Expected Value Interview Questions: 10 Worked Examples 2026

Why Expected Value Questions Keep Coming Up

Expected value is the working language of a trading desk. Every price you quote in a market making game, every "would you play this game?" and every sizing decision starts from the same question: what is this worth on average? So it is no surprise that EV questions turn up in trader, researcher and even developer interviews at quant firms.

Most of them yield to one of two techniques. The first is linearity of expectation, usually with indicator variables, which turns hard counting problems into one-line sums. The second is conditioning: split the problem on what happens first, or on what you are told, and solve the smaller pieces. The rest are traps, questions designed to see whether you notice that the obvious calculation answers the wrong question.

This guide works through 10 questions that cover all three. The answers are exact and checked. For a wider bank of probability problems, see our quant probability interview questions.

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Technique 1: Linearity and Indicator Variables

Linearity of expectation says E[X + Y] = E[X] + E[Y], always, whether or not X and Y are independent. That last part is what makes it so useful. You can break a complicated count into many small yes-or-no events, find the probability of each, and add them up without ever working out how they interact.

1. The coat check

"Ten people leave their coats at a cloakroom. The attendant hands them back completely at random. What is the expected number of people who get their own coat back?"

Answer. Let Iₖ be 1 if person k gets their own coat and 0 otherwise. Each person has a 1 in 10 chance, so E[Iₖ] = 1/10. The expected number is the sum of 10 of these, which is 1.

The answer is 1 for any number of people, 10 or 10,000. The events are not independent (if nine people have their own coats, so does the tenth), but linearity does not care. Candidates who try to compute the full distribution run out of time.

2. Runs in coin flips

"I flip a fair coin 10 times. A run is a maximal block of identical results, so HHTHH has three runs. What is the expected number of runs?"

Answer. Every sequence starts one run. After that, a new run starts at flip k whenever flip k differs from flip k − 1, which happens with probability 1/2. There are nine such boundaries.

Expected runs = 1 + 9 × 1/2 = 5.5.

3. Distinct faces in six rolls

"I roll a fair die six times. What is the expected number of different faces that appear?"

Answer. Use one indicator per face. Face j fails to appear in six rolls with probability (5/6)⁶, so it appears with probability 1 − (5/6)⁶.

Expected distinct faces = 6 × (1 − (5/6)⁶) = 6 × (1 − 15,625/46,656) ≈ 6 × 0.665 ≈ 3.99.

A good follow-up to have ready: the chance that all six faces appear is 6!/6⁶ = 720/46,656, about 1.5%. The expected count is close to four, but seeing all six is rare.

4. Seeing every face

"How many times do you expect to roll a die before you have seen all six faces?"

Answer. This is the coupon collector problem. Break the wait into stages. Once you have seen k distinct faces, each roll shows a new one with probability (6 − k)/6, so the wait for the next new face is geometric with mean 6/(6 − k).

Expected rolls = 6/6 + 6/5 + 6/4 + 6/3 + 6/2 + 6/1 = 6 × (1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6) = 6 × 2.45 = 14.7.

The last face alone takes six rolls on average, which is why the total is so much higher than most people's first guess.

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Technique 2: Conditioning

When a question gives you information, or when the answer depends on what happens first, split on it. This is also the technique behind every requote in a market making game.

5. Conditioning on the sum

"I roll two dice and tell you the total is 8. What is the expected value of the first die?"

Answer. By symmetry, both dice have the same expected value given the total, and the two expected values must add to 8. So each is 4.

You can check it directly. The outcomes with a total of 8 are (2,6), (3,5), (4,4), (5,3) and (6,2), all equally likely, and the first die averages (2 + 3 + 4 + 5 + 6)/5 = 4. Interviewers like this one because the symmetry argument takes five seconds and the brute force takes a minute.

6. Rolling for a six, given only even numbers

"You roll a die until you get a 6. Given that every roll along the way was even, what is the expected number of rolls?"

Answer. Most people say 3, reasoning that you are effectively rolling a three-sided die (2, 4 or 6). The answer is 1.5.

The mistake is treating the condition as a change to the die. It is not; it is a filter on which sequences you keep. Think of it this way: keep rolling until you see any of 1, 3, 5 or 6. Each roll stops the process with probability 4/6, so the stopping time has mean 1 ÷ (4/6) = 1.5. Which of the four numbers ends the process is independent of how long it took, so conditioning on it being a 6 does not change the expected length.

Conditioning on "all even" throws away every long sequence, because long sequences are the ones most likely to have hit an odd number. What survives is dominated by short runs.

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Technique 3: Traps

These questions have a tempting answer that is wrong, or right but beside the point.

7. The square of a die

"I roll a die and pay you the square of the result in pounds. What is the fair price to play?"

Answer. E[X²] = (1 + 4 + 9 + 16 + 25 + 36)/6 = 91/6 ≈ £15.17.

The trap is squaring the expected value: 3.5² = 12.25. For any payoff that curves upwards, the average of the payoff is higher than the payoff of the average. This is Jensen's inequality, and the gap here, 91/6 − 12.25 = 35/12, is exactly the variance of a single die.

At an options market maker this is often the bridge to option questions. An option's payoff curves upwards, which is why more volatility makes it worth more. Our options Greeks guide picks up from there.

8. The longer piece

"I break a one-metre stick at a uniformly random point. What is the expected length of the longer piece?"

Answer. If the break is at U, the longer piece is max(U, 1 − U). By symmetry, condition on U being above one half: the longer piece is then U itself, uniform between 0.5 and 1, with mean 0.75. So the expected length is 75cm.

The trap is answering 50cm, the expected length of a randomly chosen piece. The question asks about the longer one, and "the longer one" is chosen after seeing the result.

9. Two envelopes

"Two envelopes contain money, one twice as much as the other. You pick one at random and see £100. Should you switch?"

Answer. The famous argument for switching says the other envelope holds £50 or £200 with equal chances, an expected £125, so switch. And the same argument would make you switch back, forever.

The flaw is the assumption that £50 and £200 are equally likely no matter what you see. That can only hold if every amount of money is equally likely to be in the envelopes, and no such probability distribution exists. With any real prior over the amounts, whether switching helps depends on the amount you see. Before you look, switching and keeping have the same expected value, because you are equally likely to be holding the larger or smaller envelope.

The interviewer is not looking for a full treatment. They want you to say that the paradox comes from an impossible prior, and that before you look at the amount there is nothing to gain from switching.

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Expected Value Is Not the Whole Answer

10. Betting with an edge

"A biased coin lands heads 60% of the time. You start with £100 and can bet any amount at even money on each of 100 flips. How much should you bet?"

Answer. If you only care about expected wealth, bet everything every time. Each flip multiplies your expected wealth by 0.6 × 2 = 1.2, so after 100 flips the expected value is £100 × 1.2¹⁰⁰, roughly £8 billion.

It is also a terrible strategy. You only keep the money if you win all 100 flips, and the chance of that is 0.6¹⁰⁰, about 6.5 × 10⁻²³. The expected value is enormous because it is driven entirely by an outcome that will not happen.

The better question is what fraction of your wealth to bet so that your money grows fastest over time. For an even-money bet that wins with probability p, the answer is the Kelly fraction, 2p − 1, which here is 20%. Your expected log growth per flip is then 0.6 × ln 1.2 + 0.4 × ln 0.8 ≈ 0.020, about 2% a flip, with essentially no chance of going broke. Our Kelly criterion explainer goes through the derivation.

This is often the question that ends the EV section of an interview, because it tests whether you know when to stop optimising the average.

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How to Practise

Speed matters as much as correctness. In a trading interview you are expected to say the answer to questions like Examples 1, 2 and 5 within seconds, and to set up Examples 3, 4 and 6 out loud without stalling.

• Drill the two techniques separately. For every counting question you meet, ask whether indicators would make it a one-liner. For every question that gives you information, ask what to condition on.
• Keep a list of traps. Examples 6 to 9 are well-known, which means interviewers use variants of them. Being able to name the trap is half the answer.
• Keep the arithmetic sharp. Fractions like 91/6 and 35/12 need to come out cleanly. Our free mental math trainer helps with the pace.
• Work from a book. The Green Book chapters on probability and expected value remain the standard reference for this style of question.

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Where This Guide Falls Short

These 10 questions are chosen to teach the techniques, not to predict what you will be asked. Real interviews add pressure that a written page cannot: an interviewer who asks you to go faster, changes the question halfway through, or challenges a correct answer to see whether you hold your ground.

Expected value is also rarely the final answer in a trading context. Examples 9 and 10 point at this, but interviewers at many firms will push further on risk, sizing and how you would actually trade something, as our quant trader interview questions show. Get the EV right quickly, then be ready to talk about everything else.

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Recruiting Notes

This guide is based on standard probability results and the style of question candidates report from quantitative trading and research interviews. It does not represent any firm's official question bank, and nothing here guarantees a particular question, format or outcome.

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Frequently Asked Questions

What is an expected value question in a quant interview?

A question that asks for the average outcome of a random process: a dice game, a sequence of coin flips, a betting strategy. Interviewers use them to test probability fluency and speed, and trader interviews often follow up by asking what you would pay to play or how much you would bet.

What is linearity of expectation and why does it matter?

It is the rule that the expected value of a sum equals the sum of the expected values, even when the parts are dependent. Combined with indicator variables, it turns many hard counting problems, like the coat check in Example 1, into short calculations. It is the single most useful technique for EV interview questions.

How fast do I need to answer expected value questions?

In trader interviews, simple ones such as the sum of two dice or the expected number of runs should take a few seconds. Harder ones like the coupon collector problem are fine to set up out loud over 20 or 30 seconds. Interviewers generally care more about a clear method than instant recall.

Which firms ask expected value questions?

They are common across quantitative trading and research interviews, and candidate reports mention them at market makers such as Optiver, IMC and SIG as well as at hedge funds. Our firm-specific interview guides, such as the IMC Trading interview guide, describe where they appear in each process.

What is the difference between expected value and the Kelly criterion?

Expected value is the average outcome of a single bet. The Kelly criterion answers a different question: what fraction of your wealth to bet repeatedly so that it grows fastest over time. Maximising expected value on every bet can lead to near-certain ruin, as Example 10 shows; Kelly sizing avoids that.

What books cover expected value interview questions?

A Practical Guide to Quantitative Finance Interviews by Xinfeng Zhou (the Green Book) and Heard on the Street by Timothy Crack are the two most widely used. Both include sections on probability and expected value in the style that trading firms ask.