Jane Street Interview Questions

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?

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?

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?

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?

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?

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

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?