Jane Street Interview Questions

Given one fair 10-sided die and one fair 6-sided die, what is the expected value of the sum of their outcomes?

What is the expected value of the sum when two fair six-sided dice are rolled?

If there are 8 people and they all shake hands with each other once, how many handshakes are there? If there are four couples among the 8 people who do not shake hands with each other, how many handshakes are there now?

If you flip a fair coin twice, what is the expected value of the game, assuming you win $1 for each head and lose $1 for each tail?

We randomly select 4 numbers from the set of the first 20 prime numbers, without replacement. What is the probability that their sum is even? Explain your reasoning.

You are playing a game with a 6-sided die. You may roll the die once, observe the result, and choose either to stop (keeping the result) or roll again. Your final payoff is the sum of your rolls, unless this sum exceeds 9, in which case you receive nothing. What is your optimal strategy for this game? Specifically, for each possible outcome of the first roll, should you choose to stop or roll again?

A bag contains three visually indistinguishable coins: one with a 10% chance, one with a 30% chance, and one with a 60% chance of landing heads. You randomly select a coin and flip it, and it lands heads. What is the probability that if you flip the same coin again, it will land heads? Explain your reasoning.

You have two indistinguishable urns. One contains seven $1 chips and three $10 chips, and the other contains nine $1 chips and one $10 chip. You randomly draw a chip from one of the urns and it turns out to be a $10 chip (the drawn chip is not replaced). You are then offered the chance to draw and keep a chip from either urn. Should you draw from the same urn or the other urn, and what is the expected value of your draw? Explain your reasoning.

With dates written in DD/MM/YYYY format, what is the next date where no digit is repeated?

What is the probability of getting an odd number of heads in a sequence of coin flips where some coins are not fair?

What is the expected value of a die roll?

Two players play a game of coin toss with one coin. One wins if the sequence HTH occurs first, the other if HHT occurs first (H = heads, T = tails). Is the game fair? If not, who has the advantage?

There are 1000 people in a hall. One person has their hand painted. Every minute, everyone shakes hands with someone else. How much time is needed to paint all the hands? What is the best-case scenario? What is the worst-case scenario?

You and a friend are playing a coin tossing game. You toss a fair coin repeatedly and track the results. Each of you has a sequence you are watching for: your sequence is HTT and your friend's sequence is HHT. The player whose sequence appears first wins the game. Would you want to play? What is your probability of winning?

How many heads would you expect to get if you toss 4 fair coins?

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.