Quant interview preparation

Questions

100

Traced to a firm

53

Firms

15

Updated

September 2026

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All firmsOld Mission Capital12Tower Research Capital10Jump Trading7Akuna Capital5Citadel4DED.E. Shaw3Jane Street3ACAQR Capital Management2DRW2Millennium Management2Schonfeld2SCSquarepoint Capital2Susquehanna International Group2Belvedere Trading1Optiver1

Topic

All topicsProbability10Coins, cards and games6Expected value8Statistics11Market making15Estimation and mental maths4Stochastic processes4Regression5Machine learning6Time series6Programming10Options and derivatives8Fit and motivation7

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1. 025What is the expected number of fair coin flips to see two heads in a row, and how does it compare to heads followed by tails?Stochastic processesIntermediatetechnicalQuant tradingQuant research

   Say this

   Six flips for HH and four for HT. They differ because HH can destroy its own progress: a tail after a single head sends you back to nothing, while for HT a head after a head keeps you one step from done.

   Then walk it

   1. Set up states for HH. Let A be the expected flips from scratch and B from having one head. A equals 1 plus half A plus half B. B equals 1 plus half times 0 plus half A.
   2. Substitute: B equals 1 plus A/2, so A equals 1 plus A/2 plus (1 plus A/2)/2, which gives A equals 1.5 plus 0.75A, so 0.25A equals 1.5 and A equals 6.
   3. Now HT. Let A be from scratch, B from having a head. A equals 1 plus half A plus half B. But B equals 1 plus half times 0 plus half B, because another head leaves you still in state B rather than resetting. So B equals 2.
   4. Then A equals 1 plus A/2 plus 1, so A/2 equals 2 and A equals 4.
   5. The lesson worth saying out loud: patterns with self-overlap take longer. Both patterns have probability 1/4 per pair of positions, yet the waiting times differ, and that is purely about overlap structure. It generalises: the expected wait for a pattern equals the sum of 2 to the power of the length of each of its self-overlapping prefixes. HH gives 4 plus 2 equals 6, HT gives 4 plus 0 equals 4.

   Where candidates lose it

   Assuming both answers are 4 because each two-flip pattern has probability a quarter. That is the intuition the question is designed to break. Set up the state equations explicitly and pay attention to where a failed attempt lands you, because that is the only difference between the two problems.

   Expect next

   • Now do HHH.
   • In a race between HH and HT, which appears first and with what probability?
   • Derive it with the martingale approach instead.
2. 028An ant walks randomly along the edges of a cube starting at one corner. What is the expected number of steps to reach the opposite corner?Stochastic processesHardsuperdayQuant tradingQuant research

   Say this

   Ten steps. Collapse the eight vertices into four states by distance from the start, then solve three linear equations. The symmetry reduction is the whole trick.

   Then walk it

   1. By symmetry, all that matters is your graph distance from the target: state 3 is the start, then 2, then 1, then 0 which is the target. Each vertex has three neighbours.
   2. From state 3 all three neighbours are at distance 2, so E3 equals 1 plus E2.
   3. From state 2, one neighbour is at distance 3 and two are at distance 1. So E2 equals 1 plus (1/3)E3 plus (2/3)E1.
   4. From state 1, one neighbour is the target and two are at distance 2. So E1 equals 1 plus (2/3)E2.
   5. Solve: substitute E3 equals 1 plus E2 into the second equation to get E2 equals 1 plus (1 plus E2)/3 plus (2/3)(1 plus (2/3)E2). That yields E2 equal to 9, so E1 equals 7 and E3 equals 10. Sanity check with the general theorem: for a random walk on a regular graph the expected return time to a vertex is the number of vertices, 8, which is the right order of magnitude for a 10-step commute across the diagonal.

   Where candidates lose it

   Trying to track all eight vertices individually and drowning in eight equations. Say the word symmetry, lump the states by distance, and you have three unknowns. The other error is miscounting neighbours in state 2, where it is one back and two forward, not two back and one forward.

   Expect next

   • What is the expected time to return to the starting corner?
   • Do it for a tetrahedron.
   • What if the walk is on a hypercube in n dimensions?

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.