Jane Street Interview Questions

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?

Between rolling a single 20-sided die and rolling the sum of three 6-sided dice, which option gives a better chance of obtaining the highest number?

Can you generate a number from 1 to 10 uniformly at random using two dice?

You have 100 coins. How can you split them into piles such that the product of the number of coins in each pile is maximized? In the general setting, what is the optimal strategy?

What is the probability of getting an odd number of heads when 100 fair coins are flipped?

Develop a strategy for the Colonel Blotto game with 10 battlefields valued from 1 to 10 and 100 soldiers to allocate.

What is the optimal way to bet on the outcomes of a six-sided die roll?

What is the sum of the odd positive integers less than 60? What is the probability that n coins have an even number of heads, and how does this probability change if one of the coins is biased?

In a game where a fair coin is tossed repeatedly, player A wins if the sequence HTT appears first, and player B wins if the sequence TTH appears first. Is this game fair?

Starting from today's date in DD/MM/YYYY format, what is the next date in the future whose digits are all distinct?

What is 5 to the power of 5?

What is the expected value of a game where you roll a fair die and receive the number shown in dollars?

Suppose you are offered the opportunity to bet on a coin flip resulting in heads, with the odds increasing from 1.5:1 up to 7:1. Explain your reasoning for whether you would bet or not bet at each odds level.

Compute the expected value of the median when rolling a fair six-sided die three times.