Jane Street Interview Questions

Imagine I flip 100 coins in a sequence, and you need to guess the sequence in one try. You are allowed to ask one yes/no question. What do you ask to maximize the probability of guessing the sequence?

You are given an array of k linked lists, each of which is sorted in ascending order. Merge all the linked lists into one sorted linked list and return it.

You roll a standard 6-sided die (d6) and a 10-sided die (d10). Before seeing the result, you must bid on the sum of the two dice. If your bid matches the actual sum, you win that amount. What is the optimal sum to bid in order to maximize your expected winnings?

Eight people walk into a room and each person shakes hands with every other person exactly once. How many handshakes occur?

Three random points are chosen on the circumference of a circle. What is the probability that a table with legs placed at those points would stand (i.e., that all three points lie on the same semicircle)?

Eight quants from different banks want to know the average salary of the group, but each prefers not to disclose their own salary to the others. Can you design a strategy for the group to calculate the average salary without revealing individual salaries?

What is the expected number of tries needed to get 3 consecutive heads in repeated fair coin flips?

Suppose in a market making context, you are asked to make a market (set buy and sell prices, as well as quantities) on the sum or the product of two rolls of fair 6-sided dice. How would you determine appropriate prices and quantities in this scenario?

Q1: What is the smallest number whose digits multiply to 216? What about 10,000? Q2: Calculate the probability of getting 3 heads in 4 coin flips. What is the probability of getting an odd number of heads in 4 flips? What about in 9 flips? In N flips? Q3: What is the next date whose digits are all unique? Q4: After three coin flips, heads-tails-heads and heads-heads-tails are equally probable. But if I keep flipping, one combination becomes more probable than the other. Why is that?

Two symmetric dice each have two sides painted red, two sides painted black, one side painted yellow, and one side painted white. When this pair of dice is rolled, what is the probability that both dice land with the same color face up?

If you and your opponent each roll a fair six-sided die, and you win $1 if your number is larger, what is your expected winning?

The probability of raining on Saturday is 30%, and the probability of raining on Sunday is 40%. (a) What is the probability it will rain on the weekend? (b) What assumption is made in this calculation? (c) If the events are not independent, what are the maximum and minimum possible probabilities that it will rain during the weekend? (Answer: 40%, 70%)

If you earn one dollar for every head you get when flipping a fair coin, what is the expected amount you will make after 4 flips?

If you have x coins, into how many stacks, and with how many coins per stack, should you divide them in order to maximize the product of the stack sizes?

What is the expected value of the number obtained when rolling a fair six-sided die?

We toss a coin repeatedly until either the sequence HHT appears or HTH appears. What is the probability that HHT appears first? How does this change if the coin is unfair, with probability p of heads?

If you have 5 digits, what is the largest number you can create such that the product of its digits is 120? Can you make the number arbitrarily large if the restriction on the number of digits is removed?

How many coin tosses are expected before getting 5 heads in a row?

Person A and B are going to play a coin toss game. There is an initial score of 0, and whenever a head or tail appears, the score increases by 1 or decreases by 1, respectively. The coin is tossed repeatedly until one wins, that is, when the score reaches +2 or -2, A or B wins the game. There is also an initial stake of $1 for the game and person A has the option to double the stake before each coin toss. When one person wins the game, the other player needs to pay the current stake to the winner. The question is: if you are person A, what is your optimal strategy and what is your highest expected payoff in the game?

You may throw a six-sided die and, if you are dissatisfied with the result, you may re-roll it once. How should you decide when to keep the initial roll and when to re-roll in order to maximize the expected value?

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

We randomly select 3 numbers from the set of the first 9 prime numbers, without replacement. What is the probability that the sum of those numbers is even, and why?

What is the smallest positive integer whose digits multiply to 10,000?

What is the expected value of a roll of a standard six-sided die?

How many coin tosses are required to obtain at least 4 tails with a probability of at least 0.7?

You roll a 100-sided die and may either accept the face value in dollars, or pay an additional dollar to re-roll. You can play as many times as you like. What is the optimal strategy?

Solve the following mental math problems: 1) What is 14% of 42? 2) What is 36 squared? 3) What day of the week is April 15th, 2142?

What is the angle between the hour and minute hands on a clock at 9:40?

You throw 1,000 fair coins. What is the probability of getting an even number of heads?

You have a drawer with an infinite number of socks in two colors, with each color equally likely. What is the expected number of socks you must draw individually from the drawer before you obtain a matching pair?