Jane Street 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?

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.

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

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

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?

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?

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?

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

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?

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?

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?

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

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?

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?

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

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

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?

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

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?

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?

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.

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

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

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?

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

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

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?

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

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?

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