Jane Street Interview Questions

There is a bus with n people. At the first stop, 2/3 of the people get off and 7 get on. At the second stop, again 2/3 of the people get off and 7 get on. At the third stop, the same happens. What is the minimum possible number of people on the bus after the third stop?

What is the angle between the hour and minute hands of a clock at 2:30?

I drop 4 coins on the table. Given that one is heads, what is the probability that the rest are tails?

You have five coins, one of which is double-headed. You pick a coin at random without looking and flip it five times. If all five outcomes are heads, what is the probability that the coin you picked is the double-headed one?

You toss four coins in the air with the goal of maximizing the number of heads. After the first toss, you have the option to toss any subset of the coins again. What is the expected number of heads in this game, assuming you play optimally?

What is 56 squared? Calculate it using mental math.

You are sitting in front of a roulette table with a 6-sided die and a standard deck of playing cards. What is the probability that you play all three games and receive the same number in all three?

What is 37 multiplied by 43? (Solve without using pen and paper.)

I roll a single die. If I am not satisfied with the number, I can choose to reroll once (for a maximum of two rolls). What is my expected value, assuming I play optimally?

You roll a die repeatedly until the sum of the numbers rolled is greater than 13. On which number are you most likely to stop?

What is 75 squared? You have ten seconds and cannot use pen or paper.

What is the expected value of the absolute difference between the outcomes of two 30-sided dice?

Flip a coin four times. If you flip a head on the first flip, you win $1. For each consecutive head after the first, you win double your previous winnings. What is the expected value of your total winnings?

Construct a 2-point wide market for the probability that a randomly selected integer between 1 and 100 does not contain the digit 7.

You are trying to get to Orlando, which is 800 miles away. You have 2,500 apples and a truck that can hold 1,000 apples at a time. You have unlimited gas, can take as many trips as you like, and can store apples anywhere along the road to pick up later. However, for every mile you drive, one apple falls out of the truck and is lost forever. What is the maximum number of apples you can deliver to Orlando? Solve this mentally, without paper or pencil.

There exists a six-sided die. The die is rolled, and you are paid $x if the die shows x dots (e.g., if you roll a 3, then you are paid $3). What is a fair price for this game? Additional layer: After rolling the die once, you have the option of taking the rolled amount or rolling again. However, if you roll again, you must take the amount corresponding to the second roll. What is a fair price for this game?

Flip three fair coins. What is the probability that all three land heads?

Suppose you have two regular 6-sided dice. When you roll them, the sums of the faces range from 2 to 12, each with their characteristic probabilities. Now, imagine erasing the numbers on both dice and rewriting them with a new set of positive integers. Is it possible to assign numbers so that the sum distribution when rolling both dice remains unchanged? If so, how can this be done?

Find the smallest positive integer x such that x raised to the power x (x^x) contains the digits '2016' consecutively. Determine a range [n, 2n] that contains this x. Additionally, specify how much you would bet on your choice and your level of confidence.

How many throws of a fair 6-sided die are expected until each number appears at least twice?

If copying, pasting, and typing a letter each take one second, what is the fastest way to reach at least 200 letters written?

You roll a 4-sided die and sequentially accumulate the total score from each roll. What is the expected value of the first total that exceeds 100? Follow-up: What if you start at 96 instead of 0?

You and an opponent are each given a number uniformly at random between 0 and 1. The player with the higher number wins 1 point from the other. After seeing your own number (but not your opponent's), you can choose to offer 'double' odds or just continue. If you offer double, your opponent, having seen their own number, can either reject the offer (losing 1) or accept it. If they accept, the winner takes 2 points from the loser according to who has the higher number. What is the optimal strategy for playing this game?

You have a jar containing 3 balls labeled '+1$' and 2 balls labeled '-1$'. You draw balls one by one, without replacement, and you can choose to stop at any time. What is the fair price to pay to play this game?

What is the probability that the sequence HHT appears before HTT in a sequence of fair coin tosses?

Suppose there is a fair 4-sided die (numbered 1-4) and a fair 6-sided die (numbered 1-6). One die is chosen at random and rolled, resulting in a 2. What is the probability that the die rolled was the 4-sided die?

You repeatedly roll a fair six-sided die. After each roll, you add the result to your cumulative score. At any point, you may choose to stop the game and keep your current score. However, if your total score ever becomes a perfect square, you immediately lose and receive zero points. What is the optimal strategy to maximize your expected score?

How would you bet in a best-of-seven series so that you win 100 dollars if the team you support wins the series, and lose 100 dollars otherwise?

Given a phone number with one omitted digit, what is the minimum number of phone calls needed to guarantee reaching the correct number?

Two players take turns flipping a fair coin. The first player starts and has already obtained one head. They continue flipping until one player gets three heads in total. What is the probability that the first player wins?