval [0.48, 0.52]. What are all possible values for B?

Quantitative Researcher Interview

387

What is the sum of the odd numbers from 1 to 60?

Quantitative Researcher Interview

388

There are 100 coins, each flipped and the results are hidden. You can ask one yes/no question, after which you must guess the outcome of each coin flip. For each correct guess, you earn 1 dollar; for each incorrect guess, you lose 1 dollar. What is the optimal strategy for maximizing your expected return, and what is the maximum expected value?

Quantitative Researcher Interview

389

A weather report says there is a 30% chance of rain on Saturday and a 30% chance of rain on Sunday. What can you say about the probability that it rains on at least one of the days?

Quantitative Researcher Interview

390

A coin is flipped 100 times, and you may ask one yes/no question about the sequence of results. You must then guess the entire sequence. You earn one dollar for each correct guess and lose one dollar for each incorrect guess. Find a good strategy and compute its expected return.

Quantitative Researcher Interview

391

What is the expected value of rolling three six-sided dice (3d6)?

Quantitative Researcher Interview

392

A six-faced die is thrown two times. You may guess whether the sum of the two dice is even. If your guess is correct, you win one dollar. What is your expected earning?

Quantitative Research Interview

393

What is the minimum number of people required in a group so that at least two of them share a birthday in the same month?

Quantitative Researcher Interview

394

Given cards numbered from 1 to 9 placed face up and two dice, you repeatedly throw the dice. If the sum is greater than 9, you throw again. If the sum is less than or equal to 9, you have two options: flip down the card with the number equal to the sum, or flip down the two cards corresponding to each die value. For example, if you roll a 3 and a 4, you can flip down the 7, or both the 3 and 4. If the card for the sum is already flipped, you must flip down each die's card if possible; if one of those is already flipped, you must flip the sum (if possible). The game ends when no move is possible, and your score is the number of flipped cards. What is the best strategy?

Quantitative Researcher Interview

395

You play a game with a 64-sided die. On your turn, you may either take the value shown on the die in dollars, or pay $1 to roll again. What is the expected value of this game?

Quantitative Researcher Interview

396

A bear wants to catch 3 fish from a river, and will leave after catching the 3rd fish. Each time a fish comes, there is a 1/2 chance the bear will catch it. What is the probability that the 5th fish will not be caught?

Quantitative Researcher Interview

397

You have a standard 52-card deck. Cards are drawn one by one and placed face up, without replacement. At any point, you may stop and name a color (red or black). If the next two cards drawn are both of the chosen color, you win; otherwise, you lose. What is the optimal strategy?

Quantitative Research Interview

398

What is the largest number such that the product of its digits is 32?

Quantitative Researcher Interview

399

There are 200 one-dollar coins, each with an equal probability of going into a pot. You can bid for the pot (the winner gets all the coins, but does not know exactly how many coins are in it). The person who offers the highest bid wins the auction. What would your optimal bid be if there is 1 competitor? What about with 10 competitors? Now, suppose only two people are bidding and both are using their best strategies, but I have the advantage of knowing how many of the first 10 coins are in the pot. What bidding strategies should we each use, how much should you bid, and what is your expected payoff?

Quantitative Researcher Interview

400

You flip a fair coin repeatedly. What is the expected number of flips required to see the sequence 'HHT' for the first time?

Quantitative Researcher Interview

401

What is the expected length of the longest segment when a unit-length stick is broken at two random points?

Quantitative Researcher Interview

402

How many ways are there to shuffle a deck of cards?

Quantitative Research Interview

403

Build a tree data structure where each parent node can have any number of children. Then, given a quadratic polynomial and the value of y at an arbitrary x, determine the coefficient of the quadratic term.

Quantitative Researcher Interview

404

Which has a higher probability: (1) Rolling a die twice and getting two sixes, or (2) Rolling a die ten times and never getting a six?

Quantitative Researcher Interview

405

Write a program in a language of your choice that has a method to store name-sequence pairs (such as assigning the name 'jump' to the sequence 'ABC'), and another method that, given a sequence of characters as input, prints all names associated with that sequence.

Quantitative Researcher Interview

406

You have two decks of cards: a 52-card deck (26 black, 26 red) and a 26-card deck (13 black, 13 red). You randomly draw two cards and win if both are the same color. Which deck would you prefer? What if the 26-card deck was randomly drawn from the 52-card deck? Which deck would you prefer then?

Quantitative Researcher Interview

407

Given deck A (a normal 52-card deck), deck B (a 26-card deck with 50% black and 50% red cards), and deck C (a 26-card deck that is a random subset of deck A), which deck would you prefer to choose from if you need to draw two cards of the same color on consecutive draws?

Quantitative Researcher Interview

408

You play rock, paper, scissors against an opponent who cannot play rock. To maximize your expected profit, what should you play, given that you win $1 for a win, lose $1 for a loss, and win $0 for a draw?

Quantitative Researcher Interview

409

You repeatedly play rock, paper, scissors against an opponent who cannot choose 'rock.' The game continues if there is a draw, and ends when one person loses. What strategy should you use?

Quantitative Researcher Interview

410

What is the probability of picking 2 kings from a standard deck of cards?

Quantitative Researcher Interview

411

You toss two dice. If the sum is 7, you win a dollar. If the sum is even, you lose a dollar. Otherwise, roll again. What is the expected payoff?

Quantitative Researcher Interview

412

I randomly pick four numbers from the first fifteen prime numbers. What is the probability that their sum is odd?

Quantitative Researcher Interview

413

There is a solar system with three planets orbiting the sun. One has an orbital period of 60 years, another 84 years, and the third 140 years. Today, the three planets are aligned with the sun. When is the next time all three planets will be aligned with the sun together?

Quantitative Researcher Interview

414

From a standard deck of 52 cards, you randomly pick 26 cards to form a new set. From this set of 26 cards, you pick two cards. You win if both picked cards are of the same color. Is this game preferable to a game where you pick two cards at random (the first two picks) from a deck of 26 cards containing an equal number of black and red cards, and win if both are of the same color? Calculate or compare the probabilities.

Quantitative Researcher Interview

415

Given an unfair coin that lands heads with probability 2/3 and tails with probability 1/3, how can you use it to simulate a fair coin toss?

Quantitative Researcher Interview

416

You throw 1,000 darts. Each dart has a 50% chance to score. For the first 500 darts, each is worth 1 point; for the next 500 darts, each is worth 3 points. If your total score is 1,500 points, what is the most likely number of 3-point darts you have scored?

Quantitative Researcher Interview

417

If X, Y, and Z are three random variables such that X and Y have a correlation of 0.9, and Y and Z have a correlation of 0.8, what are the minimum and maximum possible values for the correlation between X and Z?

Quantitative Researcher Interview

418

A clock falls from the wall and breaks into three pieces, each of which has the same sum of the numbers printed on it. What are the three pieces?

Quantitative Researcher Interview

419

Given a standard deck of 52 poker cards, consider the following three choices for creating a deck: (A) 26 black and 26 red cards, (B) 13 black and 13 red cards, or (C) a random selection of 26 cards from the full deck. For each deck, you draw the first two cards and win $1 if they are the same color, otherwise you lose $1. Which deck provides the best odds for winning, and why? How would you simulate this scenario? Additionally, how would you select a random set of 26 cards from the deck?

Quantitative Research Interview

420

In a game, I toss a coin 5 times and you toss it 4 times. If I get more heads than you, I win; otherwise, you win. What is the probability that you will win?

Quantitative Researcher Interview

421

In a game, I throw one die four times, trying to get at least one 6. You throw two dice 24 times, trying to get at least one double six (both dice show 6 at the same time). Who has a greater probability of reaching their goal?

Quantitative Researcher Interview

422

A robot wakes up every morning and, with equal (1/4) probability, does one of the following: 1) self-destructs; 2) does nothing; 3) clones itself (resulting in 2 robots); or 4) clones itself twice (resulting in 3 robots). If you start with one robot on the first day, what is the probability that, eventually, you will have no robots remaining?

Quantitative Researcher Interview

423

A tosses n+1 coins. B tosses n coins. B wins if he has at least as many heads as A. What is the probability that B wins?

Quantitative Researcher Interview

424

What is the probability of obtaining an even number of heads when flipping 9 coins?

Trading Interview

425

What is the probability of getting an even number of heads when flipping a fair coin 100 times?

Trading Interview

426

What is the angle between the hour and minute hands at 9:20?

Trading Interview

427

What is the probability that a best-of-7 series reaches the 7th game?

Trading Interview

428

What is the expected value of throwing a six-sided die and paying an amount equal to the number rolled?

Trading Interview

429

You start with £100. We play a game where a fair coin is flipped 10 times. Each time, if the coin is heads, £1 is added to your current total. If the coin is tails, your sum is inverted, meaning if you have £x, it becomes £(1/x). For example, £100 becomes £0.01, £0.5 becomes £2, etc. What is the expected amount of money you will have after 10 consecutive coin flips?

Trading Interview

430

What is the first date in the future, written in the format YYYY-MM-DD, in which all digits are distinct?

Trading Interview

431

1. What is the angle between the hour and minute hands at 11:40? 2. Four fair coins are tossed. (a) What is the probability of getting at least 2 heads? (b) Given that there are at least 2 heads, what is the expected number of heads? 3. You roll two fair dice, one with 6 sides and one with 10 sides. You may guess the sum x of the two dice. If you guess correctly, you win $x; otherwise, you win nothing. Which number should you guess?

Trading Interview

432

What is the probability of getting a square number as the sum when you roll two 8-sided dice?

Trading Interview

433

There are 100 seats on a fully booked train. The first passenger is blind and takes a seat at random. Each subsequent passenger takes their assigned seat if it is free; if not, they choose one of the remaining empty seats at random. What is the probability that the last passenger sits in their assigned seat?

Trading Interview

434

Imagine the corners of a cube. Each second, you move to a random neighboring corner with equal probability. What is the expected number of seconds before you reach the corner opposite your starting point?

Trading Interview

435

What is the smallest number whose digits multiply to 10,000?

Trading Interview

436

If I have 4 coins and I flip them, and you get paid $1 for every head, what is the expected value of your earnings? If you had a magic wand that lets you re-flip any even number of coins (either 2 or 4), what would your expected earnings be after using it optimally?

Trading Interview

437

If two people play tic-tac-toe and both choose their moves randomly, what is the probability that the game ends in a draw?

Trading Interview

438

Estimate the total mass of all oceans on Earth and provide a 90% confidence interval for your estimate.

Trading Interview

439

A dealer runs a card game where there is a 20% chance of winning $10 and an 80% chance of losing everything. How much should the dealer charge to make the game worthwhile? (Calculate the fair expected value.)

Trading Interview

440

Estimate the number of cubic meters of water on Earth.

Trading Interview

441

What is the sum of all the digits needed to form the numbers from 1 to 1,000,000?

Trading Interview

442

You have 4 coins, and you earn $1 for each heads. If you have the option to reflip all of the coins, what is the expected value of this game?

Trading Interview

443

You have 4 coins and toss them all at the same time. You receive as many dollars as there are heads (e.g., 3 dollars for 3 heads). If you are not satisfied with your first result, you may toss again, but then you must accept whatever prize money you get. What is the expected value of your winnings under this strategy?

Trading Interview

444

On an island, there are 99 lions and one sheep. When a lion eats the sheep, he transforms into a sheep. Every lion's primary objective is survival, and their secondary goal is to eat a sheep. All lions are intelligent and make optimal decisions. How many lions will be on the island after some time?

Trading Interview

445

Sum the even numbers in the range 1 to 100.

Trading Interview

446

Consider a dice game where you get paid the number that you roll on a fair six-sided die. What is the expected value for the payout on the first roll? What is the expected value if you can choose to roll a second time (and take the better result), and what about if you can roll up to three times (always taking the best result)?

Trading Interview

447

You are going camping over the weekend. There is a 50% chance of rain on Saturday and a 60% chance on Sunday. What is the probability that you will not experience rain? (You are not told whether the days are independent, so consider this in your answer.) Then, after giving your answer, assume the probabilities are not independent and are positively correlated. Will the chance of having a 'dry' weekend increase or decrease?

Trading Interview

448

For which integers b (other than zero) is it possible to find an integer a such that the ratio a/b is contained in the interval [0.48, 0.52]?

Trading Interview

449

With one die, suppose in a round you earn the amount of dollars equal to the value that appears on the upward face of the die. After your first roll, you have the option to cancel the first roll and roll again, taking the second roll as your final value. What should your optimal strategy be?

Trading Interview

450

Game 1: You roll a 100-faced die labeled from 1 to 100. 1. You roll once and receive the amount in dollars that appears. How much would you pay for this roll? 2. How much would you pay if you can roll twice and keep the higher result? 3. If you can roll the die an infinite number of times, but each additional roll after the first costs $1, what is your optimal strategy? Game 2: You are competing with two others and a 21-faced die labeled 1 to 21. Each player picks a number, then the die is rolled. The player whose number is closest to the result wins. What is your optimal strategy? How does your strategy change if all players can communicate?

Trading Interview

451

1. I'm going to roll two dice. We both have to pick a number, and whoever's number is closest to the dice roll wins. Do you prefer to pick first or second? 2. You have the numbers between 1 and 30. What is the largest sum you can make by picking out numbers but only using each prime factor once? (For example, if you pick 6, you can't pick any other number with a factor of 2 or 3.) 3. You have a safe with a six-digit code and a light. You can input a code: if you have between 0 and 3 of the 6 digits correct, the light will turn red; if you have 4 or 5 correct, it will turn yellow; if you have all 6 correct, it will open. There is $10,000 inside. You can guess the code as many times as you want, but you have to pay for each guess. How much would you be willing to pay per guess?

Trading Interview

452

Find the angle between the hour and minute hands of a clock at 9:30. Also, what is the probability that the World Series goes to the 7th game?

Trading Interview

453

Suppose there is a game where you flip a fair coin repeatedly until you see two consecutive heads. What is the expected number of coin flips required? Follow-up: What is the expected number of tails observed in this process?

Trading Interview

454

You have a six-sided die. Each time you roll, you add the result to your current sum, starting from 0. After each roll, if the sum becomes a perfect square, the game ends and you lose all your money; otherwise, you can choose to continue rolling or stop and keep the current sum. If your current sum is 35, should you continue playing or stop?

Trading Interview

455

If you have 100 coins, at least one fair and at least one unfair, is the probability of getting an even number of heads sometimes 1/2, always 1/2, or never 1/2?

Trading Interview

456

What is the probability that the World Series goes to the 7th game?

Trading Interview

457

You have a Rubik's Cube, and you paint its exterior red before breaking it up into 27 smaller cubes. You pick one of the smaller cubes at random and roll it. What is the probability that all the faces landing upward are unpainted?

Trading Interview

458

What is the probability of obtaining the same result across three events: rolling a single die, drawing a random card from a standard 52-card deck (with aces counted as 1), and spinning a roulette wheel (consider the score as the number landed on)?

Trading Interview

459

Estimate the number of commercial airplanes purchased in the United States each year.

Trading Interview

460

What is the last digit of 3 raised to the 33rd power? How many six-digit numbers contain all the digits from 1 to 6 inclusive? What is the mean of all such numbers?

Trading Interview

461

Imagine you would receive $100 if you made more than n percent of your free throws, and would have to pay $100 if you made less than n percent of the throws. Would you prefer to have 10 throws or 100 throws?

Trading Interview

462

You begin with $100. You flip a fair coin. For each heads, you gain $1. For each tails, your current amount of money is inverted (i.e., after the first tails, if you previously had $x, your new amount is $1/x). What is the expected value of your money after 7 flips? (Hint: use recursion)

Trading Interview

463

We play a game with a 100-sided die. Each roll costs one dollar, and you can choose to stop at any time and accept the dollar amount of your most recent roll. What is the optimal stopping time to maximize your expected profit?

Trading Interview

464

In a standard pack of cards, what is the expected number of cards one must draw to obtain cards from all four suits?

Trading Interview

465

You and I play a game. There are two dice, one ten-sided and the other six-sided. You guess the sum of the numbers after I roll them. If your guess is correct, you get the sum of the numbers in dollars; otherwise, you get nothing. How would you make the best guess?

Trading Interview

466

Estimate the mass of the Earth.

Trading Interview

467

A white cube is painted red on its outer surface and then divided into 27 equal smaller cubes. If you randomly select one small cube and roll it, what is the probability that all 5 visible faces are white?

Trading Interview

468

You have a truck that can carry up to 1,000 apples and must transport 3,000 apples from your farm to a market 1,000 miles away. The truck has a hole that causes it to irrecoverably drop 1 apple per mile traveled. You may drop apples off in secure boxes along the road to pick them up later, but the truck can still hold only 1,000 apples at a time. What strategy maximizes the number of apples delivered to market?

Trading Interview

469

You ask someone to take a test in which each question has 5 possible answers and only one correct answer. You observe that the person's answer is correct. What is the probability that the test taker actually knew (derived) the answer rather than guessed?

Trading Interview

470

Flip a coin. If it lands heads, I win 1 point; if tails, you win 1 point. The first person to reach 2 points wins the game, and the loser pays the winner $1. However, I have an option to increase the stake to $2 per game. What is the value of this option?

Trading Interview

471

There are 99 lions and 1 sheep on an island. The lions want to eat the sheep but also want to stay alive. When a lion eats the sheep, it turns into a sheep. Lions can survive on other foods available on the island. The sheep cannot escape, and all creatures are rational. After some time, how many lions and how many sheep will be left?

Trading Interview

472

What is the expected number of rolls of an n-sided die required so that the cumulative total first exceeds n?

Quantitative Trader Intern Interview

473

When you roll a coin five times, what is the probability of getting an even number of heads?

Quantitative Trader Intern Interview

474

I sample p uniformly from [0,1] and flip a coin 100 times. The coin lands heads with probability p in each flip. Before each flip, you are allowed to guess which side it will land on. For each correct guess, you gain $1; for each incorrect guess, you lose $1. What would your strategy be, and would you pay $20 to play this game?

Quantitative Trader Intern Interview

475

What is the expected value when throwing a fair six-sided die once?

Quantitative Trader Intern Interview

476

What is the expected value of the number of heads when tossing a fair coin five times? Please explain your reasoning.

Quantitative Trader - Intern Interview

477

You and your friend play a betting game where you start with $1 and your friend starts with N dollars, where N is a natural number. Each round, you 'flip a fair coin for the shortest current stack' (i.e., you win the shortest stack amount from your friend if it lands Heads, and your friend wins the shortest stack amount from you if it lands Tails). You buy back in for an extra $1 every time you lose your current stack to your friend and the game continues, but if your friend loses all his stack to you, he doesn't buy back in and the game ends. (a) What is the expected number of rounds that this game will last? (b) What is the expected amount of profit that you walk away with? (c) What is the expected number of times you expect to buy back in for an iteration of the game for very large N? (d) In the real world, a U.S. penny has about a 51% chance of landing the same side up as before it was flipped, and about an 80% chance of landing Tails if spun on edge. Now, you may choose to use your real U.S. penny in the game: flip it with Heads up (51% Heads), Tails up (49% Heads), or spin it (20% Heads). Alternatively, you can use the perfectly fair coin (50% Heads). Your goal is always to maximize your expected profit. What is your optimal strategy and the expected profit? (e) The game also ends if you lose N dollars (i.e., you are down N dollars from your original $1 buy-in), in which case your friend wins. What is the minimum probability of landing Heads the coin must have for you and your friend to have equal chances of winning the game?

Quantitative Trader Intern Interview

478

If you throw two dice, which value has the maximum expected value (EV)?

Quantitative Trader Intern Interview

479

Suppose you have two urns that are indistinguishable from the outside. One urn contains 3 one-dollar coins and 7 ten-dollar coins. The other urn contains 5 one-dollar coins and 5 ten-dollar coins. You choose an urn at random and draw a coin at random. You find that it is a $10 coin. Now you have the option to draw again (without replacing the first coin) from either the same urn or the other urn. Should you draw from the same urn or switch to the other urn to maximize the probability of drawing another $10 coin?

Quantitative Trader Intern Interview

480

What is the probability of getting an even number of heads when tossing 4 coins?

Quantitative Trader Intern Interview

481

Given one fair 10-sided die and one fair 6-sided die, what is the expected value of the sum of their outcomes?

Quantitative Trader Intern Interview

482

What is the expected value of the sum when two fair six-sided dice are rolled?

Quantitative Trader - Intern Interview

483

If there are 8 people and they all shake hands with each other once, how many handshakes are there? If there are four couples among the 8 people who do not shake hands with each other, how many handshakes are there now?

Quantitative Trader Intern Interview

484

If you flip a fair coin twice, what is the expected value of the game, assuming you win $1 for each head and lose $1 for each tail?

Quantitative Trader Intern Interview

485

What is the sum of the odd numbers from 1 to 60?

Quantitative Trader - Intern Interview

486

We randomly select 4 numbers from the set of the first 20 prime numbers, without replacement. What is the probability that their sum is even? Explain your reasoning.

Quantitative Trader Intern Interview

487

You are playing a game with a 6-sided die. You may roll the die once, observe the result, and choose either to stop (keeping the result) or roll again. Your final payoff is the sum of your rolls, unless this sum exceeds 9, in which case you receive nothing. What is your optimal strategy for this game? Specifically, for each possible outcome of the first roll, should you choose to stop or roll again?

Quantitative Trader Intern Interview

488

A bag contains three visually indistinguishable coins: one with a 10% chance, one with a 30% chance, and one with a 60% chance of landing heads. You randomly select a coin and flip it, and it lands heads. What is the probability that if you flip the same coin again, it will land heads? Explain your reasoning.

Quantitative Trader Intern Interview

489

You have two indistinguishable urns. One contains seven $1 chips and three $10 chips, and the other contains nine $1 chips and one $10 chip. You randomly draw a chip from one of the urns and it turns out to be a $10 chip (the drawn chip is not replaced). You are then offered the chance to draw and keep a chip from either urn. Should you draw from the same urn or the other urn, and what is the expected value of your draw? Explain your reasoning.

Quantitative Trader Intern Interview

490

With dates written in DD/MM/YYYY format, what is the next date where no digit is repeated?

Quantitative Trader Intern Interview

491

What is the probability of getting an odd number of heads in a sequence of coin flips where some coins are not fair?

Quantitative Trader Intern Interview

492

What is the expected value of a die roll?

Quantitative Trader Intern Interview

493

Two players play a game of coin toss with one coin. One wins if the sequence HTH occurs first, the other if HHT occurs first (H = heads, T = tails). Is the game fair? If not, who has the advantage?

Quantitative Trader - Intern Interview

494

There are 1000 people in a hall. One person has their hand painted. Every minute, everyone shakes hands with someone else. How much time is needed to paint all the hands? What is the best-case scenario? What is the worst-case scenario?

Quantitative Trader - Intern Interview

495

You and a friend are playing a coin tossing game. You toss a fair coin repeatedly and track the results. Each of you has a sequence you are watching for: your sequence is HTT and your friend's sequence is HHT. The player whose sequence appears first wins the game. Would you want to play? What is your probability of winning?

Quantitative Trader Intern Interview

496

Design a real-time system to process millions of trades per second. How would you ensure low latency, fault tolerance, and exactly-once processing?

Intern Interview

497

If I roll two dice, what is the expected value of the product of the two faces?

Intern Interview

498

Suppose we play a game with a die where we roll and sum our rolls. We can stop at any time, and the sum is our score. However, if our sum is ever a multiple of 10, our score becomes zero and our game is over. What strategy will yield the greatest expected score? What about if the target multiple is a value other than 10?

Intern Interview

499

Implement the game Tetris in 30 minutes.

Intern Interview

500

What is the optimal strategy for winning in rock-paper-scissors if your opponent can only choose between rock and paper?

Intern Interview

501

In n coin tosses, what is the probability that the number of heads is even? Prove your result rigorously.

Intern Interview

502

If it rains today, will that affect the probability that it rains tomorrow? Explain your reasoning using probability theory.

Intern Interview

503

What is the sum of the first 30 even numbers?

Intern Interview

504

Which of the following has the highest expected value: the square of a single die roll, the product of two dice, or the square of the median of three dice rolls?

Intern Interview

505

What is the smallest number whose digits multiply to give 96?

Intern Interview

506

Sum all the odd numbers between 1 and 100.

Intern Interview

507

There is a p% chance of raining on Saturday and a q% chance of raining on Sunday. What are the maximum and minimum probabilities of it raining on both days?

Intern Interview

508

Two dice are rolled. What is the probability that the sum of the numbers shown is a perfect square?

Intern Interview

509

Given two dice, where it is known that the first die shows a 6, what is the probability that both dice show a 6?

Intern Interview

510

Three fair coins are tossed. What is the probability of getting at least two heads?

Intern Interview

511

Calculate the angle between the hour and minute hands of a clock at a given time.

Intern Interview

512

I'm flipping three coins. If all three are the same (all heads or all tails), then I will receive $10 and may finish the game. If not, I may choose to flip any number of the coins again. What is the expected gain from playing this game?

Intern Interview

513

What is the expected value of the number of heads if you flip 4 coins and, after the initial flips, you can flip over any pair of coins that both show tails?

Intern Interview

514

What is the expected waiting time to get three consecutive heads when flipping a fair coin repeatedly?

Intern Interview

515

What is the final digit of 17 raised to the 17th power (17^17)?

Intern Interview

516

You have a shuffled deck of 26 red and 26 black cards. You play a game by repeatedly looking at the top card and either discarding it or ending the game. At the end, if the color of the next card matches the top card, you win; otherwise, you lose. What is the optimal strategy?

Intern Interview

517

Find the sum 1 + 3 + 5 + ... + 99 (the sum of all odd numbers below 100).

Intern Interview

518

A grey cube of size 3x3x3 is painted red on all its edges. Then, it is split into 1x1x1 cubes. One of these small cubes is chosen uniformly at random and placed randomly on a table. What is the probability that no red edges are visible on this small cube?

Intern Interview

519

What is the last digit of 17 raised to the 17th power (17^17)?

Intern Interview

520

I'm thinking of an integer with no 0s or 1s. When you multiply all the digits of the number together, you get 10,000. What is the largest number I could be thinking of? What is the smallest such number?

Intern Interview

521

Find the probability that a randomly chosen 3-digit number has its first digit even, its second digit odd, and its third digit different from the first two digits.

Intern Interview

522

You have a number whose digits multiply to 96, and none of the digits is 1. What are the largest and smallest such numbers you can form?

Intern Interview

523

Three coins are in a bag. The first coin flips heads with probability 50%, the second coin flips heads with probability 60%, and the third coin flips heads with probability 70%. I pull out a coin at random and flip it; it lands heads. If I flip this coin again, what is the probability it will land heads?

Intern Interview

524

Flip a fair coin 8 times. What is the probability that the number of heads is a multiple of 3?

Intern Interview

525

What is the probability of getting exactly 2 heads when tossing a fair coin five times?

Intern Interview

526

What is the probability that the sum of the first eight prime numbers is odd?

Intern Interview

527

You have two dice: one 10-sided and one 6-sided. You may guess any number between 2 and 16. If the sum of the two dice equals your guess, you win that number of dollars. What number should you guess to maximize your expected winnings?

Intern Interview

528

Suppose you are standing at one-third the length of a bridge when you hear a train coming from behind. You have just enough time to run back and get off the bridge before the train reaches you, and you also have just enough time if you run forward. What is the relative speed between you and the train?

Intern Interview

529

1) What is 12% of 47? 2) If I pick 2 cards from a shuffled deck (with no jokers), what is the probability that both are queens? 3) i) If I toss 4 fair coins and earn a dollar for every head, what is the expected value of this game? ii) If I can re-toss all 4 coins and must take the value from the second round, what is the expected value now? 4) How confident are you that you got 0, 1, 2, 3, or all questions correct, giving percentages? 5) If two teams play a best 4 out of 7 series, where each team has a 50% chance to win each round, what is the probability that the series lasts all 7 games?

Intern Interview

530

What is one million minus one hundred eleven?

Intern Interview

531

What is the result of multiplying 567 by 39?

Intern Interview

532

What is the probability that if you roll two dice, the sum is greater than 7?

Intern Interview

533

You roll a six-sided die and receive the amount you roll in cash. How much should you pay to play this game? What if you do not like your first roll and are allowed to roll exactly once more (but you must take what you get the last time)? What if you can roll up to three times, always keeping the last roll? What if you can roll infinitely many times?

Intern Interview

534

A cube that is white on the inside is painted blue on the outside and then cut into thirds along each dimension, resulting in 27 smaller cubes. Blindfolded, I randomly pick one of the smaller cubes and toss it so it lands with one side down. I observe that all of the visible sides are white. What is the probability that the side facing down is blue?

Intern Interview

535

You have a tricycle and plan to travel one thousand miles. You also have two spare tires with you. If you want each of the five tires to be worn equally by the end of the journey, what is the minimum number of stops you must make to achieve this?

Intern Interview

536

It is currently 10:02. What is the angle between the hour and minute hands on a clock?

Intern Interview

537

In a best-of-7 game between two players, you wish to place 50/50 bets before each point such that if player A wins the series you win £1000, and if player B wins the series you lose £1000. How can you structure your bets to achieve this outcome?

Intern Interview

538

How many times does the digit '1' appear in all the numbers from 1 to 1,000,000 inclusive?

Intern Interview

539

Sum all the odd numbers between 1 and 100.

Intern Interview

540

Estimate the weight of Mount Kilimanjaro.

Intern Interview

541

Consider a game where you roll a fair six-sided die. What is the expected value of a single roll? What is the expected value if you can roll twice and either keep the result of the first roll, or discard it and are forced to keep the second roll? How does this extend to three rolls or an infinite number of rolls?

Intern Interview

542

What is the result when you subtract 12 from 1,000,000?

Intern Interview

543

What is 2 to the power of 10?

Intern Interview

544

If I spin a roulette wheel, roll a die, and pick a card from a 52-card deck, what is the probability that all three show the same number? How confident are you in your answer?

Intern Interview

545

If you flip ten coins, what is the expected value of the product of the number of heads and the number of tails?

Intern Interview

546

If you have only 5-cent and 11-cent stamps, what is the smallest amount that cannot be formed using any combination of these stamps?

Intern Interview

547

a) What is the expected value of a die? b) Suppose you play a game where you receive a dollar amount equivalent to the number of dots that show up on a die. You roll once; if you don't like the result, you may reroll, but you must keep the second roll. What is the fair value of this game? c) Same as (b), but now you may reroll up to twice.

Intern Interview

548

What is the probability of rolling a sum of 7 with two dice?

Quantitative Trading Intern Interview

549

In a game, you have a standard six-faced die, starting with the $1 face up. Each turn, you may either roll the die (randomly changing the upface) or 'take' by cashing out the current upface for its dollar amount. The game does not end when you take, and you may take as many times as you want, including multiple times in a row. For example, you could take 100 times on the initial $1 upface for $100 total. Your strategy is to roll until you see a face of at least n for the first time, then 'take' on that face and continue taking as much as you like. Assuming you choose n optimally, what is your expected payout in this game?

Quantitative Trading Intern Interview

550

What is the probability of getting exactly 3 heads when tossing 5 fair coins?

Quantitative Trading Intern Interview

551

What is the expected value of the number of heads when you flip two fair coins?

Quantitative Trading Intern Interview

552

You are given a stack of 10 chips. Arrange the chips into piles so that the product of the number of chips in each pile is maximized.

Quantitative Trading Intern Interview

553

1. If you have 100 chips, how can you split them into piles in order to maximize the product of the number of chips in all piles? 2. You are flipping four coins and you get $1 for each head that shows up. a) You are given the choice to reflip all 4 coins once. What is your optimal strategy? b) You can reflip all 4 coins as many times as you want, but you must pay $1 for each reflip. What is your optimal strategy?

Quantitative Trading Intern Interview

554

What is the expected value of a fair six-sided die roll, given that after observing the result of your first roll, you can choose whether to keep it or reroll once and must keep the second result?

Quantitative Trading Intern Interview

555

What is the probability that the sum of the outcomes when rolling two 6-sided dice is an odd number?

Quantitative Trading Intern Interview

556

What is the smallest integer whose digits multiply to 108?

Quantitative Trading Intern Interview

557

If you roll a fair six-sided die twice, what is the probability that the sum of the two rolls is 10?

Quantitative Trading Intern Interview

558

What is the probability that exactly half of the coins show heads when n fair coins are flipped?

Quantitative Trading Intern Interview

559

What is the expected value and optimal strategy for a game where you have 3 blue balls and 2 red balls in a bag, and you get +$1 when you draw a blue ball and -$1 when you draw a red ball?

Trading Intern Interview

560

You have 5 buckets and infinitely many balls. You earn $1 each time you throw a ball into an empty bucket and lose $1 if you throw the ball into a bucket that already contains at least one ball. What is your optimal exit strategy, and what is the corresponding expected return?

Trading Intern Interview

561

What is the optimal strategy and the expected value when rolling a 20-sided die, where you can pay 1 dollar to re-roll and always receive the face value of your final roll?

Trading Intern Interview

562

What is the expected value of the money you would receive if you receive one dollar for every 'head' of a die and get to throw the die four times?

Trading Intern Interview

563

What is the sum of all the digits in the numbers from 1 to 100?

Trading Intern Interview

564

What is the sum of all odd numbers up to sixty?

Trading Intern Interview

565

Four 50-sided dice are rolled so that the numbers are all different, then assigned randomly to players A, B, C, and D. The players with the highest and lowest numbers pair up, as do the two with the middle numbers. The team with the higher total pays the team with the lower total the difference. You are one of the players. Someone offers you a way to fix the game so that you start with a given number from 1 to 50. Question 1: What is the best number to pick? Question 2: Assuming you choose X as the best number, and there is an auction to sell the hack that lets you start with X, how should you bid? (Is this just asking for the expected value of the hack, or is there more to consider about bidding strategies?)

Trading Intern Interview

566

What is the probability that there will be an even number of heads in n coin flips?

Trading Intern Interview

567

There is some amount of money in a box, determined as follows: 200 fair coins are flipped. Let the number of heads that come up be H. The amount of money in the box is H*(100-H)/100. How much would you pay for the box?

Trading Intern Interview

568

Find the number of digits in 99 raised to the power of 99.

Trading Intern Interview

569

If you roll two dice, what is the probability that the sum is a square number?

Trading Intern Interview

570

What is the probability of getting at least 2 heads when tossing a fair coin 5 times?

Trading Intern Interview

571

If I roll two dice and tell you that at least one is a 6, what is the probability that both are 6s?

Trading Intern Interview

572

You have 100 blank cards and can write a single positive integer on each card. After assigning numbers, the interviewer shuffles the deck and guesses the top card. If the interviewer guesses correctly, they earn the amount written on the card. What numbers should you write on the cards to minimize the expected return of the interviewer?

Trading Intern Interview

573

You have 4 fair coins. If you flip all of them, what is the probability of getting at least 2 heads?

Trading Intern Interview

574

We're going to play a game. You go first. You flip a coin; if you get heads, I give you $30. If you get tails, you give me the coin and I flip. If I get heads, you give me $30; if I get tails, I give it back to you. We keep going until one of us gets heads. What is the maximum amount you would be willing to pay to go first? Give a 50% confidence range and a 90% one for your answer. Why?

Trading Intern Interview

575

You play a game where you roll a 100-sided die. You can either accept the value of the roll as your payout in dollars, or pay $1 to reroll the die. What is the optimal strategy for playing this game, and what is the fair value of the game?

Trading Intern Interview

576

If I roll two dice and multiply the two outcomes, what is the probability that the product is a perfect square?

Trading Intern Interview

577

What is the result of 235 minus 438?

Trading Intern Interview

578

You have a 6-sided die and a 10-sided die. You roll both dice together, guess the sum, and if you guess correctly, you win that amount in dollars. What sum should you pick to maximize your expected winnings?

Internship Interview

579

Six cups and saucers come in pairs: there are two red, two white, and two with stars. If the cups are placed randomly onto the saucers (one on each), what is the probability that no cup is placed on a saucer of the same pattern?

Internship Interview

580

You have 4 fair coins. When you toss the coins, you win an amount in dollars equal to the total number of tails. Additionally, you have the option to re-toss all the coins, but this costs you 1 dollar. What is the expected value of this game?

Internship Interview

581

What is the probability of getting exactly 2 heads in 4 coin tosses?

Internship Interview

582

One player rolls three six-sided dice (3d6), while the other player rolls one twenty-sided die (1d20). Which player has the higher probability of obtaining the greatest score?

Internship Interview

583

What is the probability of getting an even number of heads after tossing a fair coin n times?

Internship Interview

584

What is the expected value of the result of a single fair six-sided die roll?

Internship Interview

585

What is the probability of getting an even number of heads when tossing 4 coins? How does this change with 100 coins? What if one coin is unfair? What if all coins are unfair? What if only one coin is fair? How many fair coins are needed to ensure a 50% chance of getting an even number of heads?

Internship Interview

586

Player A rolls three six-sided dice (d6) and sums the values, while Player B rolls one twenty-sided die (d20). Which player has a greater probability of getting a higher number?

Internship Interview

587

You have 100 white balls and your opponent has 100 black balls. Each of you may put any number of your balls into a common pot. A third party randomly draws one ball from the pot. Whoever's ball is drawn wins an amount of money equal to the number of balls they have left over. If you know your opponent will put in 99 balls, how many balls should you put in to maximize your expected winnings?

Internship Interview

588

What is the expected number of heads when tossing 6 coins, given that the number of heads is greater than 2?

Internship Interview

589

Consider a list of all the integers from 0 to 1,000,000. What is the sum of all the digits of these numbers?

Internship Interview

590

There are 8 people in a room. Everyone shakes hands with each other exactly once. Calculate the total number of handshakes.

Internship Interview

591

There are one hundred doors, each with one dollar behind it. You roll a one-hundred-sided die one hundred times. After rolling, you may take the dollar behind the door corresponding to any number that was rolled. What is the expected amount of money you can obtain? Explain why.

Internship Interview

592

You are given a ten-sided die (values 1-10) and are allowed to roll it once or twice. After your first roll, you may choose to roll again. If you roll a second time, you add both values for your final score. If your total is 13 or less, you receive that amount in pounds as a payout. If your total exceeds 13, you receive nothing. What is the optimal strategy, and how did you arrive at your answer?

Internship Interview

593

What is 253 multiplied by 387? Solve this without using pen, paper, or a calculator.

Internship Interview

594

You are bidding on a car whose true price is uniformly distributed between 0 and 100. If your bid exceeds the actual price, you win the car and can resell it for 1.5 times its actual price. What bid maximizes your expected profit?

Internship Interview

595

You are given two identical eggs and a 100-story building. Your task is to determine the highest floor from which you can drop an egg without it breaking.

Quantitative Analyst Interview

596

How can you invert a pyramid of coins by moving only three coins?

Quantitative Analyst Interview

597

You flip a fair coin repeatedly and stop flipping after three heads in a row occur. What is the expected number of flips required?

Quantitative Analyst Interview

598

How many handshakes occur if every person in a room shakes hands with every other person exactly once?

Quantitative Analyst Interview

599

Given that the probability it rains on Sunday is 40% and the probability it rains on the weekend (Saturday or Sunday) is 60%, what is the probability it rains on Saturday?

Quantitative Analyst Interview

600

There are 10 castles, numbered 1 through 10, with respective values of 1 to 10 points. You have 100 soldiers to distribute among the castles in any way you choose, and your opponent does the same independently. For each castle, the player with more soldiers wins that castle's points; in the event of a tie, no one receives points for that castle. Additionally, for each castle you win, you lose 0.2 points for every soldier you have more than your opponent at that location. All 100 soldiers must be deployed. Formulate a strategy to maximize your expected score.

Quantitative Analyst Interview

601

1. How many shortest paths exist from one corner of a chessboard to the opposite corner? 2. What is the smallest positive integer that has exactly 28 divisors?

Quantitative Analyst Interview

602

What is the probability that the sum of the numbers is even when tossing two dice?

Quantitative Analyst Interview

603

How many heads would you expect to get if you toss 4 fair coins?

Trader Intern Interview

604

Two players each have a die. I have a 20-sided die numbered 1-20, and the other player has a 30-sided die numbered 1-30. Both players roll their respective die. If my number is greater, the other player pays me the value of my die roll in dollars. If the other player's number is greater, I pay them the value of their die roll in dollars. If we roll the same number, I pay the other player that number in dollars. What is the expected value of my winnings or losses for a single round of this game?

Quantitative Analyst Interview

605

Suppose two players play a game where Player A and then Player B each pick an integer between 1 and 30. Then, a 30-sided die is rolled. Whoever guessed closer to the value of the roll wins an amount of money equal to the value of the roll from the other player. Given the choice, should you go first or second? What number should you choose? What is the expected value of your position?

Quantitative Analyst Interview

606

Given a 4x4 chessboard, can a knight start from any square and visit every other square exactly once without revisiting any square?

Quantitative Analyst Interview

607

Design a heap data structure that supports adding elements and removing the top element, with the following constraints: (1) The topmost element must always be less than the topmost element in the left and right child subtrees; (2) The left child subtree must have as many or one more element than the right child subtree. The heap should also support a 'min' function (returns the value at the top of the tree) and an 'empty' function (returns an empty tree). Provide a description and implementation of such a data structure.

Quantitative Analyst Interview

608

You have two decks of cards. Each deck contains both red and black cards in equal proportion. One deck has 52 cards, and the other has 104 cards, both consisting of half red and half black cards. You may choose which deck to play with. Then, you draw two cards at random from your chosen deck. If both cards are red, you win a prize. Which deck should you choose to maximize your chance of winning, and why?

Quantitative Analyst Interview

609

There are 30 blue balls and 30 red balls, and two urns. Your opponent may arrange the balls in the two urns in any way he chooses, without telling you the arrangement. You then select one urn and draw a ball at random from it. You win $10 if you draw a blue ball, and $0 otherwise. How much should you be willing to pay to play this game?

Quantitative Analyst Interview

610

Find the lowest positive integer such that the product of its digits equals n.

Trader Intern Interview

611

What is the probability of getting at least one head in 4 coin tosses? What about in 9 coin tosses?

Trader Intern Interview

612

How much would you pay for a game where your payoff equals the number shown on a die, with an option to reroll once? Generalize to n opportunities to reroll.

Trader Intern Interview

613

Walk me through the solution to the birthday problem (i.e., calculate the probability that at least two people in a group share the same birthday).

Trader Intern Interview

614

If you randomly pick a 3-digit number, what is the probability that all three digits are even numbers?

Trader Intern Interview

615

What is the set of numbers between 2 and 30, where no two numbers share a common factor greater than 1 (i.e., the set is pairwise coprime), that gives the maximum possible sum? Using the same rules, what is the highest possible number you can have in a set of 1000?

Trader Intern Interview

616

A and B play a game. Each chooses a different integer between 2 and 12. Two dice are rolled, and the sum is calculated. The player whose chosen number is closer to the dice sum wins. If you can choose first or second, which position should you choose, and what is your optimal strategy?

Trader Intern Interview

617

Basketball players A and B each play in Game 1 and Game 2. In both games, A has a higher shooting average than B. Is it possible for B to have a higher overall shooting average than A? If so, provide an example.

Trader Intern Interview

618

You have two bowling balls of the same density. One has a radius of 8 and a weight of 16; the other has a radius of 12. What is the weight of the second ball?

Trader Intern Interview

619

What is the expected value of an optional reroll of a fair six-sided die, given that your payout is the number of pips shown on the die? In other words, how much would you pay for the option to reroll once, given that your final payout is the number shown on the die?

Trader Intern Interview

620

Two people each bid a number before rolling a 30-sided die. Whoever bids closer to the number the die shows wins, and wins an amount of money equal to the number rolled. For example, if I bid 15 and you bid 16, and the die lands on 10, then I win 10 from you. What is the optimal bidding strategy and the expected payoff?

Trader Intern Interview

621

Two players toss a fair coin repeatedly. Player A wins if the sequence HHT appears first; player B wins if HTT appears first. What is the probability that player A wins the game?

Trader Intern Interview

622

What is the minimum number of people required in a group to ensure that at least 7 people share the same birthday month?

Trader Intern Interview

623

What is 1,000 to the 1,000th power?

Trader Intern Interview

624

What is the expected outcome when rolling two 10-sided dice? Please explain why.

Trader Intern Interview

625

You have 4 coins. You throw them in the air. For every coin that lands heads, you get 1 dollar. How much would you pay to play this game?

Trader Intern Interview

626

You are playing a game in which four fair coins are flipped, and the amount of money you receive in dollars equals the number of heads that appear. If you do not like the outcome of the first four flips, you may re-flip all four coins, but you must accept the result of the second set. Determine the fair value for the game.

Trader Intern Interview

627

In Russian Roulette, there is a revolver with 4 blanks and 2 bullets loaded consecutively in adjacent chambers. If the person before you fires a blank (the gun did not discharge), should you take the next shot as-is, or spin the cylinder before shooting? Justify your answer with probabilities.

Trader Intern Interview

628

What is the smallest number made up of only 1s and 0s that is divisible by 225?

Trader Intern Interview

629

What is the expected value of a die roll? Additionally, create a market (bid-ask spread) based on your expectation.

Trader Intern Interview

630

You flip a fair coin an unlimited number of times. If the sequence HHT appears before HTT, you win. What is the probability that you win?

Junior Trader Interview

631

Two fair 6-sided dice are rolled. What is the probability that the product of the top numbers is a perfect square?

Junior Trader Interview

632

What is the probability of getting an even number of heads in seven coin tosses?

Junior Trader Interview

633

A train is approaching a bridge, and you are standing at the 1/4 position along the length of the bridge. If running in either direction allows you to escape just in time, what is the ratio of your speed to the speed of the train?

Junior Trader Interview

634

What is the cube root of 169381?

Junior Trader Interview

635

Choose a subset of numbers from 1 to 30 such that no two elements in the subset share a common factor (i.e., they are pairwise coprime). What is the subset with the largest possible sum?

Junior Trader Interview

636

I have a number in mind such that the product of each of its digits is 96, and no digit is 1. What are the largest and smallest numbers I could think of?

Junior Trader Interview

637

Imagine an infinite chessboard. If a knight starts from a particular square, in how many different squares can it possibly end up after 10 moves? You do not need to calculate an exact number, but instead provide a 95% confidence interval for the number of possible ending squares. Pen and paper allowed.

Junior Trader Interview

638

If the chance of rain on Saturday is 70% and on Sunday is 60%, what is the probability that it will rain at least once during the weekend?

Junior Trader Interview

639

You have 3,000 apples to transport from city A to city B, which are 1,000 miles apart. When there are apples on the truck, one apple is consumed per mile driven. Assume unlimited gasoline. What is the maximum number of apples that can be transported to city B? Given your answer is x, what market would you make on a lottery: you win $10 if your answer is correct, otherwise $0. (All calculations must be done without pen and paper.)

Junior Trader Interview

640

Starting with 13 red and blue cards labeled from 1 to 13, remove 7 blue cards at random. What is the probability that a card drawn is blue, given that its number is 3?

Summer Intern Interview

641

I have a 12-sided die and you have a 20-sided die. Each of us gets up to two rolls, and on either roll, we can choose to stop and keep the number from that roll. Whoever has the higher number wins, with ties going to the person with the 12-sided die. What is the probability that the person with the 20-sided die wins this game?

Summer Intern Interview

642

You play a 100-round game: in each round, you can either roll a fair 20-sided die (with outcomes 1 to 20) or 'lock in' the current face value and collect that same reward for all remaining rounds. You must roll in the first round. What is your optimal stopping or locking strategy, and what is your expected value?

Quant Trading Intern Interview

643

You have a drawer with an infinite supply of socks in a 1:1 ratio of two colors. What is the expected number of draws needed to get a matching pair?

Quant Trading Intern Interview

644

For an odd number of fair coin flips, what is the probability of getting an even number of heads? What about for an even number of flips? How does this probability change if some of the coins are unfair?

Quant Trading Intern Interview

645

1. One bus leaves. At each stop, three-quarters of the people get off the bus, while seven get on. It continues this way until the end of the route. What is the minimum number of passengers at the beginning? 2. Roll two dice and multiply their results. What is the probability that the product is a perfect square?

Intern Quant Trading Interview

646

What is the probability of getting an even number of heads when tossing 4 coins?

Quant Trader Interview

647

You are playing a one-player game with two opaque boxes. At each turn, you can choose to either 'place' or 'take.' 'Place' puts $1 from a third party into one box randomly, while 'take' empties out one box randomly and the money is yours. The game consists of 100 turns, and at each turn you must either place or take. Assuming optimal play, what is the expected payoff of this game? Note: You do not know how much money you have taken until the end of the game.

Quant Trader Interview

648

If you flip a fair coin, what is the probability of getting a head? What is the probability of getting a certain number of heads if you flip two coins? How does this generalize for n coins?

Quant Trader Interview

649

What is the weight of each of five melons if their pairwise weights sum to 16, 17, 18, 19, 20, 21, 22, 24, 25, and 26?

Quant Trader Interview

650

How can you simulate the roll of a fair 15-sided die using a standard 6-sided die?

Quant Trader Interview

651

How many people do you need to have in a room before you can be certain that at least 5 of them were born in the same month?

Quant Trader Interview

652

Two teams, A and B, played football twice in independent matches. In match 1, the average number of shots by team A is greater than that of team B. In match 2, the average number of shots by team A is also greater than that of team B. Can we conclude that, when combining data from both matches, the average number of shots by team A is still greater than that of team B? If so, why? If not, provide an example.

Quant Trader Interview

653

There are 30 people, consisting of 15 couples. Everyone shakes hands with other people, except nobody shakes hands with their own partner. You are told that 29 people each shook hands with a different number of people (for example, one person shook hands with 4 people, another with 5, etc.). How many people did the 30th person shake hands with?

Quant Trader Interview

654

You roll a 6-sided die and a 10-sided die. I will pay you the sum of the numbers on the dice only if you correctly guess the sum before the roll. What number should you guess to maximize your expected payout?

Quantitative Trading Interview

655

What is the minimum number of people required to guarantee that there exist seven people who are born in the same month?

Quantitative Trading Interview

656

Find the smallest integer such that the product of its digits is 3,000,000.

Quantitative Trading Interview

657

Devise a strategy for playing a game where a fair 100-sided die is rolled, and you are paid the value shown on the die in pounds. Each roll of the die costs £1.

Quantitative Trading Interview

658

If there is a 0.3 probability it will rain on Saturday and a 0.4 probability it will rain on Sunday, what is the range of possible probabilities that it will rain on at least one day?

Quantitative Trading Interview

659

It rains 30% of the time on Saturday and 40% of the time on Sunday. How often does it rain on the weekend?

Quantitative Trading Interview

660

You flip a coin 4 times. What is the probability of getting exactly 2 heads?

Quantitative Trading Interview

661

I am thinking of a number that does not contain the digit 1. The product of its digits is 96. What is the smallest possible number? What is the largest?

Quantitative Trading Interview

662

Player 1 chooses a number, then Player 2 chooses a different number. Two dice are tossed, and whoever is closer to the outcome (the sum of the two dice) wins. Would you rather be Player 1 or Player 2? What number would you choose?

Quantitative Trading Interview

663

You have 3 buckets, each able to hold one ball. Every time you throw a ball, you are guaranteed to hit a bucket; if the bucket is already filled, the ball bounces out. Each ball that goes in gives +1, each ball that bounces out gives -1. If you have 2 balls to throw, what is the expected gain?

Quant Intern Interview

664

What is the expected number of rolls of a six-sided die to observe each face at least once? What is the expected number of rolls to observe each face at least twice?

Quant Intern Interview

665

What is the sum of the even numbers from 1 to 60?

Quant Intern Interview

666

An escalator travels upwards at X steps per second. You walk up the escalator at Y steps per second, and the escalator is Z steps tall. How long will it take you to reach the top?

Quant Intern Interview

667

What is the smallest positive integer whose digits multiply to 108?

Analyst Interview

668

What is the probability of drawing two red cards in succession from a standard deck of 52 cards and from a double deck of 104 cards?

Analyst Interview

669

An unfair die with 12 faces has the number 11 with a probability of 40%, while the other faces are equally likely. You and another player are playing a game where whoever is closer to the correct answer wins. What is your optimal strategy, and should you choose to go first or second?

Analyst Interview

670

How much would you pay to play a game where you roll a die and receive the number of dots rolled in dollars?

Analyst Interview

671

You pick n points independently and uniformly at random on the circumference of a circle. 1. What is the probability P(n) that all n points lie within some semicircle of the circle? 2. Simplify your answer to a closed form in terms of n.

Quantitative Researcher Interview

Quant Interview Questions

In a group of 10 people standing in a circle, how many handshakes occur if each person shakes hands with every other person exactly once?

You flip 4 fair coins and receive $1 for each coin that lands heads. What is the expected value of your total winnings?

If you roll two dice, and you have the option to roll again and get paid the highest roll, when is it optimal to reroll?

What is 2,000,000 minus 2,022?

How many digits does 99 raised to the 99th power have?

There are four coins. For each heads you get, you receive $1. After flipping all four coins, you may re-flip one coin of your choice. What is the maximum amount you would pay to play this game?

Given the cards that are already on the table, make a market on the probability that the next card drawn is a face card.

In a best-of-seven World Series, what are the odds that the series goes to seven games if each team has an equal chance of winning each game?

If you have an asset that increases or decreases by 20% each day with probability 1/2, what is the probability that after n days, the asset's price is unchanged?

What is the probability of drawing two kings from a standard deck of cards?

What is the expected value when rolling two fair dice? Also, if I offer you a game where I receive X dollars for outcome set A and you receive Y dollars for outcome set B, would you play? Explain your reasoning.

Two teams, each with an equal probability of winning, are playing a best-of-7 tournament series. What is the probability that the series lasts until the 7th game?

On a multiple-choice exam with three possible answers for each of the five questions, what is the probability that a student would get four or more correct answers just by guessing?

I flip four coins and give you a dollar for each head. After all four coins are flipped, you may choose to re-flip one coin that landed tails. What is the expected value of this game?

You roll a 30-sided die (numbered 1 to 30) repeatedly and sum all the outcomes. You stop rolling when the sum is greater than or equal to 300. What is the most likely final value of the sum?

What day of the week will it be exactly 10 years from today?

You roll a fair six-sided die and receive a payoff equal to the number shown. a) What is the expected payoff? b) If you are given the option to re-roll after seeing the result, under what conditions should you choose to re-roll? What is your expected payoff with this re-roll option?

You have 4 coins. I flip them, and you receive one dollar for each heads. How much are you willing to pay to play this game? What if, after the initial flip, I give you the option to reflip any number of coins? What if I require you to reflip any number of coins of your choice?

If I write down all the numbers from 1 to 1,000,000 on a page, how many times do I write the digit 2?

If there are N points on a circle and you draw a chord between each pair of points, how many regions is the circle subdivided into?

If you have four coins and I have four coins, we both toss our four coins. If the outcome of your four coins matches exactly with mine, I will give you $2; otherwise, you give me $1. Should you take this bet?

What is 37 multiplied by 43?

If two teams are equally talented, what is the probability that a 7-game series between them reaches the 7th game?

First, throw a die with numbers 1 to 8. After seeing the result, you can either stop or throw a second die with numbers 1 to 12. You win the amount of money equal to the final number shown. What strategy should you use to maximize your expected winnings?

What is the expected number of heads when you toss a fair coin 6 times?

Given n unbiased coins, what is the probability that exactly half of them are heads? Answer the question for n = 2, 3, and 20,000.

Given two fair dice, what is the probability that both show six if it is observed that at least one die shows a six?

What is 15% of 115?

A has 5 fair coins and B has 4 fair coins. A wins only if he flips more heads than B does. What is the probability that A wins?

Given 5 pumpkins and the sum of the weights of every pair of two pumpkins, determine the weight of each individual pumpkin.

Suppose there are n people on a bus. At each of three stops, 2/3 of the people on the bus get off and 7 people get on. What is the minimum number of people that were initially on the bus before the first stop, given that the number of people on the bus remains an integer after each stop?

You flip 4 fair coins and receive x dollars, where x is the square of the number of heads. What are your expected winnings?

What is the sum of the odd numbers between 1 and 100?

What is 2,000,000 minus 1,011?

What is 37 times 37?

If there are N points on a circle and a chord is drawn between each pair of points, how many regions does the circle get subdivided into?

What is 38 squared?

In Linux, how can you find the total number of files in the current folder, including all subfolders?

If two equally talented teams are playing in the World Series, what is the probability that the series lasts seven games?

A bus pulls up to a stop. Three-quarters of the people get off and seven people get on. The bus goes to two more stops with the same results. What is the smallest number of people that could have initially been on the bus?

On an infinite chessboard, to how many possible positions can a knight move after 10 moves? Provide a 90% confidence interval. Is the actual answer even or odd?

Player A had a better batting average than Player B in both the first and second halves of the season. Is it possible for Player B to have a better average than Player A over the entire season?

Apples cost 27 cents each. How many apples can you purchase with $10? (Solve this in under one minute.)

If you roll a 10-sided die and a 20-sided die, what is the probability that the 10-sided die shows a higher number than the 20-sided die?

A steel ball with an 8 cm diameter weighs 8 ounces. How much does a steel ball with a 12 cm diameter weigh, assuming both balls are made of the same material?

You throw a die two times. What is the probability of getting at least one six?

We play a game where we each guess the sum of two dice rolls. Whoever's guess is closest to the actual outcome wins. What would your optimal strategy be?

You have two packs of poker cards: one with 52 cards, the other with 104 cards. You will pick two cards separately from the same pack. If both cards are red, you win. Which pack should you choose to maximize your chances of winning?

I roll a die and will give you an amount in pounds equal to the number on the die. For example, if I get 5, I will give you 5 pounds. What is the price you are willing to pay for this game?

What is the sum of the digits of all the numbers from 1 to 1,000,000? This is different from the sum of the numbers. For instance, the sum of the numbers from 1 to 10 is 55, whereas the sum of the digits is 46.

What is 87 multiplied by 46?

What is the expected value of a fair six-sided die?

What is the expected value of the maximum of two fair six-sided dice?

You have two ropes, each of which takes 1 hour to burn from end to end, but they burn at non-uniform rates along their lengths. How can you use them to measure exactly 45 minutes?

Given a fair six-sided die, what is the expected value of the difference between two independent die rolls?

There are three pancakes: one with both sides burned, one with only one side burned, and one with neither side burned. The pancakes are stacked in a plate, and the top visible side is burned. What is the probability that the other side of this pancake is also burned?

What is 45 multiplied by 69?

What is 34 multiplied by 82? Solve within ten seconds without using a pen or pencil.

I have a square and place three dots randomly along the four edges. (1) What is the probability that the dots, when connected, do not form a triangle? (2) What is the probability that the dots lie on distinct edges?

What is 91 squared (91^2)?

Suppose you have n points on a circle arranged so that, after connecting every pair of points with straight chords, the number of regions inside the circle is maximized. How many regions are there?

We are going to play a game. We flip a fair coin repeatedly until either the sequence HHT or HTT appears. Which sequence would you choose, HHT or HTT, and why?

In Russian Roulette, a six-chamber gun contains two adjacent bullets. You pull the trigger once, and do not get shot. What is the probability that you will get shot on the next trigger pull, without spinning the cylinder?

What is the expected value of rolling one die?

A number between 0 and 1 is written on a piece of paper. You and Person X are playing a game. You must choose a number less than the number on the paper, but greater than X's guess. X picks a number at random. What is the lowest number you can choose to maximize your probability of winning?

Is it possible to relabel the numbers on two six-sided dice with other positive numbers so that the probability distribution of their sum remains unchanged?

You are offered a game where you flip a fair coin. Each time it comes up heads, you win $1 and can choose to continue playing or stop. Each time it comes up tails, you lose $1. When is the optimal time to stop?

You and another player each independently sample a number uniformly from [0,1] (each number is private information). Both numbers are put into a pot, and you take turns bidding for the pot. What is your optimal bidding strategy?

You have 100 light bulbs labeled from 1 to 100, all initially off. You perform 100 rounds of toggling: on the nth round, you toggle every nth bulb. Toggling means turning a bulb on if it's off and off if it's on. After all 100 rounds, which bulbs are on?

Two players each roll a die. Each wins if their number is higher than the other's, but they are not obligated to bet. If the first player declines the bet, that round ends. The second player knows whether the first player made a bet or not. Which player has a higher expected payout?

Sum all odd numbers from 1 to 2n + 1.

What is the smallest number whose digits multiply to 10,000?

You play rock, paper, scissors with a friend, but your friend cannot play rock. What is the optimal strategy for each player (the winner gets $1 from the loser), and what is the expected pay-off?

A coin has a probability of 0.8 of landing heads. What is the expected number of coin tosses needed to observe two consecutive heads?

You have a fair coin and continually toss it. What is the expected number of tosses needed to obtain the sequence TTT (three tails in a row)?

If you flip 9 fair coins, what is the probability of getting an even number of heads? What about for n fair coins?

How much should you be willing to pay to play a game where you roll two 6-sided dice (die A and die B), and can then choose to be paid out either (A/B) or (B/A), whichever is higher?

How many handshakes occur in a group of 25 people if each person shakes hands with every other person exactly once?

There are 52 cards: 26 red and 26 black. You draw cards one at a time, indefinitely, with replacement. For each red card drawn, you lose 1 point; for each black card, you gain 1 point. What strategy maximizes your total points?