Jane Street Interview Questions

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 expected number of rolls of an n-sided die required so that the cumulative total first exceeds n?

When you roll a coin five times, what is the probability of getting an even number of heads?

I sample p uniformly from [0,1] and flip a coin 100 times. The coin lands heads with probability p in each flip. Before each flip, you are allowed to guess which side it will land on. For each correct guess, you gain $1; for each incorrect guess, you lose $1. What would your strategy be, and would you pay $20 to play this game?

What is the expected value when throwing a fair six-sided die once?

What is the expected value of the number of heads when tossing a fair coin five times? Please explain your reasoning.

You and your friend play a betting game where you start with $1 and your friend starts with N dollars, where N is a natural number. Each round, you 'flip a fair coin for the shortest current stack' (i.e., you win the shortest stack amount from your friend if it lands Heads, and your friend wins the shortest stack amount from you if it lands Tails). You buy back in for an extra $1 every time you lose your current stack to your friend and the game continues, but if your friend loses all his stack to you, he doesn't buy back in and the game ends. (a) What is the expected number of rounds that this game will last? (b) What is the expected amount of profit that you walk away with? (c) What is the expected number of times you expect to buy back in for an iteration of the game for very large N? (d) In the real world, a U.S. penny has about a 51% chance of landing the same side up as before it was flipped, and about an 80% chance of landing Tails if spun on edge. Now, you may choose to use your real U.S. penny in the game: flip it with Heads up (51% Heads), Tails up (49% Heads), or spin it (20% Heads). Alternatively, you can use the perfectly fair coin (50% Heads). Your goal is always to maximize your expected profit. What is your optimal strategy and the expected profit? (e) The game also ends if you lose N dollars (i.e., you are down N dollars from your original $1 buy-in), in which case your friend wins. What is the minimum probability of landing Heads the coin must have for you and your friend to have equal chances of winning the game?

If you throw two dice, which value has the maximum expected value (EV)?

Suppose you have two urns that are indistinguishable from the outside. One urn contains 3 one-dollar coins and 7 ten-dollar coins. The other urn contains 5 one-dollar coins and 5 ten-dollar coins. You choose an urn at random and draw a coin at random. You find that it is a $10 coin. Now you have the option to draw again (without replacing the first coin) from either the same urn or the other urn. Should you draw from the same urn or switch to the other urn to maximize the probability of drawing another $10 coin?

What is the probability of getting an even number of heads when tossing 4 coins?

Given one fair 10-sided die and one fair 6-sided die, what is the expected value of the sum of their outcomes?

What is the expected value of the sum when two fair six-sided dice are rolled?

If there are 8 people and they all shake hands with each other once, how many handshakes are there? If there are four couples among the 8 people who do not shake hands with each other, how many handshakes are there now?

If you flip a fair coin twice, what is the expected value of the game, assuming you win $1 for each head and lose $1 for each tail?