Jane Street interview questions and how to prepare

A fair coin is tossed twice. Given that at least one toss is heads, what is the probability that both are heads?

100 passengers board a full 100-seat plane. The first passenger lost their ticket and sits in a uniformly random seat. Each later passenger takes their own seat if free, else a random free seat. What is the probability the 100th passenger sits in their own seat?

A stick is broken at two independent, uniformly random points. What is the probability the three pieces can form a triangle?

Two independent Uniform(0, 1) random variables are drawn. What is the expected value of the larger one?

n people throw their hats into a pile and each takes one back at random. What is the expected number of people who get their own hat?

A family has two children. You learn that at least one of them is a boy born on a Tuesday (assume boys and girls, and all birth days, are equally likely and independent). What is the probability that both children are boys?

100 prisoners’ numbers are placed randomly in 100 boxes, one per box. Each prisoner may open at most 50 boxes, looking for their own number; all must succeed or all die, and they cannot communicate once the search begins. With the optimal strategy (each prisoner starts at their own box and follows the number found to the next box), what is the probability that every prisoner finds their number? Answer to two decimals.

You roll a fair die and are paid its face value in dollars, but you may re-roll up to two times, keeping only the final roll. Playing optimally, what is this game worth?

Cards from a shuffled standard 52-card deck are revealed one at a time. Your running total gains $1 for every red card revealed and loses$1 for every black card. You may stop whenever you like and collect the running total (you may also stop immediately at $0). Playing optimally, what is this game worth?

Everyone in a large group simultaneously picks a number from 0 to 100. The winner is whoever is closest to two-thirds of the group average. If every player is perfectly rational and knows everyone else is too, what number does game theory say everyone picks?

Two bidders compete in a sealed-bid first-price auction. Each privately values the item as an independent uniform draw on [0, 1], and the higher bid wins and pays its own bid. In the symmetric equilibrium each bidder bids half their value. Your value is 0.8 — what do you bid?

A pot starts at $2 and doubles after every heads; the game ends at the first tails and pays the pot. In dollars, what is the expected payout of this game?

Two envelopes hold money, one exactly twice the other. You pick one, open it, and see $100. A voice argues: “The other holds$50 or $200 with equal probability, expected value$125 — always switch!” The same argument applies before opening ANY envelope, implying you should switch forever. Where exactly does the argument break?

Five rational pirates split 100 gold coins. The most senior proposes a split; if at least half (including the proposer) approve, it passes — otherwise he is thrown overboard and the next-senior proposes. Pirates prioritise survival, then gold, then bloodthirst. How many coins does the most senior pirate keep?

12 identical-looking balls include exactly one of a different weight — you don’t know if it’s heavier or lighter. Using only a balance scale, what is the minimum number of weighings that guarantees finding the odd ball AND whether it is heavy or light?

You have 1000 bottles of wine, exactly one of which is poisoned. A test strip reacts (after one hour) if it touches any poison. With only one hour before a feast, what is the minimum number of test strips needed to guarantee finding the poisoned bottle?

Three jars are labeled “Apples”, “Oranges”, and “Apples + Oranges”. Every label is wrong. Drawing one fruit at a time without looking, what is the minimum number of draws to correctly relabel all three jars?

On an island live 100 blue-eyed people and some brown-eyed people. Nobody knows their own eye colour, everyone sees everyone else’s, and anyone who deduces their own colour must leave that midnight. All islanders are perfect logicians and this is common knowledge. One day a visitor announces to everyone: “At least one of you has blue eyes.” On which night do the blue-eyed islanders leave?

A camel must transport 3,000 bananas across a 1,000 km desert to market. It can carry at most 1,000 bananas at a time and eats 1 banana per km travelled (in either direction). Bananas may be cached anywhere along the route. What is the maximum whole number of bananas that reach the market?

Four people must cross a bridge at night with one flashlight. They take 1, 2, 5 and 10 minutes to cross. At most two cross at once, and a pair moves at the slower person’s pace; the flashlight must be walked back each time. What is the minimum total time (minutes)?

What is the expected number of fair-coin flips to get two heads in a row (HH)?

What is the expected number of fair-coin flips to first see the pattern H-T-H?

You start with $30 and bet$1 on a fair coin each round, stopping when you reach $0 or$100. What is the probability you reach $100 before going broke?

In an election, candidate A receives 60 votes and candidate B receives 40. The ballots are counted one at a time in uniformly random order. What is the probability that A is strictly ahead throughout the entire count?

A symmetric random walk starts at 3 and stops when it hits 0 or 10 (each step ±1, equally likely). What is the expected number of steps until it stops?

What value of x > 0 maximizes $x^{1/x}$? (Enter a decimal.)

What is the derivative of $x^x$ evaluated at $x = 1$?