Quant interview questions, with worked solutions

I work as a high-frequency trading quant, and I have been through these interviews from the candidate's side. The specific questions change every season and firms rotate them precisely because they end up online. The types do not change, and neither does what the interviewer is listening for.

So this is not a list to memorise. For each type I give one clean example, the solution, and the thing that separates a strong answer from a correct one.

1. Expected value with states: how many rolls until two sixes in a row?

Warm-up first. The expected number of rolls of a fair die until the first six is 6, because the number of rolls is geometric with success probability 1/6. Say that in one sentence and move on. Then comes the real question: how many rolls, on average, until you see two sixes in a row?

Define states by what matters for the future. Let E₀ be the expected number of further rolls when the last roll was not a six (or you have not started), and E₁ when the last roll was a six.

• From state 0: roll once. With probability 1/6 you move to state 1, otherwise you stay in state 0. So E₀ = 1 + (5/6)E₀ + (1/6)E₁.
• From state 1: roll once. With probability 1/6 you are done, otherwise you are back to state 0. So E₁ = 1 + (5/6)E₀.

Substitute the second into the first: E₀ = 1 + (5/6)E₀ + 1/6 + (5/36)E₀, so E₀(1/36) = 7/6 and E₀ = 42.

What earns the offer: naming the states before writing anything, and a sanity check at the end. 42 is much more than 2 × 6 = 12, and you should be able to say why: every time you roll a six and then miss, you lose the progress. Candidates who guess 36 (thinking "probability 1/36, so 36 tries") are showing exactly the reflex interviewers are screening out.

2. Martingales in disguise: gambler's ruin

You start with £k and bet £1 on fair coin flips until you reach £N or go broke. What is the probability you reach £N?

Your wealth is a martingale: a fair bet does not change its expected value. The game stops at a bounded time with wealth either 0 or N. By optional stopping, expected wealth at the end equals wealth at the start:

p · N + (1 − p) · 0 = k, so p = k / N.

The follow-up is almost always the expected length of the game. The trick is a second martingale: W_t² − t. Applying optional stopping again gives an expected duration of k(N − k).

What earns the offer: recognising the shape. Random walks on graphs, first passage problems and "expected time to absorption" questions are all this one in different clothes. Solving it by writing a recurrence works too, but it takes three times as long and the interviewer will usually ask whether you see a faster way.

3. Conditioning: the question that is secretly about information

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

The four equally likely families are BB, BG, GB and GG. "At least one boy" removes GG, leaving three. One of them is BB, so the answer is 1/3, not 1/2.

The interviewer then changes one word: "The older child is a boy." Now only BB and BG are possible, and the answer is 1/2. The mathematics barely changed. The information did.

What earns the offer: asking how the information was obtained before answering. That instinct, "what exactly do I know, and how did I come to know it?", is the whole job on a trading desk, and a surprising number of probability questions exist only to test it.

4. Market making: quote me a market

"I am going to roll two dice. Make me a market on the sum."

The fair value is 7. A reasonable opening quote is something like 6.5 bid, 7.5 offered: wide enough to be paid for your risk, narrow enough to trade. Say why you chose the width.

Then the game starts. The interviewer buys from you at 7.5. Then again. Then again. What should you do?

The dice have not changed, so on the pure maths the fair value is still 7. But the right response is to ask why someone keeps buying. Maybe they are just testing you. Maybe they know something: in some versions of this game the interviewer has seen one die. Moving your quote up, and saying out loud that you are protecting yourself against someone better informed, is the answer they want. That is adverse selection, and it is the central idea of market making.

What earns the offer: updating, explaining the update, and keeping track of your position as you go. Candidates who hold 6.5 at 7.5 forever because "the expected value is 7" have the mathematics right and the job wrong.

5. Mental maths, and why it is not a gimmick

17 × 23? Write it as (20 − 3)(20 + 3) = 400 − 9 = 391. That took two seconds.

Rounds of rapid arithmetic are common in trading interviews, and people resent them. They matter for two reasons. In a market-making game, arithmetic that uses up your attention leaves none for the actual question. And people who compute fluently notice when an answer is off by a factor of ten, which is worth a great deal when real money is involved.

It is also the most trainable skill on this page. Twenty minutes a day for six weeks changes it completely.

What every one of these has in common

In each case the answer is the smallest part of what is being assessed. The interviewer is watching whether you find structure, whether you check yourself, and what you do when you are told you are wrong. I wrote about that at more length in what a quant interview is actually testing.

If you have a live process, the most useful preparation is not more questions. It is doing these out loud with someone who interrupts you, pushes back, and does not let you stay silent while you think.

Common questions

For 200 more, with answers, see the quant interview question bank, or test yourself against the clock in the timed quiz.

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