Expected Value Interview Questions
Expectation is the single most leveraged idea in quant interviews: almost every pricing, sizing, and risk question reduces to 'what's the average outcome?'. This playlist builds the full toolkit -- linearity of expectation with indicator variables (count anything without touching the joint distribut
How to think about expected value questions
Most expectation problems look like they need a hard sum or integral, but the winning move is almost always to avoid one. Break the quantity into pieces whose averages you already know, add them up, and the scary problem dissolves — no joint distribution required.
LINEARITY FIRST
The single most reused fact in the bank: the average of a sum is the sum of the averages, whether or not the parts are independent. Write your target as a sum of simple indicators — “did event k happen?” — and the messy dependence between them stops mattering, because you never multiply the pieces together.
CONDITION TO RECURSE
When the process has memory or a first step that resets it, condition on that first step and let the unknown expectation appear on both sides. One line of the tower rule turns “expected time until…” into an equation you solve for the answer instead of a sum you grind out.
Work this set top to bottom and one reflex takes over: before computing anything, ask — can I write this as a sum of things I already average, or condition on the first step?
Expected Value questions (100)
- Expected Sum of Two Dice Rolls
- Conditional Expectation and Variance of Bivariate Normal
- Expected Stopping Time for Uniform Sum Exceeding One
- Expected Fixed Points of a Random Permutation
- Minimizing Expected Squared Deviation
- Bus Waiting Paradox
- Expected Value of a Single Die Roll
- Second Moment of a Gaussian Random Variable
- Expected Sum of Rolls in the Coupon Collector Problem
- Expected Position of the Ace of Spades in a Shuffled Deck
- Covariance of a Standard Normal and Its Square
- Coupon Collector's Problem
- Fair Price to Enter a Coin Flip Game
- Expected Value of the Maximum of Uniform Random Variables
- Tower Property of Conditional Expectation
- Expected Draws to First Ace
- Expected Product of Two Dice
- Mean and Variance of Nested Uniform Random Variables
- Expected Minimum of n Uniform Random Variables
- Expected Unique Pokemon Cards in Happy Meals
- Expected Value of the Minimum of Two Geometric Random Variables
- Expected Maximum and Minimum of Three Dice Rolls
- Expected Elevator Stops in a Ten-Floor Building
- Mean and Variance of a Mixture of Normals
- First Fill Time Across Multiple Venues
- Expected Rolls Until an Odd Number
- Expected Value of the Maximum of Two Dice
- Value of a Dice Re-Roll Option
- Ten-Sided Die Game: Expected Payoff with Multiple Rolls
- Expected Number of Uniforms to Exceed One
- Single Die vs. Average of 100 Dice: Which Game to Play?
- Expected Flips Until Both Heads and Tails Appear
- Expected Number of Fixed Points in a Random Permutation
- Half-Kelly Betting With Equal Wins and Losses
- Expected Tosses for Three Consecutive Heads
- Expected Rolls with a Fair Die
- Covariance of a Coordinate with Squared Distance
- Expected Waiting Time for Three Consecutive Heads
- Expected Crossing Handshakes in a Circle
- Optimal Stopping: Three-Roll Dice Game
- Random Walk on a Table -- Expected Steps to Fall Off
- Expected Number of Local Maxima in a Circle
- Wallet Bid: Conditional Expected Total
- Optimal Stopping for Uniform Draws
- Expected Adjacent Couples at a Round Table
- Expected Sum Until Rolling a 1
- Optimal Reroll Strategy for a Die Game
- Expected Number of Loops from Tying Rope Ends
- Expected Flips for Consecutive Heads
- Choosing Among Mutually Exclusive Coin-Flip Bets
- Expected Number of Surviving Sharks
- Expected Rolls to Get a 4 Then a 5 in Order
- Expected Total Payment in a Dice Rolling Game
- Optimal Stopping in a Dice Sum Game
- Expected Flips to See Tails-Then-Heads
- Expected Stopping Time and Sum When Rolling Until a Six
- Optimal Stopping in a Bill Drawing Game
- Conditional Expectation of a Linear Combination Given a Sum
- Time Until the Second Ace
- Expected Flips to Get k Heads
- Expected Absorption Time for an Asymmetric Random Walk
- Expected Sum of Dice Rolls Until a Target Face
- Expected Distance to Escape a Room
- Random Walk on a Cube to the Opposite Vertex
- The St. Petersburg Paradox
- Conditional Expectation of a Partial Sum Given the Total
- Expected Time for All Ants to Fall Off a Stick
- Expected Rolls to Reach Position 10 in a Symmetric Random Walk
- Expected Insertions to Pay a Parking Meter
- Airplane Boarding Problem
- Urn of Red and Blue Orbs
- Doubling Coin Game: Expected Balance
- Waiting Time to Patterns HTH and HHH
- Bayesian Posterior and Break-Even Price Under Information Asymmetry
- Optimal Stopping for Maximum of Uniforms
- Pattern Waiting Time
- Order Statistics: Expectations and Densities for Uniform Samples
- Expected Payoff of the Red-Black Card Game
- Random Walk Exit Time and Cover Time on a Cycle
- Expected Maximum of Standard Normal Random Variables
- Linear MMSE Estimator Derivation
- Expected Value of an Optimal Stopping Card Game
- Sharpe Ratio Selection Bias
- Adverse Selection and the Break-Even Spread
- The Secretary Problem
- Expected Maximum and Second Maximum of Standard Normals
- Limit Order Fill Probability
- Optional Stopping and the Doubling Strategy
- Expected Maximum of Two Standard Normals with Correlation
- St. Petersburg Paradox and Log Utility
- Expected Flips to Reach a Head-Tail Boundary
- Sum-of-Three-Cards Market Making
- Expected Cards Between First Two and First Ace
- Expected Hitting Time to a Linear Boundary
- Expected Rolls to See Each Die Face Twice
- Expected Waiting Time With Random Bus Delays
- Expected First Passage Time for a Biased Random Walk
- Expected Cards Until King and Queen-or-Ace
- Expected Value of the Running Total When It First Exceeds 100
- Expected Hitting Time with Uniform Starting Point
Expected Value interview questions FAQ
What kind of expected value questions show up in quant interviews?
This page collects 100 expected value problems that recur in quant trading and research interviews, each with a full worked solution and the intuition behind it. They range from quick warmups to the harder variants firms use to separate candidates.
How hard are expected value interview questions?
The set spans 22 easy, 57 medium and 21 hard problems. Most sit at medium difficulty — a few minutes of clean reasoning — with a harder tail that rewards knowing the canonical approach rather than grinding.
How should I practice expected value for quant interviews?
Work through them by difficulty, starting just below your level, and write the solution out before checking. 29 are free to open with the full worked solution, so you can judge the quality first. Focus on the recurring patterns rather than memorizing answers — the same handful of ideas generate most variants.
Are these real quant interview questions?
They are a curated set drawn from our problem bank — the kind of expected value question that actually appears in quant interviews, rewritten for clarity with solutions we author ourselves. We don't claim any single wording is verbatim, and every problem carries a full solution.