Jane Street Interview Questions

In a Rock-Paper-Scissors game where your opponent is not allowed to play rock, you win $1 for each win, lose $1 for each loss, and draw $0 for a tie. What should you play to maximize your expected profit?

You are rolling a die. Each time you roll, you can either stop the game and receive payment equal to the face-up value, or choose to roll again. You may roll up to three times in total. What is the expected gain from playing this game optimally?

Suppose you are playing Russian roulette. What is the probability that the person who fires first wins?

Given the probabilities with which your opponent will play Rock, Paper, and Scissors, how should you respond to maximize your expected winnings?

You have 3,000 apples in Edinburgh and want to transfer as many as possible to London. You have a truck with a maximum capacity of 1,000 apples. London is 1,000 miles away from Edinburgh. For every mile the truck drives while carrying apples, it consumes (or drops) one apple. What is the maximum number of apples you can deliver to London?

You are given a biased coin with a known probability p of landing heads. How would you determine the expected number of flips until it shows heads twice in a row?

You toss 4 fair coins. You earn one dollar for each head. Compute the expected payoff.

Twenty-five people each shake hands with every other person exactly once. How many handshakes take place in total?

Given the probability of rain on Saturday and Sunday, find the probability that it is sunny for the whole weekend.

I toss 10 fair coins. What is the probability of getting an even number of heads?

a) Flip 4 coins and you get paid $1 for every head. What is the expected value of your earnings? b) If you have a magic wand that allows you to tap on every pair of coins with opposite faces (one head and one tail) to flip both of them again, what is the expected value of your earnings after using the wand?

Two people each make a bid from 1 to 100. The highest bid is reduced by 10. What should be your strategy to maximize your probability of winning the auction?

Compare the expected value of (1) the square of the outcome when rolling one die, (2) the product of the outcomes when rolling two dice, and (3) the square of the median outcome when rolling five dice.

Given 3 black and 2 white balls in a bag, you may take out several balls and stop whenever you choose. How should you determine when to stop in order to maximize the expected value of the number of black balls minus the number of white balls (b - w) you draw?

What is the probability that a seventh game will be needed in a best-of-seven game series?

What is the probability that the sum of two prime numbers between 1 and 20 is even?

Person A walks up an escalator at 2 steps/second. Person B walks up the same escalator at 3 steps/second and gets on 5 seconds after Person A, arriving at the top X seconds after Person A. If the escalator moves at 1 step/second, how many steps are visible on the escalator at any given moment?

If I throw two regular dice and tell you that one of them is a six, what is the probability that both dice are sixes?

Given that the probability of rain on Saturday is p1 and the probability of rain on Sunday is p2, what is the probability that it rains on at least one day of the weekend?

What is the expected value of the sum when rolling two dice, one with 11 sides and the other with 7 sides?

You play a game where you roll two dice. If you get two sixes (6,6), you win $100. If you get one six and the other die is not a six, you lose $x. In all other cases, you can choose to roll both dice again. When is it optimal to play the game, in terms of the value of x?

A deck of cards is shuffled and placed face down on a table. The cards are turned over one by one. If a black card is turned up, you win $1.00; if a red card is turned up, you lose $1.00. What is your strategy, and what are the expected winnings of your strategy?

We have two integers A and B, and we know that A divided by B (A/B) is in the interval [0.48, 0.52]. What are all possible values for B?

What is the sum of the odd numbers from 1 to 60?

There are 100 coins, each flipped and the results are hidden. You can ask one yes/no question, after which you must guess the outcome of each coin flip. For each correct guess, you earn 1 dollar; for each incorrect guess, you lose 1 dollar. What is the optimal strategy for maximizing your expected return, and what is the maximum expected value?

A weather report says there is a 30% chance of rain on Saturday and a 30% chance of rain on Sunday. What can you say about the probability that it rains on at least one of the days?

A coin is flipped 100 times, and you may ask one yes/no question about the sequence of results. You must then guess the entire sequence. You earn one dollar for each correct guess and lose one dollar for each incorrect guess. Find a good strategy and compute its expected return.

What is the expected value of rolling three six-sided dice (3d6)?

A six-faced die is thrown two times. You may guess whether the sum of the two dice is even. If your guess is correct, you win one dollar. What is your expected earning?

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?