Expected Value Interview Questions

Expected value questions in quant interviews come in five recurring flavors: counting via linearity and indicator variables, waiting times via first-step recursion, continuous expectations over uniforms and normals, keep-or-fold games priced by optimal stopping, and market EV — sizing, spreads and order flow.
The common thread: ask whether the target splits into pieces you can already average — or into an equation from conditioning on the first step.

Last updated 3 July 2026 · sub-areas, difficulty mix and firm attributions are compiled from the 427 expected-value problems in the QuantVault bank; firm tags are candidate-reported, not employer-verified.

Core sub-areas
Linearity & indicators · waiting times & recursion · continuous & order statistics · optimal stopping · betting & market EV
Typical difficulty
Centers on medium — 115 easy, 232 medium, 80 hard across 427 problems
Who leans on it
Every quant loop — top candidate tags: Jane Street, Citadel, Optiver, Two Sigma, SIG
Practice pool
427 problems in the bank · 75 free to open with the full worked solution

Where expected value shows up: at every stage, from online assessments to trader phone screens and research onsites — pricing a game, sizing a bet and quoting a spread are all expectation computations. For specific funnels, see the Jane Street interview questions guide and the Citadel interview questions guide.

Practice expected value questions interactively → Browse the full problem bank All interview topics

The surfaceThe expected value sub-areas quant interviews test

Five families cover essentially every expected-value question in the bank. Each row pairs a sub-area with the recurring question shape and a representative flavor — drawn from real problems, not a leaked wording.

The expected value sub-areas tested in quant interviews, with the recurring question shape and a representative question type for each.
Sub-areaRecurring shapeRepresentative type (flavor, not a real question)
Linearity & indicator variablesWrite the count as a sum of 0/1 indicators; expectations add even under dependenceExpected fixed points in a shuffled deck, adjacent couples at a round table, or runs in a card sequence.
Waiting times & first-step recursionCondition on the first step and let the unknown expectation appear on both sidesExpected flips to see a pattern, rolls until a cumulative target, or steps for a walk to hit a boundary.
Continuous expectation & order statisticsIntegrate against the density — or dodge the integral with symmetry and known order-statistic resultsExpected minimum of two uniforms, absolute difference of uniforms, or the mean of a clipped or mixed normal.
Optimal stopping & game valuePrice the option to continue, then stop exactly when the current holding beats itA dice game with limited rerolls or a keep-or-play draw — the answer is a threshold policy plus its EV.
Betting, sizing & market EVTurn a stated edge into an expectation, then ask what size or spread survives repetitionKelly-style sizing, the break-even spread under adverse selection, or the expected mid move from random flow.

What's confirmed vs. what varies: the sub-areas and difficulty mix come straight from the problem set; firm attributions are candidate-reported through tags — treat them as directional. Trading tracks weight the stopping and market-EV families; research tracks the continuous computations.

The patternsSignature expected value question patterns

Three moves generate most correct answers here. Each worked box is a 60–90 second micro-example — the reasoning template on a generic setup, not any firm's wording.

Linearity of expectation — count with indicators

Takeaway: when a question asks only for an average count, never build the distribution — write the count as a sum of indicators and add their expectations, dependence and all.

Shape. Roll a fair die six times. How many distinct faces do you expect?

1. Decompose. Let \(I_k=1\) if face \(k\) appears at least once. The distinct-face count is \(I_1+\cdots+I_6\), so its expectation is \(6\,\mathbb{P}(I_k=1)\) by linearity and symmetry.

2. Average one piece. Face \(k\) is missed with probability \((5/6)^6\), so \(\mathbb{P}(I_k=1)=1-(5/6)^6\approx 0.665\).

3. Add. \(6\times 0.665\approx 3.99\). The move: the indicators are heavily correlated and it does not matter — linearity never asks for independence.

First-step recursion — solve an equation, not a series

Takeaway: when a process resets after failure, condition on the first step so the unknown expectation appears on both sides — then solve a one-line equation instead of an infinite series.

Shape. How many fair-coin flips until you see two heads in a row?

1. Name the states. Let \(E_0\) be the answer from no progress and \(E_1\) from “just flipped a head.” Then \(E_0 = 1 + \tfrac12 E_1 + \tfrac12 E_0\) and \(E_1 = 1 + \tfrac12 \cdot 0 + \tfrac12 E_0\).

2. Solve. The first equation gives \(E_0 = 2 + E_1\); substituting the second yields \(E_0 = 3 + \tfrac12 E_0\), so \(E_0 = 6\).

3. Pressure-test. The follow-up interviewers love: HT takes only 4 flips, because HT cannot overlap itself while HH can. The move: one line per state, then algebra — the series never gets written down.

Optimal stopping — the threshold is the continuation value

Takeaway: in keep-or-reroll games the threshold is never a guess — it is exactly the expected value of playing on, computed from the last decision backward.

Shape. Roll a fair die; you may keep the result or reroll once. What is the game worth?

1. Price the fallback. A forced reroll is worth its mean, \(3.5\) — your continuation value.

2. Derive the rule. Keep the first roll exactly when it beats \(3.5\) — a 4, 5 or 6; reroll otherwise.

3. Price the game. With probability \(\tfrac12\) you keep an average of \(5\); otherwise you take the reroll's \(3.5\): value \(\tfrac12(5)+\tfrac12(3.5)=4.25\). A second reroll raises the threshold to \(4.25\), giving \(\tfrac13(5.5)+\tfrac23(4.25)=\tfrac{14}{3}\approx 4.67\). The move: each extra option iterates the same backward step.

Free practiceExpected value practice questions by sub-area

The bank holds 427 expected-value problems; 75 are free. Below: a curated 30 spanning every sub-area — 8 easy, 14 medium, 8 hard. The full free pool runs 16 easy, 48 medium and 11 hard; the interactive playlist serves the rest, and the full problem bank goes deeper on every family.

Linearity & indicator variables

Waiting times & first-step recursion

Continuous expectation & order statistics

Optimal stopping & the value of a game

Betting, sizing & market EV

The planHow to prepare for expected value questions

Five techniques to make reflexive, in the order they pay off.

  1. Make linearity your first instinct. Any “expected number of…” question gets the same reflex: one indicator per potential occurrence, average, multiply — if you find yourself building a joint distribution, restart with indicators.
  2. Name states and write one equation per state. Waiting-time and hitting-time problems collapse once you condition on the first step — practice until setting up the \(E = 1 + \sum p_i E_i\) system takes under a minute.
  3. Memorize the small continuous library. The minimum of \(n\) uniforms averages \(\tfrac{1}{n+1}\), the \(k\)-th order statistic \(\tfrac{k}{n+1}\), and mixture moments come from the tower rule — a dozen such results cover most live continuous questions.
  4. Price the option to continue. In keep-or-reroll games, compute the continuation value first and stop exactly when your current holding beats it — interviewers listen for that sentence.
  5. Translate EV into market language. Edge becomes size through Kelly-style arguments, and a counterparty's willingness to trade moves your conditional expectation. Then run the free practice set and take the interactive expected value playlist on a clock.

FAQExpected value interview questions — frequently asked

How important is expected value for quant interviews?

It is the closest thing quant interviews have to a universal topic — most probability questions end by asking for an expectation, and market questions are expectations under a different name. In our bank it is the largest math topic after general probability, at 427 problems.

What expected value topics should I focus on?

Linearity of expectation with indicator variables, first-step recursion for waiting times, and optimal stopping cover most of what gets asked. Round that out with the standard continuous results and the market-facing trio of Kelly sizing, adverse selection and break-even spreads.

How hard are expected value interview questions?

The mix centers on medium: of the 427 problems in this set, 115 are easy, 232 medium and 80 hard. The difficulty is rarely the arithmetic — it is spotting the decomposition that makes the computation short.

Are these real quant interview questions?

They are representative, not verbatim. The problems are curated from our bank to match the expected-value shapes candidates report, rewritten for clarity with worked solutions we author ourselves — we never claim any wording is a leaked question.