Jane Street Interview Guide

A magician and her assistant perform with an ordinary $52$-card deck. A spectator picks any five cards and hands them to the assistant. The assistant secretly examines them, keeps one hidden, and passes the remaining four to the magician one at a time in an order of the assistant's choosing. From nothing but the identities and sequence of those four cards, the magician names the hidden card — correctly, every single time. Design a scheme that makes this trick infallible, and explain why it works.

A boardwalk arcade stall runs a game built around a single fair six-sided die. Two wagers are on offer. Wager 1: put down $4 on the roll landing even; a winning ticket pays $6 back in total. Wager 2: put down $6 on the roll landing odd; a win returns $9 in total. You can buy as many tickets of either type as you like. If your goal is to maximize expected profit, what is the right amount to wager?

A casino hands you a coin whose probability of landing heads is some fixed $p$, which nobody tells you. The coin will be tossed $100$ times; before each toss you must call a side, collecting $\$1$ whenever your call matches the outcome and nothing otherwise. You are free to adapt your calls based on everything you've observed so far. Playing as well as possible, what is your expected total winnings, expressed in terms of $p$?

A bakery sells $11$ varieties of donuts. An office manager wants to bring back a box of $6$ donuts for the team and is happy to include repeats of any variety. How many distinct boxes (counted by variety composition, ignoring order) are possible?

You keep rolling a fair six-sided die, adding up the results, and stop the moment the running total climbs to $9{,}000$ or beyond. Which face value is most likely to show on that final, threshold-crossing roll?

Take the sixteen smallest primes,$$\{2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53\},$$and draw $4$ of them at random without replacement. What is the probability that the four drawn numbers add up to an even total?

Three contestants take turns publicly committing to integers. Contestant A goes first, naming any integer from 1 to 100. Contestant B, having seen A's number, names a different integer in the same range. Contestant C, having seen both, names a third distinct integer from 1 to 100. A referee then draws a target $R$ uniformly at random from 1 to 100, and whoever's number lies closest to $R$ wins; exact ties in distance are settled by a fair coin. If B and C both play optimally, what should A announce to maximize the chance of winning?

A carnival sack holds $20$ tokens: $19$ gold coins and $1$ cursed token, thoroughly mixed. You draw tokens one at a time without replacement. Each gold coin adds $\$1$ to your pot, but drawing the cursed token ends the game and wipes out the entire pot. You may stop after any draw and walk away with the pot. What is the optimal stopping rule, and what is the game's expected value under it?

A dealer shuffles a standard 52-card deck (26 red, 26 black) and turns cards face up one at a time. At some point you must interrupt and declare that the very next card will be red — and you are required to make this call before the deck runs out, so if you wait until 51 cards are shown you are calling the last card. You win if the card right after your call is red. You see every card as it is dealt and may base your timing on the running counts. Can any strategy make your win probability exceed $50\%$?

A boardwalk hustler shuffles a full 52-card deck and peels off the top two cards over and over, 26 pairs in all. Every red–red pair goes into your pile, every black–black pair into hers, and every mixed pair goes into the trash. Whoever ends the deck with more pairs collects $\$10$ from the other; equal piles push. How much should you be willing to pay for the right to play?

At an arcade, Rui challenges Sana to a series of one-token games: each round the loser hands one token to the winner, and Rui wins any given round with probability $\frac{2}{3}$, independently of the rest. Rui starts with 1 token and Sana with 2, and they keep playing until one of them holds all 3 tokens. What is the probability that Rui wins everything?

You and a friend flip a fair coin for $\$1$ a flip, and you track your net winnings starting from $0$. You agree in advance to quit the moment you are either down $\$2$ or up $\$3$. What is the probability the session ends with you down $\$2$? Would the answer change if your running P&L instead followed a driftless Brownian motion stopped at the same two levels?

You and a rival each spin a dial that lands uniformly on $[0,1]$, each seeing only your own result. Each player may keep their number or scrap it and spin exactly once more, being forced to keep the second spin. The higher final number wins. Both players play optimally to maximize their probability of winning. Below what first-spin value should you take your mulligan?

A game show has a three-digit board with a hundreds, tens, and units slot. A fair six-sided die is rolled three times; after each roll you must immediately lock the result into one empty slot, with no rearranging later. You win the final three-digit number in dollars. What is the optimal placement strategy, and what does it earn on average?

A charity raffle hopper contains $52$ tickets, of which exactly $4$ are winners, mixed uniformly. Tickets are pulled out one at a time without replacement. What is the expected number of pulls up to and including the first winning ticket?

Five security guards each work one shift during a day. All $10$ shift endpoints are drawn independently and uniformly over the day, then paired off: the first two draws bound guard 1's shift, the next two bound guard 2's, and so on (each guard's earlier draw is the start, the later one the end). What is the probability there is some moment when all five guards are on duty simultaneously?

At a fair booth you first roll a fair six-sided die. Whatever number $N$ comes up, you then flip a fair coin exactly $N$ times and collect one token for every head. What is the expected number of tokens you walk away with?

Mira and Noah flip separate fair coins, one flip each per round, starting together. Mira stops the first round her coin has just shown two heads in a row; Noah stops the first round his coin has just shown three heads in a row. Let $X$ and $Y$ be their stopping rounds. What is the probability that Noah finishes strictly first, $\Pr(Y < X)$?

A carnival booth lays out two dice that look identical. One is fair; the other has had its six ground off and can never show a 6. Nobody tells you which is which. Each roll costs $\$1$: you pick up either die (switching between rolls is free and unlimited), roll it, and collect $\$20$ if it shows a 6. You may quit whenever you like. How should you play, and is the game worth playing?

At a quant off-site, three participants each pick a number in $[0, 100]$. Whoever picks the strictly largest number must pay each of the other two an amount equal to that person's own pick; the two lower pickers just collect. Priya picks first, publicly. Marco sees her number and picks second. The third pick comes from a random number generator, uniform on $[0, 100]$, as everyone knows. Priya and Marco are rational expected-profit maximizers. What number should Priya pick?