Jane Street Interview Questions

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?

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

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

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

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

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

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?

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

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?

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

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?

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?

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?

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

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

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.)

Estimate the number of cubic meters of water on Earth.

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

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?

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?

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?

Sum the even numbers in the range 1 to 100.

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)?

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?

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]?

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?

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?

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?

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?

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?