Jane Street Interview Questions

What is the result of 235 minus 438?

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?

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?

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?

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

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?

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

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

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?

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?

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?

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

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

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

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.

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?

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

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?

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.

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

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

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

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?

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.

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?

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

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

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?

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?

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