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 1–2 of 2 · 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. 003Using that procedure with p equal to one third, what is the expected number of biased flips you need to produce one fair flip?Expected valueIntermediatetechnicalD.E. ShawResearch · New York · 2026Tower Research CapitalProp Trading · New York · 2019

    Say this

    Four and a half. Each pair succeeds with probability 2p(1-p), which is 4/9 here, so the number of pairs is geometric with mean 9/4, and each pair costs two flips. Two times 9/4 is 4.5 flips.

    Then walk it

    1. A geometric with success probability q has mean 1/q. Here q is 4/9, so you expect 2.25 pairs before one is usable.
    2. Two flips per pair gives 4.5 flips per fair bit. Say the arithmetic out loud so the interviewer sees the two-step structure: geometric on pairs, then a constant multiplier.
    3. Sanity check the extremes. At p equal to a half, q is 1/2 and the cost is 4 flips per fair bit, which is the cheapest this method ever gets. As p goes to zero the cost blows up like 1/p, which matches the intuition that a near-deterministic coin carries almost no information.
    4. Compare that to the theoretical floor. A p equal to 1/3 coin carries about 0.918 bits of entropy per flip, so in principle you need only about 1.09 flips per fair bit. Von Neumann at 4.5 is four times worse than optimal.
    5. The gap is the interesting part, and it is where the follow-up goes: you are throwing away the information in the discarded HH and TT pairs, and better extractors recycle it.

    Where candidates lose it

    Forgetting to double. Candidates compute 9/4 as the number of trials and stop, when a trial is a pair of flips. Also worth stating the entropy bound unprompted, because the interviewer is almost certainly going to ask whether you can do better, and knowing the floor is how you answer that credibly.

    Expect next

    • What is the information-theoretic minimum number of flips?
    • Describe a scheme that gets closer to that bound.
    • What is the variance of the number of flips, not just the mean?

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