Jane Street Interview Questions

Design a heap data structure that supports adding elements and removing the top element, with the following constraints: (1) The topmost element must always be less than the topmost element in the left and right child subtrees; (2) The left child subtree must have as many or one more element than the right child subtree. The heap should also support a 'min' function (returns the value at the top of the tree) and an 'empty' function (returns an empty tree). Provide a description and implementation of such a data structure.

You have two decks of cards. Each deck contains both red and black cards in equal proportion. One deck has 52 cards, and the other has 104 cards, both consisting of half red and half black cards. You may choose which deck to play with. Then, you draw two cards at random from your chosen deck. If both cards are red, you win a prize. Which deck should you choose to maximize your chance of winning, and why?

There are 30 blue balls and 30 red balls, and two urns. Your opponent may arrange the balls in the two urns in any way he chooses, without telling you the arrangement. You then select one urn and draw a ball at random from it. You win $10 if you draw a blue ball, and $0 otherwise. How much should you be willing to pay to play this game?

Find the lowest positive integer such that the product of its digits equals n.

What is the probability of getting at least one head in 4 coin tosses? What about in 9 coin tosses?

How much would you pay for a game where your payoff equals the number shown on a die, with an option to reroll once? Generalize to n opportunities to reroll.

Walk me through the solution to the birthday problem (i.e., calculate the probability that at least two people in a group share the same birthday).

If you randomly pick a 3-digit number, what is the probability that all three digits are even numbers?

What is the set of numbers between 2 and 30, where no two numbers share a common factor greater than 1 (i.e., the set is pairwise coprime), that gives the maximum possible sum? Using the same rules, what is the highest possible number you can have in a set of 1000?

A and B play a game. Each chooses a different integer between 2 and 12. Two dice are rolled, and the sum is calculated. The player whose chosen number is closer to the dice sum wins. If you can choose first or second, which position should you choose, and what is your optimal strategy?

Basketball players A and B each play in Game 1 and Game 2. In both games, A has a higher shooting average than B. Is it possible for B to have a higher overall shooting average than A? If so, provide an example.

You have two bowling balls of the same density. One has a radius of 8 and a weight of 16; the other has a radius of 12. What is the weight of the second ball?

What is the expected value of an optional reroll of a fair six-sided die, given that your payout is the number of pips shown on the die? In other words, how much would you pay for the option to reroll once, given that your final payout is the number shown on the die?

Two people each bid a number before rolling a 30-sided die. Whoever bids closer to the number the die shows wins, and wins an amount of money equal to the number rolled. For example, if I bid 15 and you bid 16, and the die lands on 10, then I win 10 from you. What is the optimal bidding strategy and the expected payoff?

Two players toss a fair coin repeatedly. Player A wins if the sequence HHT appears first; player B wins if HTT appears first. What is the probability that player A wins the game?

What is the minimum number of people required in a group to ensure that at least 7 people share the same birthday month?

What is 1,000 to the 1,000th power?

What is the expected outcome when rolling two 10-sided dice? Please explain why.

You have 4 coins. You throw them in the air. For every coin that lands heads, you get 1 dollar. How much would you pay to play this game?

You are playing a game in which four fair coins are flipped, and the amount of money you receive in dollars equals the number of heads that appear. If you do not like the outcome of the first four flips, you may re-flip all four coins, but you must accept the result of the second set. Determine the fair value for the game.

In Russian Roulette, there is a revolver with 4 blanks and 2 bullets loaded consecutively in adjacent chambers. If the person before you fires a blank (the gun did not discharge), should you take the next shot as-is, or spin the cylinder before shooting? Justify your answer with probabilities.

What is the smallest number made up of only 1s and 0s that is divisible by 225?

What is the expected value of a die roll? Additionally, create a market (bid-ask spread) based on your expectation.

You flip a fair coin an unlimited number of times. If the sequence HHT appears before HTT, you win. What is the probability that you win?

Two fair 6-sided dice are rolled. What is the probability that the product of the top numbers is a perfect square?

What is the probability of getting an even number of heads in seven coin tosses?

A train is approaching a bridge, and you are standing at the 1/4 position along the length of the bridge. If running in either direction allows you to escape just in time, what is the ratio of your speed to the speed of the train?

What is the cube root of 169381?

Choose a subset of numbers from 1 to 30 such that no two elements in the subset share a common factor (i.e., they are pairwise coprime). What is the subset with the largest possible sum?

I have a number in mind such that the product of each of its digits is 96, and no digit is 1. What are the largest and smallest numbers I could think of?