Quant Interview Questions and Answers (2026): Probability, Brainteasers, and Options

Quant interviews at trading firms and hedge funds test how you reason about uncertainty under time pressure. Most rounds mix probability puzzles, expected value games, mental math, some statistics, and a few questions on options or code. Below are 21 quant interview questions with worked answers. Every number here was checked with exact arithmetic or a simulation before publishing.

What quant interviews test

Trader and researcher roles weight these differently. Trading interviews lean on the first three rows. Research interviews spend more time on statistics and code.

Probability questions

1. You roll two dice. What is the probability the sum is 7?

There are 36 equally likely outcomes, and six of them sum to 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1). The probability is 6/36 = 1/6.

A common follow-up: what about a sum of 10 or more? The outcomes are (4,6), (5,5), (6,4), (5,6), (6,5), (6,6), which is also 6 out of 36, so 1/6.

2. A family has two children, and at least one is a boy. What is the probability both are boys?

List the equally likely cases: boy-boy, boy-girl, girl-boy, girl-girl. "At least one boy" removes girl-girl and leaves three cases. One of them is boy-boy, so the answer is 1/3.

Interviewers use this to see whether you jump to 1/2. If the question were "the older child is a boy," the answer would be 1/2, because that leaves only boy-boy and boy-girl.

3. The Monty Hall problem: should you switch?

You pick one of three doors. The host, who knows where the car is, opens another door to show a goat and offers you the switch. Switch. Your first pick is right 1/3 of the time, and the host's action doesn't change that. The other 2/3 of the time, the car is behind the remaining door. Switching wins 2/3 of the time.

4. How many people do you need in a room for a better than 50% chance that two share a birthday?

Work with the complement. The chance that n people all have different birthdays is (365/365) × (364/365) × ... × ((365 − n + 1)/365). With 22 people the chance of a match is 47.6%. With 23 it's 50.7%. The answer is 23.

5. A disease affects 1% of people. A test catches 99% of real cases and gives a false positive 5% of the time. You test positive. What is the chance you have it?

Use Bayes' rule with 10,000 people. 100 have the disease and 99 of them test positive. 9,900 don't have it, and 5% of them, which is 495, test positive anyway. Of the 594 positives, 99 are real: 99/594 = 1/6, about 16.7%.

The point is that a rare condition produces mostly false positives even with a good test.

6. You draw five cards from a deck. What is the probability of at least one ace?

Complement again. The number of five-card hands with no ace is C(48,5), out of C(52,5) total. The probability of no ace is 65.9%, so at least one ace is 34.1%.

7. You break a stick at two random points. What is the probability the three pieces form a triangle?

Three lengths form a triangle only if no piece is longer than half the stick. The probability is 1/4. The quick argument: the chance that one specific piece exceeds half is 1/4, at most one piece can exceed half, and there are three pieces, so the chance that some piece is too long is 3/4.

Expected value questions

8. How many coin flips do you expect until the first head? Until two heads in a row?

First head: each flip succeeds with probability 1/2, so the expected number is 2.

Two heads in a row: let E be the answer. If the first flip is tails (probability 1/2), you've used one flip and start over. If it's heads then tails (1/4), you've used two and start over. If it's heads then heads (1/4), you're done in two. So E = 1/2 × (1 + E) + 1/4 × (2 + E) + 1/4 × 2, which gives E = 6.

Follow-up: heads then tails takes only 4 flips on average. After the first head, you can't lose progress: more heads just keep you waiting for a tail.

9. How many rolls of a die do you expect before you've seen all six faces?

This is the coupon collector problem. After you've seen k faces, the chance of a new one is (6 − k)/6, so the expected wait is 6/(6 − k). Add them up: 6/6 + 6/5 + 6/4 + 6/3 + 6/2 + 6/1 = 14.7 rolls.

10. You roll a die and win the number shown in dollars. You may re-roll once. What is the game worth?

A single roll is worth 3.5. So keep a 4, 5, or 6 and re-roll a 1, 2, or 3. Half the time you keep an average of 5, and half the time you take the re-roll worth 3.5: 0.5 × 5 + 0.5 × 3.5 = 4.25.

With two re-rolls allowed, the game after your first roll is worth 4.25, so you now keep only a 5 or 6: (2/6) × 5.5 + (4/6) × 4.25 = 4.67.

11. What is the expected value of the larger of two dice?

The maximum is at most k in k² of the 36 outcomes, so it equals exactly k in k² − (k − 1)² = 2k − 1 outcomes. The expected value is (1×1 + 2×3 + 3×5 + 4×7 + 5×9 + 6×11)/36 = 161/36, about 4.47.

12. You have $3. Each round you win or lose $1 on a fair coin flip. You stop at $0 or $10. What is the chance you reach $10?

For a fair game, your expected wealth doesn't change, so 10 × p + 0 × (1 − p) = 3 and p = 3/10.

A follow-up is how long the game lasts on average. For a fair walk starting at k with barriers at 0 and N, the expected number of rounds is k × (N − k) = 3 × 7 = 21.

13. A coin lands heads 60% of the time and pays even money. What fraction of your bankroll should you bet each round?

Betting everything maximizes the expected value of one round and guarantees ruin over many. The Kelly criterion maximizes the long-run growth rate. For an even-money bet it says to bet 2p − 1 of your bankroll, which is 2 × 0.6 − 1 = 20%.

A good answer adds that many traders bet less than Kelly, because the true probability is never known exactly and overbetting is costly.

Mental math and market making

14. What is 17 × 23?

Look for structure. 17 × 23 = (20 − 3)(20 + 3) = 400 − 9 = 391. Interviewers care that you have a method and say it aloud. Practice two-digit multiplication, fractions to decimals, and percentages until they're quick.

15. Make me a market on the sum of three dice

You're asked for a bid and an offer. Start with the expected value: 3 × 3.5 = 10.5. The standard deviation of one die is about 1.71, and for three dice it's about 2.96, so there's real spread around the middle.

A reasonable quote is "10 at 11." If the interviewer buys at 11, ask yourself what they might know, and move your next quote up. The test is whether you quote quickly around a sensible fair value, keep the width reasonable, and adjust after trades instead of defending your first number.

Statistics questions

16. The correlation between A and B is 0.8, and between B and C is 0.8. What can the correlation between A and C be?

It doesn't have to be high. The bounds are ρ(A,B) × ρ(B,C) ± √((1 − ρ(A,B)²)(1 − ρ(B,C)²)), which is 0.64 ± 0.36. So the correlation between A and C can be anywhere from 0.28 to 1.

17. A strategy has a daily mean return of 0.05% and a daily standard deviation of 1%. What is its annualized Sharpe ratio?

The daily Sharpe ratio is 0.05/1 = 0.05, ignoring the risk-free rate. Assuming independent daily returns and 252 trading days, the mean scales with time and the standard deviation scales with its square root, so multiply by √252, about 15.9. The annualized Sharpe ratio is about 0.79.

Say the assumption out loud. Autocorrelated returns make this scaling wrong.

18. What is a p-value, and why do so many backtests look better than they are?

A p-value is the probability of seeing a result at least as extreme as yours if the null hypothesis were true. It isn't the probability that the hypothesis is true.

Backtests look too good for a few reasons. If you test 100 strategies, about 5 will pass a 5% significance test by chance. Lookahead bias uses information that wasn't available at the time. Survivorship bias drops the companies that failed. And parameters tuned on the same data used for evaluation fit noise. Good answers mention out-of-sample testing, correcting for the number of trials, and realistic trading costs.

Options questions

19. What is put-call parity?

For European options on a stock that pays no dividends, with the same strike K and expiry T: C − P = S − K × e^(−rT). A call minus a put has the same payoff as owning the stock and owing the strike price at expiry.

Example: the stock is at 100, the strike is 100, the rate is 5%, and expiry is one year away. The present value of the strike is 95.12. If the call trades at 10.45, the put must be worth 10.45 − 100 + 95.12 = 5.57. Any other price is an arbitrage.

20. Explain delta, gamma, and vega

Delta is how much the option price moves when the stock moves by 1. A call's delta runs from 0 to 1. An at-the-money call has a delta near 0.5, and somewhat higher when rates and time to expiry are large. In the example above, with 20% volatility, it's about 0.64.

Gamma is how fast delta changes as the stock moves. It's highest for at-the-money options close to expiry.

Vega is how much the price changes when implied volatility changes. Both calls and puts gain value when volatility rises, and longer-dated options have more vega.

Coding question

21. Write a function that returns the maximum drawdown of a price series

Maximum drawdown is the largest drop from a peak to a later low, as a fraction of the peak. Track the highest price so far and the worst drop from it in one pass.

def max_drawdown(prices):
    peak = prices[0]
    worst = 0.0
    for p in prices:
        if p > peak:
            peak = p
        drawdown = (peak - p) / peak
        if drawdown > worst:
            worst = drawdown
    return worst

print(max_drawdown([100, 120, 90, 110, 80, 130]))  # 0.3333333333333333

The answer is 33.3%: the fall from 120 to 80. The function runs in O(n) time with O(1) extra space. We tested it against a brute-force version that checks every pair of days on 2,000 random series. For more practice, see our Python interview questions and the Big O cheat sheet.

How to prepare

1. Drill the classics until the setup is automatic: conditional probability, expected value by recursion, and the complement trick.
2. Practice mental math daily for a few weeks. Speed here is trainable.
3. Talk through every problem aloud. Interviewers grade the reasoning, and a wrong answer with clear logic often survives.
4. When you're stuck, try a smaller case. Two doors, two dice, a stick broken once.
5. For trading roles, practice making markets with a friend and updating your quote after each trade.
6. Prepare your own story too. Firms still ask why trading or research, and what you've built. Our guide to tell me about yourself helps with that part.

If you're interviewing at trading firms, also see our notes on the DRW online assessment. For bank roles, the investment banking interview questions cover a different set of topics.