Jane Street Interview Questions

a) What is the expected value of a die? b) Suppose you play a game where you receive a dollar amount equivalent to the number of dots that show up on a die. You roll once; if you don't like the result, you may reroll, but you must keep the second roll. What is the fair value of this game? c) Same as (b), but now you may reroll up to twice.

What is the probability of rolling a sum of 7 with two dice?

In a game, you have a standard six-faced die, starting with the $1 face up. Each turn, you may either roll the die (randomly changing the upface) or 'take' by cashing out the current upface for its dollar amount. The game does not end when you take, and you may take as many times as you want, including multiple times in a row. For example, you could take 100 times on the initial $1 upface for $100 total. Your strategy is to roll until you see a face of at least n for the first time, then 'take' on that face and continue taking as much as you like. Assuming you choose n optimally, what is your expected payout in this game?

What is the probability of getting exactly 3 heads when tossing 5 fair coins?

What is the expected value of the number of heads when you flip two fair coins?

You are given a stack of 10 chips. Arrange the chips into piles so that the product of the number of chips in each pile is maximized.

1. If you have 100 chips, how can you split them into piles in order to maximize the product of the number of chips in all piles? 2. You are flipping four coins and you get $1 for each head that shows up. a) You are given the choice to reflip all 4 coins once. What is your optimal strategy? b) You can reflip all 4 coins as many times as you want, but you must pay $1 for each reflip. What is your optimal strategy?

What is the expected value of a fair six-sided die roll, given that after observing the result of your first roll, you can choose whether to keep it or reroll once and must keep the second result?

What is the probability that the sum of the outcomes when rolling two 6-sided dice is an odd number?

What is the smallest integer whose digits multiply to 108?

If you roll a fair six-sided die twice, what is the probability that the sum of the two rolls is 10?

What is the probability that exactly half of the coins show heads when n fair coins are flipped?

What is the expected value and optimal strategy for a game where you have 3 blue balls and 2 red balls in a bag, and you get +$1 when you draw a blue ball and -$1 when you draw a red ball?

You have 5 buckets and infinitely many balls. You earn $1 each time you throw a ball into an empty bucket and lose $1 if you throw the ball into a bucket that already contains at least one ball. What is your optimal exit strategy, and what is the corresponding expected return?

What is the optimal strategy and the expected value when rolling a 20-sided die, where you can pay 1 dollar to re-roll and always receive the face value of your final roll?

What is the expected value of the money you would receive if you receive one dollar for every 'head' of a die and get to throw the die four times?

What is the sum of all the digits in the numbers from 1 to 100?

What is the sum of all odd numbers up to sixty?

Four 50-sided dice are rolled so that the numbers are all different, then assigned randomly to players A, B, C, and D. The players with the highest and lowest numbers pair up, as do the two with the middle numbers. The team with the higher total pays the team with the lower total the difference. You are one of the players. Someone offers you a way to fix the game so that you start with a given number from 1 to 50. Question 1: What is the best number to pick? Question 2: Assuming you choose X as the best number, and there is an auction to sell the hack that lets you start with X, how should you bid? (Is this just asking for the expected value of the hack, or is there more to consider about bidding strategies?)

What is the probability that there will be an even number of heads in n coin flips?

There is some amount of money in a box, determined as follows: 200 fair coins are flipped. Let the number of heads that come up be H. The amount of money in the box is H*(100-H)/100. How much would you pay for the box?

Find the number of digits in 99 raised to the power of 99.

If you roll two dice, what is the probability that the sum is a square number?

What is the probability of getting at least 2 heads when tossing a fair coin 5 times?

If I roll two dice and tell you that at least one is a 6, what is the probability that both are 6s?

You have 100 blank cards and can write a single positive integer on each card. After assigning numbers, the interviewer shuffles the deck and guesses the top card. If the interviewer guesses correctly, they earn the amount written on the card. What numbers should you write on the cards to minimize the expected return of the interviewer?

You have 4 fair coins. If you flip all of them, what is the probability of getting at least 2 heads?

We're going to play a game. You go first. You flip a coin; if you get heads, I give you $30. If you get tails, you give me the coin and I flip. If I get heads, you give me $30; if I get tails, I give it back to you. We keep going until one of us gets heads. What is the maximum amount you would be willing to pay to go first? Give a 50% confidence range and a 90% one for your answer. Why?

You play a game where you roll a 100-sided die. You can either accept the value of the roll as your payout in dollars, or pay $1 to reroll the die. What is the optimal strategy for playing this game, and what is the fair value of the game?

If I roll two dice and multiply the two outcomes, what is the probability that the product is a perfect square?