Jane Street Interview Questions

A car has a uniformly distributed value between 0 and 1000. If you bid less than this value, you get nothing but keep your bid. If you bid more than this value, you can sell the car for 1.5 times the value. What should you bid? The surface of a 3x3 block is painted. The block is split into 27 cubes and one is dropped on the floor. What is the probability that no visible face is painted? If no visible face is painted, what is the probability that it is the center cube?

You flip 4 fair coins and receive $1 for each coin that lands heads. What is the expected value of your total winnings?

If you roll two dice, and you have the option to roll again and get paid the highest roll, when is it optimal to reroll?

What is 2,000,000 minus 2,022?

How many digits does 99 raised to the 99th power have?

You and I play a game: we both pick a number before rolling two dice. The one whose chosen number is closer to the sum of the dice wins. The sum is not uniformly distributed—there is a 40% chance of getting a sum of 12, and the remaining probability is distributed uniformly among all other possible sums. What is your optimal strategy, and should you go first or second?

There are four coins. For each heads you get, you receive $1. After flipping all four coins, you may re-flip one coin of your choice. What is the maximum amount you would pay to play this game?

Given the cards that are already on the table, make a market on the probability that the next card drawn is a face card.

In a best-of-seven World Series, what are the odds that the series goes to seven games if each team has an equal chance of winning each game?

If you have an asset that increases or decreases by 20% each day with probability 1/2, what is the probability that after n days, the asset's price is unchanged?

What is the probability of drawing two kings from a standard deck of cards?

What is the expected value when rolling two fair dice? Also, if I offer you a game where I receive X dollars for outcome set A and you receive Y dollars for outcome set B, would you play? Explain your reasoning.

Two teams, each with an equal probability of winning, are playing a best-of-7 tournament series. What is the probability that the series lasts until the 7th game?

On a multiple-choice exam with three possible answers for each of the five questions, what is the probability that a student would get four or more correct answers just by guessing?

I flip four coins and give you a dollar for each head. After all four coins are flipped, you may choose to re-flip one coin that landed tails. What is the expected value of this game?

You roll a 30-sided die (numbered 1 to 30) repeatedly and sum all the outcomes. You stop rolling when the sum is greater than or equal to 300. What is the most likely final value of the sum?

There are infinitely many cards numbered 1 to 5. You construct a deck by selecting any composition of these cards. A player guesses a value between 1 and 5 and draws a card at random from the deck. If they guessed the correct number, they receive a payout equal to that value; otherwise, they get nothing. What composition of the deck minimizes the player's expected payout?

What day of the week will it be exactly 10 years from today?

You roll a fair six-sided die and receive a payoff equal to the number shown. a) What is the expected payoff? b) If you are given the option to re-roll after seeing the result, under what conditions should you choose to re-roll? What is your expected payoff with this re-roll option?

You have 4 coins. I flip them, and you receive one dollar for each heads. How much are you willing to pay to play this game? What if, after the initial flip, I give you the option to reflip any number of coins? What if I require you to reflip any number of coins of your choice?

Five pirates of different ages have a treasure of 100 gold coins. On their ship, they decide to split the coins as follows: The oldest pirate proposes how to share the coins, and all pirates (including the oldest) vote for or against the proposal. If 50% or more of the pirates vote for it, the coins are shared accordingly. Otherwise, the proposing pirate is thrown overboard, and the process repeats with the remaining pirates. If a pirate would receive the same number of coins whether he votes for or against the proposal, he votes against so that the proposer is thrown overboard. Assuming all five pirates are intelligent, rational, greedy, do not wish to die, and are skilled in mathematics, what will happen?

If I write down all the numbers from 1 to 1,000,000 on a page, how many times do I write the digit 2?

If there are N points on a circle and you draw a chord between each pair of points, how many regions is the circle subdivided into?

Part 1: We are playing a game where we start at 0 and flip a coin until someone wins. Every heads adds +1 to the number and every tails adds -1. If the number reaches -10, the interviewer wins; if it reaches 20, you win. What is the probability that the interviewer will win? Part 2: I am thinking of a 10-digit number. The first digit is the number of 0s in the number, the second is the number of 1s, and so on. What is the number?

If you have four coins and I have four coins, we both toss our four coins. If the outcome of your four coins matches exactly with mine, I will give you $2; otherwise, you give me $1. Should you take this bet?

What is 37 multiplied by 43?

If two teams are equally talented, what is the probability that a 7-game series between them reaches the 7th game?

First, throw a die with numbers 1 to 8. After seeing the result, you can either stop or throw a second die with numbers 1 to 12. You win the amount of money equal to the final number shown. What strategy should you use to maximize your expected winnings?

What is the expected number of heads when you toss a fair coin 6 times?

Given n unbiased coins, what is the probability that exactly half of them are heads? Answer the question for n = 2, 3, and 20,000.