Guide · probability
Probability interview questions for quant roles
27 problems · 7 min read · updated 2026-09-07
Probability is the one subject every quant interview tests, at every firm, in every round. The questions look like puzzles but they are not: almost all of them are solved by one of four moves — condition on the first step, count the sample space honestly, use linearity of expectation, or recognise a memoryless process.
This page sorts the questions that keep coming up by the move that solves them. Learn the move, and the variant you have never seen becomes the variant you have.
Conditional probability and Bayes
The information you are given changes the sample space. Recount it. Monty Hall, the two-children problem and the Tuesday boy are the same lesson escalating; the disease-test question is the same lesson with base rates that matter in real trading.
Three doors: one hides a car, two hide goats. You pick a door; the host (who knows) opens a different door revealing a goat, then offers you the switch. What is your probability of winning the car if you switch?
Answer: 2/3 · Worked solution →
A family has two children. Given that at least one is a boy, what is the probability that both are boys?
Answer: 1/3 · Worked solution →
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?
Answer: 13/27 · Worked solution →
A disease affects 1% of a population. A test is 99% sensitive (true-positive) and has a 1% false-positive rate. Given a positive test, what is the probability the person actually has the disease?
Answer: 0.5 · Worked solution →
A standard 52-card deck is shuffled. What is the probability the top two cards are both aces? (Enter as a fraction or decimal.)
Answer: 1/221 · Worked solution →
Expected value and stopping times
Condition on the first trial and write the recursion; linearity of expectation does the rest. Expected flips until a pattern, expected rolls until all faces, expected maximum of a sample — these are asked at every trading firm and are pure technique.
What is the expected number of flips of a fair coin until the first head appears?
Answer: 2 · Worked solution →
What is the expected number of rolls of a fair die to get the first 6?
Answer: 6 · Worked solution →
You roll a fair six-sided die repeatedly. What is the expected number of rolls until you have seen all six faces?
Answer: 14.7 · Worked solution →
Two fair six-sided dice are rolled. What is the expected value of the larger of the two (ties count as that value)?
Answer: 4.472 · Worked solution →
Two fair dice are rolled. What is the expected value of the smaller of the two (ties count as that value)?
Answer: 91/36 · Worked solution →
Two independent Uniform(0, 1) random variables are drawn. What is the expected value of the larger one?
Answer: 2/3 · Worked solution →
A stick of length 1 is broken at a point chosen uniformly at random. What is the expected length of the shorter piece?
Answer: 0.25 · Worked solution →
Counting and combinatorics
Complements, inclusion–exclusion and honest enumeration. The birthday problem is asked because candidates guess; Galileo’s dice is asked because unordered outcomes are not equally likely; the hat-check problem hides 1/e.
What is the smallest number of people in a room for which the probability that at least two share a birthday exceeds 50%? (Ignore leap years.)
Answer: 23 · Worked solution →
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?
Answer: 1 · Worked solution →
Three fair dice are rolled. Gamblers noticed sums of 10 appear more often than sums of 9, though both can be written as 6 unordered combinations. What is the probability of rolling a sum of exactly 10?
Answer: 0.125 · Worked solution →
Two fair six-sided dice are rolled. What is the probability their sum equals 7?
Answer: 1/6 · Worked solution →
You roll a fair die 4 times. What is the probability of getting at least one 6? (Round to 3 decimals.)
Answer: 0.518 · Worked solution →
The classic hard problems
Each of these has a famous elegant solution and a brute-force route that runs out of time. Interviewers accept either, but they promote candidates who find the symmetry (the drunk passenger), the structure (cycles in the 100 prisoners problem) or the threshold rule (the secretary problem).
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?
Answer: 1/2 · Worked solution →
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.
Answer: 0.31 · Worked solution →
You interview 100 candidates in random order, must accept or reject each on the spot, and want the single best. The optimal rule observes and rejects an initial fraction, then takes the first candidate better than everyone seen so far. Roughly what percentage should you observe first?
Answer: 37 · Worked solution →
A stick is broken at two independent, uniformly random points. What is the probability the three pieces can form a triangle?
Answer: 1/4 · Worked solution →
A 6-chamber revolver has 2 bullets in adjacent chambers. The cylinder is spun, the trigger is pulled, and it’s empty. If you do NOT re-spin, what is the probability the next pull is also empty?
Answer: 3/4 · Worked solution →
Processes: memorylessness and random walks
Exponential and Poisson questions test whether you understand memorylessness; random-walk questions test whether you can set up a boundary-value recursion. Gambler’s ruin is the bridge into stochastic calculus and is asked in both trading and research interviews.
Calls arrive as a Poisson process at a rate of 3 per hour. What is the probability of receiving no calls in the next hour? (Round to 4 decimals.)
Answer: 0.0498 · Worked solution →
A component’s lifetime is Exponentially distributed with rate λ = 2 per year. What is the probability it lasts beyond 1 year? (Round to 3 decimals.)
Answer: 0.135 · Worked solution →
What is the expected number of fair-coin flips to get two heads in a row (HH)?
Answer: 6 · Worked solution →
You start with 1 on a fair coin each round, stopping when you reach 100. What is the probability you reach $100 before going broke?
Answer: 0.3 · Worked solution →
What is the expected number of fair-coin flips to first see the pattern H-T-H?
Answer: 10 · Worked solution →
How to approach these in the interview
Write the sample space down. Most conditional-probability mistakes come from answering before the space is enumerated. Two children with "at least one boy" is four equally likely pairs minus one — say that, and the answer is unavoidable.
Condition on the first step. For any "expected number of trials" question, let E be the answer, condition on what happens first, and solve the resulting linear equation. It works for flips, rolls, random walks and patterns alike.
Check the limits. Before you say a number, ask what happens as the probability goes to 0 or 1. If your formula breaks there, it is wrong.
Know the three famous constants. 1/e (hat check, secretary), 1/2 (drunk passenger), and ln 2 (median of an exponential) appear so often that recognising them saves minutes.
Frequently asked
What level of probability do quant interviews require?
Undergraduate probability done fluently: conditional probability and Bayes, expectation and variance, the common distributions (binomial, geometric, Poisson, exponential, normal), and the ability to set up and solve small recursions. Measure theory is not required for trading; some research interviews touch martingales.
What is the most common probability interview question?
Variants of "expected number of flips until two heads in a row" and "probability of at least one six in N rolls" are asked more than any other, closely followed by Monty Hall-style conditioning questions.
How should I answer when I do not know the trick?
Set up the recursion or enumerate the small case out loud. Interviewers reward a correct method that runs slowly far more than a memorised answer, and the small case very often reveals the trick anyway.
Do I need to know stochastic calculus for probability rounds?
Not for trading. Research and derivatives roles will follow probability with Brownian motion and Itô questions; the random-walk and gambler’s-ruin problems above are the natural bridge.
Practise these with answer checking and spaced review
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