Quant interview preparation

Prop market making and quantitative research, weighted the way the interviews actually are: probability and expected value, statistics and machine learning, market making logic, programming and options. Every question is either traced to a named firm from a public candidate report, or tagged at desk level when we could not trace it, and every probability answer shows the reasoning path rather than just the number.

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Question bank

100 questions, mapped to the firms that asked them

Questions
100
Traced to a firm
53
Firms
15
Updated
September 2026
Asked at
All firmsOld Mission Capital12Tower Research Capital10Jump Trading7Akuna Capital5Citadel4D.E. Shaw3Jane Street3AQR Capital Management2DRW2Millennium Management2Schonfeld2Squarepoint Capital2Susquehanna International Group2Belvedere Trading1Optiver1
Topic
All topicsProbability10Coins, cards and games6Expected value8Statistics11Market making15Estimation and mental maths4Stochastic processes4Regression5Machine learning6Time series6Programming10Options and derivatives8Fit and motivation7
Level
AnyCoreIntermediateHard
Type
AnyBrainteaserTechnicalCaseMarket viewFit
Showing 1–3 of 3 · filtered from 100Clear filters
  1. 002I hand you a coin that comes up heads one third of the time. How do you generate a fair coin flip from it?Coins, cards and gamesIntermediatephone / first roundD.E. ShawResearch · New York · 2026Tower Research CapitalProp Trading · New York · 2019

    Say this

    Flip it twice. Call heads-then-tails a fair heads, tails-then-heads a fair tails, and if you get HH or TT throw the pair away and start again. HT and TH both have probability p times one minus p, so they are equally likely whatever p is.

    Then walk it

    1. With p equal to a third: HT is 1/3 times 2/3 which is 2/9, and TH is 2/3 times 1/3 which is also 2/9. Identical, so conditioning on one of the two having happened gives you exactly a half.
    2. That is von Neumann's trick. The point is that it needs no knowledge of p at all, which is what makes it useful. You never have to estimate the bias.
    3. It does need two things: the flips are independent, and p is strictly between zero and one. A coin that is genuinely two-headed breaks it, and so does a coin whose bias drifts flip to flip.
    4. Probability a given pair is useful is 2 times 2/9, which is 4/9. So you discard more than half your pairs at p equal to a third.
    5. The honest limitation: it is unbiased but wasteful. If the bias drifts slowly, you can protect yourself by pairing adjacent flips rather than flips far apart, so the drift cancels locally.

    Where candidates lose it

    Trying to estimate p first and then correct for it. That introduces estimation error and gives you an approximately fair coin, not a fair one. The elegant answer is exactly fair with zero knowledge of p, and the interviewer is looking for that symmetry argument.

    Expect next

    • What is the expected number of flips of the biased coin per fair flip?
    • How would you get a uniform random number on one to three from the same coin?
    • Can you do better than throwing HH and TT away entirely?

    Reported by candidates at D.E. Shaw (Research, New York, 2026); Tower Research Capital (Prop Trading, New York, 2019). Source: Wall Street Oasis.

  2. 006You want to draw a black card followed by a red card. One deck is a full 52-card deck, another has had some cards removed. Which deck do you choose and why?Coins, cards and gamesIntermediatetechnicalOld Mission CapitalQuantitative Research · New York · 2014

    Say this

    Write the probability down before you pick. For a deck with b blacks and r reds, drawing black then red is b/(b+r) times r/(b+r-1). Then just compare the candidate decks on that expression, and you will find you want the deck that is as balanced as possible and as small as possible.

    Then walk it

    1. Full deck: 26/52 times 25/51, which is 0.5 times 0.490, about 24.5 percent.
    2. Now try a tiny balanced deck, one black and one red. That is 1/2 times 1/1, which is 50 percent. Far better.
    3. So the direction is clear. Removing cards helps if it keeps the deck balanced, because the second draw's conditional probability improves once the black card you removed is a bigger fraction of a smaller deck.
    4. Unbalancing hurts. A deck of 26 blacks and 1 red gives 26/27 times 1/26, which is 1/27, about 3.7 percent. Almost all your probability mass dies on the second draw.
    5. So: balanced beats unbalanced, small beats large, and the extreme is one black plus one red at fifty percent. Say the formula first, then test the corners. That is faster and less error-prone than trying to reason about it verbally.

    Where candidates lose it

    Reasoning in words about whether removing cards helps or hurts, and getting tangled. Write b/(b+r) times r/(b+r-1) immediately, then plug in three corner cases. Also do not forget the minus one in the denominator, because sampling without replacement is the entire content of the question.

    Expect next

    • What deck maximises the probability of black then red then black?
    • What if you wanted two cards of the same colour instead?
    • Now make me a market on the probability for the standard deck.

    Reported by candidates at Old Mission Capital (Quantitative Research, New York, 2014). Source: Wall Street Oasis.

  3. 007What is the probability of being dealt four of a kind in a five-card poker hand?Coins, cards and gamesIntermediatetechnicalOld Mission CapitalQuantitative Research · New York · 2014

    Say this

    624 hands out of 2,598,960, which is about 0.024 percent, or one in roughly 4,165. Thirteen choices of rank for the quad, times 48 remaining cards for the fifth card.

    Then walk it

    1. Denominator: 52 choose 5 is 2,598,960. Worth memorising, it comes up constantly.
    2. Numerator: pick the rank of the four of a kind, 13 ways. All four suits are forced. Then the fifth card is any of the 48 cards left, so 13 times 48 is 624.
    3. 624 over 2,598,960 simplifies to 1 over 4,165. Call it one in four thousand.
    4. The counting discipline that matters: the kicker is 48, not 12. If you write 13 times 12 you are counting ranks not cards, and you would be off by a factor of four.
    5. Quick cross-check against a fact you might already know: a full house is 3,744 hands and a straight flush is 40. Four of a kind sitting between them at 624 is consistent with the standard hand ranking, which is ordered by exactly this rarity.

    Where candidates lose it

    Double counting, or using 12 instead of 48 for the fifth card. The other classic error is dividing by 5 factorial somewhere by accident. Use combinations consistently in both numerator and denominator, and state the denominator before you start so the interviewer can follow.

    Expect next

    • Now do a full house.
    • What is the probability of a flush, excluding straight flushes?
    • How would that change in a seven-card game like Texas hold'em?

    Reported by candidates at Old Mission Capital (Quantitative Research, New York, 2014). Source: Wall Street Oasis.

Firm tags come from public, anonymous candidate reports on Wall Street Oasis: strong signal, not sworn testimony. Firms are named as the places a question was reported, not as partners of Fin Maverick. Answers are written for this page to show how to think out loud; they are not scripts to recite.