Jane Street Interview Questions

If you flip 4 fair coins, what is the expected number of heads?

You throw a fair die and are paid the number you roll. If you are not satisfied with the first result, you may choose to throw a second time and must accept the result of the second throw. What is the expected payoff?

Is Simpson's paradox possible? If so, provide an example illustrating the paradox.

What day of the week will it be exactly one year from today?

What is the probability of drawing two queens from a standard deck of 52 playing cards?

A cube painted with red paint is divided into 27 smaller cubes (3×3×3). If one small cube is randomly selected and tossed, what is the probability that at least one of the sides that are visible has paint on it?

You are given a six-sided die. What is the expected value of the difference between two dice rolls?

If I roll a pair of dice and know that one die shows a six, what is the probability that both dice show six?

I am thinking of two positive integers, a and b. I can tell you that a/b belongs to the closed interval [0.48, 0.52]. Give all possible values for b, no matter what value a takes.

What is the expected value of rolling a standard six-sided die, if you can reroll up to 3 times and take the highest number obtained?

What is the probability of obtaining an even number of heads when flipping 9 coins?

What is the probability of getting an even number of heads when flipping a fair coin 100 times?

What is the angle between the hour and minute hands at 9:20?

What is the probability that a best-of-7 series reaches the 7th game?

What is the expected value of throwing a six-sided die and paying an amount equal to the number rolled?

You start with £100. We play a game where a fair coin is flipped 10 times. Each time, if the coin is heads, £1 is added to your current total. If the coin is tails, your sum is inverted, meaning if you have £x, it becomes £(1/x). For example, £100 becomes £0.01, £0.5 becomes £2, etc. What is the expected amount of money you will have after 10 consecutive coin flips?

What is the first date in the future, written in the format YYYY-MM-DD, in which all digits are distinct?

1. What is the angle between the hour and minute hands at 11:40? 2. Four fair coins are tossed. (a) What is the probability of getting at least 2 heads? (b) Given that there are at least 2 heads, what is the expected number of heads? 3. You roll two fair dice, one with 6 sides and one with 10 sides. You may guess the sum x of the two dice. If you guess correctly, you win $x; otherwise, you win nothing. Which number should you guess?

What is the probability of getting a square number as the sum when you roll two 8-sided dice?

There are 100 seats on a fully booked train. The first passenger is blind and takes a seat at random. Each subsequent passenger takes their assigned seat if it is free; if not, they choose one of the remaining empty seats at random. What is the probability that the last passenger sits in their assigned seat?

Imagine the corners of a cube. Each second, you move to a random neighboring corner with equal probability. What is the expected number of seconds before you reach the corner opposite your starting point?

If I have 4 coins and I flip them, and you get paid $1 for every head, what is the expected value of your earnings? If you had a magic wand that lets you re-flip any even number of coins (either 2 or 4), what would your expected earnings be after using it optimally?

If two people play tic-tac-toe and both choose their moves randomly, what is the probability that the game ends in a draw?

Estimate the total mass of all oceans on Earth and provide a 90% confidence interval for your estimate.

A dealer runs a card game where there is a 20% chance of winning $10 and an 80% chance of losing everything. How much should the dealer charge to make the game worthwhile? (Calculate the fair expected value.)

Estimate the number of cubic meters of water on Earth.

What is the sum of all the digits needed to form the numbers from 1 to 1,000,000?

You have 4 coins, and you earn $1 for each heads. If you have the option to reflip all of the coins, what is the expected value of this game?

You have 4 coins and toss them all at the same time. You receive as many dollars as there are heads (e.g., 3 dollars for 3 heads). If you are not satisfied with your first result, you may toss again, but then you must accept whatever prize money you get. What is the expected value of your winnings under this strategy?

On an island, there are 99 lions and one sheep. When a lion eats the sheep, he transforms into a sheep. Every lion's primary objective is survival, and their secondary goal is to eat a sheep. All lions are intelligent and make optimal decisions. How many lions will be on the island after some time?