Jane Street Interview Questions

We randomly select 4 numbers from the set of the first 20 prime numbers, without replacement. What is the probability that their sum is even? Explain your reasoning.

You are playing a game with a 6-sided die. You may roll the die once, observe the result, and choose either to stop (keeping the result) or roll again. Your final payoff is the sum of your rolls, unless this sum exceeds 9, in which case you receive nothing. What is your optimal strategy for this game? Specifically, for each possible outcome of the first roll, should you choose to stop or roll again?

A bag contains three visually indistinguishable coins: one with a 10% chance, one with a 30% chance, and one with a 60% chance of landing heads. You randomly select a coin and flip it, and it lands heads. What is the probability that if you flip the same coin again, it will land heads? Explain your reasoning.

You have two indistinguishable urns. One contains seven $1 chips and three $10 chips, and the other contains nine $1 chips and one $10 chip. You randomly draw a chip from one of the urns and it turns out to be a $10 chip (the drawn chip is not replaced). You are then offered the chance to draw and keep a chip from either urn. Should you draw from the same urn or the other urn, and what is the expected value of your draw? Explain your reasoning.

With dates written in DD/MM/YYYY format, what is the next date where no digit is repeated?

What is the probability of getting an odd number of heads in a sequence of coin flips where some coins are not fair?

What is the expected value of a die roll?

Two players play a game of coin toss with one coin. One wins if the sequence HTH occurs first, the other if HHT occurs first (H = heads, T = tails). Is the game fair? If not, who has the advantage?

There are 1000 people in a hall. One person has their hand painted. Every minute, everyone shakes hands with someone else. How much time is needed to paint all the hands? What is the best-case scenario? What is the worst-case scenario?

You and a friend are playing a coin tossing game. You toss a fair coin repeatedly and track the results. Each of you has a sequence you are watching for: your sequence is HTT and your friend's sequence is HHT. The player whose sequence appears first wins the game. Would you want to play? What is your probability of winning?

Design a real-time system to process millions of trades per second. How would you ensure low latency, fault tolerance, and exactly-once processing?

If I roll two dice, what is the expected value of the product of the two faces?

Suppose we play a game with a die where we roll and sum our rolls. We can stop at any time, and the sum is our score. However, if our sum is ever a multiple of 10, our score becomes zero and our game is over. What strategy will yield the greatest expected score? What about if the target multiple is a value other than 10?

Implement the game Tetris in 30 minutes.

What is the optimal strategy for winning in rock-paper-scissors if your opponent can only choose between rock and paper?

In n coin tosses, what is the probability that the number of heads is even? Prove your result rigorously.

If it rains today, will that affect the probability that it rains tomorrow? Explain your reasoning using probability theory.

What is the sum of the first 30 even numbers?

Which of the following has the highest expected value: the square of a single die roll, the product of two dice, or the square of the median of three dice rolls?

What is the smallest number whose digits multiply to give 96?

There is a p% chance of raining on Saturday and a q% chance of raining on Sunday. What are the maximum and minimum probabilities of it raining on both days?

Two dice are rolled. What is the probability that the sum of the numbers shown is a perfect square?

Given two dice, where it is known that the first die shows a 6, what is the probability that both dice show a 6?

Three fair coins are tossed. What is the probability of getting at least two heads?

Calculate the angle between the hour and minute hands of a clock at a given time.

I'm flipping three coins. If all three are the same (all heads or all tails), then I will receive $10 and may finish the game. If not, I may choose to flip any number of the coins again. What is the expected gain from playing this game?

What is the expected value of the number of heads if you flip 4 coins and, after the initial flips, you can flip over any pair of coins that both show tails?

What is the expected waiting time to get three consecutive heads when flipping a fair coin repeatedly?

What is the final digit of 17 raised to the 17th power (17^17)?

You have a shuffled deck of 26 red and 26 black cards. You play a game by repeatedly looking at the top card and either discarding it or ending the game. At the end, if the color of the next card matches the top card, you win; otherwise, you lose. What is the optimal strategy?