Jane Street Interview Questions

You have a 3x3x3 cube painted yellow on the outside, which is then split into 1x1x1 cubes. You randomly pick one of these cubes and observe that the side facing you is painted. Without knowing if any other sides are painted, what is the probability that this cube is a corner cube?

If two dice are rolled, what is the probability that the sum is greater than 7?

1. Four fair coins are tossed: a) What is the probability of getting more than 2 heads? b) What are the expected winnings if you receive $1 for each head? c) What are the expected winnings if you may re-toss one coin of your choice? 2. A 20-sided die is rolled. Player A and B each choose a number before the roll. Whichever player's number is closest to the value of the die wins the amount shown on the die. What is the optimal strategy? Is it better to choose first? What are the expected winnings?

What is the smallest number whose digits have a product of 10,000?

What is the probability of getting an even number of heads when flipping 39 fair coins?

I have one fair coin and one biased two-headed coin, and I put both in my pocket. I randomly choose one coin and flip it. It shows heads. What is the probability that the coin has tails on the other side?

If you have $10 and each mint costs 27 cents, how many mints can you buy?

If you have $20 in $1 and $5 denominations, how many bills do you expect to have?

You have two dice and two players. Each player picks a number. The winner is the player who picks the number closest to, or equal to, the sum of the rolls of the dice. What is the optimal strategy to maximize your chances of winning? Is it always better to pick the number that has the highest probability of being rolled? Discuss whether this strategy also applies if the two dice numbers are multiplied instead of summed.

The value of a car is uniformly distributed between 0 and 1000. You can sell the car for 1.5 times its actual value, provided you win the car by bidding at least its value. What is the optimal bid amount?

What is the sum of the odd numbers from 1 to 100?

Suppose you own a car worth an unknown amount between $0 and $1000. You can make a single bid to buy the car. If your bid exceeds the car's value, you pay your bid and receive the car; if your bid is less than the value, you do not get the car. 1) What should your optimal bid be? 2) If you win, how much do you stand to lose? 3) If you have a mechanic who can increase the value of the car by a constant multiple of the car's current value, what should this constant be to make bidding worthwhile?

How many digits are in 99 raised to the 10th power (99^10)?

1. A larger cube is made up of 27 smaller cubes. If the exterior of the larger cube is painted red and then a small cube is chosen at random and rolled like a die, what is the probability that no red faces are showing? Given that no red faces are showing, what is the probability that the rolled cube is the center cube (which has no red faces)? 2. If I roll two ten-sided dice, which is more likely: rolling two odd numbers or rolling two even numbers?

A company has a value V uniformly distributed between 0 and 1. You are planning to place a bid B for the company. If B is less than V, your bid loses and you get nothing. If B is greater than or equal to V, you purchase the company at price B, and the company ends up being worth 1.5 × V. What bid B should you place to maximize your expected profit?

Person A has a 30-sided die and person B has a 20-sided die. Both players roll, and the person with the highest roll wins (on a draw, B wins). The loser pays the winner the value shown on the winner's die. 1. Calculate the expected value of this game for player A. 2. How does this value change if player B can re-roll, and when should B re-roll? 3. Now, how much is it worth for player A to get a re-roll option in this scenario? 4. If player A cannot re-roll, how many re-rolls would player B need in order to be favored in the game?

We are playing a game with an unbiased 30-sided die. Before rolling, both players pick distinct numbers between 1 and 30. After rolling the die, if the number rolled is closer to your number, I pay you the number on the die in dollars. If it is closer to my number, you pay me the number on the die in dollars. If it is equidistant, neither pays. What is the optimal strategy: should you go first or second, and how should you pick your number? What is your expected value?

If you could ask one true or false question to all Americans and wanted to minimize the number of people who answer correctly, what would you ask?

Four fair coins are flipped. You receive one dollar each time a head appears. What is the expected payoff? If you are told that there are at least two heads among the four coins, what is the expected payoff under this condition?

A seller is offering you a car whose value is uniformly distributed between 0 and 1000, but you do not know the real value and must bid for the car. If your bid is higher than the car's actual value, you buy the car at your bid price and can then resell it elsewhere for 1.5 times its real value. Otherwise, the transaction does not occur. You can only bid once. What is your optimal bid price?

The value of a box is uniformly distributed between 0 and 100. You place a bet. If your bet is higher than the value, you get the box and can sell it at 1.5 times its value. What is the optimal bet amount?

What is the largest positive integer that cannot be expressed as a sum of non-negative multiples of 7 and 11?

You are given a die with 100 sides, numbered from 1 to 100. When you roll the die, you can choose to either (a) take the number of dollars equal to the number of dots that show up, or (b) pay $1 to reroll. You can continue to reroll as many times as you like, but you can only keep the result of your final roll. What is the optimal strategy, and what is the expected value of playing optimally?

Given two dice—one 10-sided and one 6-sided—you must guess the roll of each die. If you guess under or exactly, there is no penalty. You receive the total amount shown on the dice if your guess matches the rolls exactly. What is the expected value of your winnings? What guessing strategy would maximize your expected payout?

You have a dartboard that is split in half. If you hit the left half, you get 2 points; if you hit the right half, you get 3 points. You have an 80% chance of hitting the dartboard on any given throw and, if you hit the board, a 70% chance of hitting the side you are aiming for (and a 30% chance of hitting the other side). With an unlimited number of throws, what is the probability that you achieve a score of exactly 7?

Given a black box that generates random numbers uniformly from 0 to 1, with independent draws and the probability of drawing any exact number being zero, design an algorithm to simulate a fair coin flip (with exactly 50% probability of heads or tails).

If you toss 4 coins and receive 1 dollar for each heads, how much are you willing to pay to play the game? What is your answer if you have the option to toss one coin again, or two coins again?

You have a die and are told that when you roll it, you receive the amount rolled. What is the expected value? What if you have the option to roll again and choose the higher result, what is the expected value? What if you have the option to roll a third time and choose the highest roll, what is the expected value?

Suppose you have a sequence of 36 integers consisting only of 1's and 0's (e.g., 010101..., 10000100...). Can you construct a sequence where every subsequence of length 5 is unique? Assume the sequence starts with 00000. How would you construct such a sequence? For example, if you start with 00000100001..., it is not valid because it contains two identical subsequences (00001, 00001) of length 5.

What is the probability of getting exactly 2 heads in 4 flips of a fair coin?