Jane Street Interview Questions

What is the minimum number of people required in a group so that at least two of them share a birthday in the same month?

Given cards numbered from 1 to 9 placed face up and two dice, you repeatedly throw the dice. If the sum is greater than 9, you throw again. If the sum is less than or equal to 9, you have two options: flip down the card with the number equal to the sum, or flip down the two cards corresponding to each die value. For example, if you roll a 3 and a 4, you can flip down the 7, or both the 3 and 4. If the card for the sum is already flipped, you must flip down each die's card if possible; if one of those is already flipped, you must flip the sum (if possible). The game ends when no move is possible, and your score is the number of flipped cards. What is the best strategy?

You play a game with a 64-sided die. On your turn, you may either take the value shown on the die in dollars, or pay $1 to roll again. What is the expected value of this game?

A bear wants to catch 3 fish from a river, and will leave after catching the 3rd fish. Each time a fish comes, there is a 1/2 chance the bear will catch it. What is the probability that the 5th fish will not be caught?

You have a standard 52-card deck. Cards are drawn one by one and placed face up, without replacement. At any point, you may stop and name a color (red or black). If the next two cards drawn are both of the chosen color, you win; otherwise, you lose. What is the optimal strategy?

What is the largest number such that the product of its digits is 32?

There are 200 one-dollar coins, each with an equal probability of going into a pot. You can bid for the pot (the winner gets all the coins, but does not know exactly how many coins are in it). The person who offers the highest bid wins the auction. What would your optimal bid be if there is 1 competitor? What about with 10 competitors? Now, suppose only two people are bidding and both are using their best strategies, but I have the advantage of knowing how many of the first 10 coins are in the pot. What bidding strategies should we each use, how much should you bid, and what is your expected payoff?

You flip a fair coin repeatedly. What is the expected number of flips required to see the sequence 'HHT' for the first time?

What is the expected length of the longest segment when a unit-length stick is broken at two random points?

How many ways are there to shuffle a deck of cards?

Build a tree data structure where each parent node can have any number of children. Then, given a quadratic polynomial and the value of y at an arbitrary x, determine the coefficient of the quadratic term.

Which has a higher probability: (1) Rolling a die twice and getting two sixes, or (2) Rolling a die ten times and never getting a six?

Write a program in a language of your choice that has a method to store name-sequence pairs (such as assigning the name 'jump' to the sequence 'ABC'), and another method that, given a sequence of characters as input, prints all names associated with that sequence.

You have two decks of cards: a 52-card deck (26 black, 26 red) and a 26-card deck (13 black, 13 red). You randomly draw two cards and win if both are the same color. Which deck would you prefer? What if the 26-card deck was randomly drawn from the 52-card deck? Which deck would you prefer then?

Given deck A (a normal 52-card deck), deck B (a 26-card deck with 50% black and 50% red cards), and deck C (a 26-card deck that is a random subset of deck A), which deck would you prefer to choose from if you need to draw two cards of the same color on consecutive draws?

You play rock, paper, scissors against an opponent who cannot play rock. To maximize your expected profit, what should you play, given that you win $1 for a win, lose $1 for a loss, and win $0 for a draw?

You repeatedly play rock, paper, scissors against an opponent who cannot choose 'rock.' The game continues if there is a draw, and ends when one person loses. What strategy should you use?

What is the probability of picking 2 kings from a standard deck of cards?

You toss two dice. If the sum is 7, you win a dollar. If the sum is even, you lose a dollar. Otherwise, roll again. What is the expected payoff?

I randomly pick four numbers from the first fifteen prime numbers. What is the probability that their sum is odd?

There is a solar system with three planets orbiting the sun. One has an orbital period of 60 years, another 84 years, and the third 140 years. Today, the three planets are aligned with the sun. When is the next time all three planets will be aligned with the sun together?

From a standard deck of 52 cards, you randomly pick 26 cards to form a new set. From this set of 26 cards, you pick two cards. You win if both picked cards are of the same color. Is this game preferable to a game where you pick two cards at random (the first two picks) from a deck of 26 cards containing an equal number of black and red cards, and win if both are of the same color? Calculate or compare the probabilities.

Given an unfair coin that lands heads with probability 2/3 and tails with probability 1/3, how can you use it to simulate a fair coin toss?

You throw 1,000 darts. Each dart has a 50% chance to score. For the first 500 darts, each is worth 1 point; for the next 500 darts, each is worth 3 points. If your total score is 1,500 points, what is the most likely number of 3-point darts you have scored?

If X, Y, and Z are three random variables such that X and Y have a correlation of 0.9, and Y and Z have a correlation of 0.8, what are the minimum and maximum possible values for the correlation between X and Z?

A clock falls from the wall and breaks into three pieces, each of which has the same sum of the numbers printed on it. What are the three pieces?

Given a standard deck of 52 poker cards, consider the following three choices for creating a deck: (A) 26 black and 26 red cards, (B) 13 black and 13 red cards, or (C) a random selection of 26 cards from the full deck. For each deck, you draw the first two cards and win $1 if they are the same color, otherwise you lose $1. Which deck provides the best odds for winning, and why? How would you simulate this scenario? Additionally, how would you select a random set of 26 cards from the deck?

In a game, I toss a coin 5 times and you toss it 4 times. If I get more heads than you, I win; otherwise, you win. What is the probability that you will win?

In a game, I throw one die four times, trying to get at least one 6. You throw two dice 24 times, trying to get at least one double six (both dice show 6 at the same time). Who has a greater probability of reaching their goal?

A robot wakes up every morning and, with equal (1/4) probability, does one of the following: 1) self-destructs; 2) does nothing; 3) clones itself (resulting in 2 robots); or 4) clones itself twice (resulting in 3 robots). If you start with one robot on the first day, what is the probability that, eventually, you will have no robots remaining?