Probability Lab

Quant probability interview questions, with worked solutions

285 curated probability questions from real quant trading and research interviews, each with a full solution that teaches the method, not just the answer. Free, in your browser, no sign-up.

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Quant interviews at firms like Jane Street, Optiver, IMC, SIG and Citadel lean hard on fast, correct probability reasoning: conditional probability and Bayes, expected value, dice and card setups, random walks and gambler's ruin, combinatorics, and the market-making intuition that turns a fair value into a two-sided quote. The Probability Lab drills exactly those, sorted into 13 topics and three difficulty tiers.

Every question ships with a written solution that shows the reasoning a good interviewer wants to hear out loud. You can work in Study mode (see the solution after each), Gauntlet mode (10 questions, no feedback until the end, like the real thing), Odds Sprint mode (Beat-the-Odds style: 15 quick probability calls, 60 seconds each, timeouts count as misses), or Review mode, which uses spaced repetition to resurface the questions you got wrong.

Example questions

A disease affects 1% of a population. A test is 99% sensitive and 99% specific. You test positive. What is the probability you are actually sick?

50%. By Bayes, P(sick | +) = (0.99 × 0.01) / (0.99 × 0.01 + 0.01 × 0.99) = 0.0099 / 0.0198 = 1/2. With a rare condition, the false positives from the huge healthy majority match the true positives. Always weigh the base rate.

A family has two children. You learn at least one is a boy. What is the probability both are boys?

1/3. Of the equally likely outcomes BB, BG, GB, GG, conditioning on "at least one boy" leaves BB, BG, GB. Only one is two boys. If instead you were told "the eldest is a boy", the answer would be 1/2: the information matters.

Flipping a fair coin, does HH or HT tend to appear first, and what is the expected wait for each?

Neither appears first more often (each 1/2), but the waits differ. After the first head, the next flip decides the HH-vs-HT race 50/50. Yet E[time to HH] = 6 and E[time to HT] = 4: when HH fails it resets fully, so it is slower on average even though it wins the head-to-head exactly half the time.

You start with $30 and bet $1 on fair coin flips until you reach $100 or go broke. What is the probability you reach $100?

30%. In a fair gambler's ruin, P(reach N from i) = i/N = 30/100. Fastest proof: wealth is a martingale, so E[final] = 30 = 100·p, giving p = 0.3.

An asset is worth 90 or 110, equally likely. 20% of traders are informed (buy iff worth 110); the rest trade 50/50. What ask should a zero-profit market maker quote?

102. P(buy | high) = 0.6, P(buy | low) = 0.4, so P(high | buy) = 0.6 and the fair ask conditional on a buy is 0.6·110 + 0.4·90 = 102. Symmetrically the bid is 98. This is the Glosten-Milgrom model: the spread is literally the price of adverse selection.

What is covered

Conditional probability and Bayes: disease tests, the two-child problem, coins from a bag, the taxi-cab problem.
Expected value: dice games, optimal rerolls, the St. Petersburg paradox, the Kelly criterion.
Coins, dice and cards: waiting times for patterns, Penney's game, expected extremes, poker-hand odds.
Random walks, ruin and Markov chains: hitting times, Poisson processes, stationary distributions, the ballot problem.
Combinatorics: derangements, Catalan numbers, stars and bars, the birthday problem.
Geometric and continuous: uniform order statistics, breaking a stick into a triangle, Buffon's needle.
Statistics and game theory: variance and correlation, regression to the mean, auctions, the pirate game, Nim.
Market intuition: quoting two-sided markets, inventory skew, the winner's curse, adverse selection.

Miss a question? Drill it with fresh numbers

Get one wrong and the Lab can regenerate it with new numbers, over and over, until you get three in a row. The answers are computed by code, never memorizable, so you practice the technique rather than the specific figure. This drill mode is free and always will be.

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