Jane Street Interview Questions

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?

You have a six-sided die. Each time you roll, you add the result to your current sum, starting from 0. After each roll, if the sum becomes a perfect square, the game ends and you lose all your money; otherwise, you can choose to continue rolling or stop and keep the current sum. If your current sum is 35, should you continue playing or stop?

If you have 100 coins, at least one fair and at least one unfair, is the probability of getting an even number of heads sometimes 1/2, always 1/2, or never 1/2?

What is the probability that the World Series goes to the 7th game?

You have a Rubik's Cube, and you paint its exterior red before breaking it up into 27 smaller cubes. You pick one of the smaller cubes at random and roll it. What is the probability that all the faces landing upward are unpainted?

What is the probability of obtaining the same result across three events: rolling a single die, drawing a random card from a standard 52-card deck (with aces counted as 1), and spinning a roulette wheel (consider the score as the number landed on)?

Estimate the number of commercial airplanes purchased in the United States each year.

What is the last digit of 3 raised to the 33rd power? How many six-digit numbers contain all the digits from 1 to 6 inclusive? What is the mean of all such numbers?

Imagine you would receive $100 if you made more than n percent of your free throws, and would have to pay $100 if you made less than n percent of the throws. Would you prefer to have 10 throws or 100 throws?

You begin with $100. You flip a fair coin. For each heads, you gain $1. For each tails, your current amount of money is inverted (i.e., after the first tails, if you previously had $x, your new amount is $1/x). What is the expected value of your money after 7 flips? (Hint: use recursion)

We play a game with a 100-sided die. Each roll costs one dollar, and you can choose to stop at any time and accept the dollar amount of your most recent roll. What is the optimal stopping time to maximize your expected profit?

In a standard pack of cards, what is the expected number of cards one must draw to obtain cards from all four suits?

You and I play a game. There are two dice, one ten-sided and the other six-sided. You guess the sum of the numbers after I roll them. If your guess is correct, you get the sum of the numbers in dollars; otherwise, you get nothing. How would you make the best guess?

A white cube is painted red on its outer surface and then divided into 27 equal smaller cubes. If you randomly select one small cube and roll it, what is the probability that all 5 visible faces are white?

You have a truck that can carry up to 1,000 apples and must transport 3,000 apples from your farm to a market 1,000 miles away. The truck has a hole that causes it to irrecoverably drop 1 apple per mile traveled. You may drop apples off in secure boxes along the road to pick them up later, but the truck can still hold only 1,000 apples at a time. What strategy maximizes the number of apples delivered to market?

You ask someone to take a test in which each question has 5 possible answers and only one correct answer. You observe that the person's answer is correct. What is the probability that the test taker actually knew (derived) the answer rather than guessed?

Flip a coin. If it lands heads, I win 1 point; if tails, you win 1 point. The first person to reach 2 points wins the game, and the loser pays the winner $1. However, I have an option to increase the stake to $2 per game. What is the value of this option?

There are 99 lions and 1 sheep on an island. The lions want to eat the sheep but also want to stay alive. When a lion eats the sheep, it turns into a sheep. Lions can survive on other foods available on the island. The sheep cannot escape, and all creatures are rational. After some time, how many lions and how many sheep will be left?

What is the probability of rolling a sum of 7 with two dice?

In a game, you have a standard six-faced die, starting with the $1 face up. Each turn, you may either roll the die (randomly changing the upface) or 'take' by cashing out the current upface for its dollar amount. The game does not end when you take, and you may take as many times as you want, including multiple times in a row. For example, you could take 100 times on the initial $1 upface for $100 total. Your strategy is to roll until you see a face of at least n for the first time, then 'take' on that face and continue taking as much as you like. Assuming you choose n optimally, what is your expected payout in this game?

What is the probability of getting exactly 3 heads when tossing 5 fair coins?

What is the expected value of the number of heads when you flip two fair coins?