Jane Street Interview Questions

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?

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?

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 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?

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?

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?

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?