Two Sigma Interview Questions
The real Two Sigma quant-interview problems candidates report — expected value, probability and coding — each with a full worked solution.
Inside the Two Sigma interview
Two Sigma is a systematic, research-first hedge fund that runs its strategies on data and statistical models, hiring Quant Researchers and engineers. Its interviews lean hard on probability and expectation, with a strong statistics / regression flavor and clean algorithmic coding alongside.
What they test
The single largest block is expectation (38 problems) — optimal-stopping games, hitting times, conditional survival, and dice/coin reasoning — backed by a deep probability core (26) of distributions, Bayesian updates, and combinatorial counting. Around it sit a steady stream of algorithmic coding (10), a game-theory strand (9) of two-player and pricing games, and a thinner layer of optimization (4) and random-variable (3) questions where the research-first culture shows.
The recurring shapes
Most expectation problems reduce to conditioning on the first step and solving a fixed point — E = 1 + (chance you must restart) × E. The probability questions reward setting up the right sample space and applying Bayes' rule cleanly, and the game-theory items want you to find the equilibrium or fair value by backward induction rather than simulation.
How to approach
State your assumptions, then lean on symmetry and linearity of expectation before grinding algebra, and sanity-check against small cases and extreme limits. For probability, write the sample space and condition explicitly; for the coding problems, narrate the one-pass or streaming invariant before you write it; and treat the game-theory questions as fixed-point / backward-induction arguments.
The mix is mostly medium (56) with a heavy hard tail of 31 — the tougher game-theory, optimization, and stochastic problems — plus 13 easy warm-ups, so build fluency across expectation, probability, and stats rather than betting on one.
Two Sigma expected value questions (38)
- Expected Arc Length Between Random Points on a Sphere
- Expected Balls Left in the Banach Match Problem
- Expected Number of Uniforms to Exceed a Threshold
- Order Statistics: Expectations and Densities for Uniform Samples
- Three Frogs on an Equilateral Triangle
- Unbiased Coin from a Biased Coin
- Linear MMSE Estimator Derivation
- The Secretary Problem
- Pricing a Game with Infinite Expected Payoff
- Expected Absorption Time for an Asymmetric Random Walk
- Expected Number of Gold Heads Given Total Payoff
- Expected Number of Mutual Nearest Neighbors on the Unit Interval
- Expected Rolls Until Divisible by Six
- Expected Tosses for Consecutive Heads
- Nearest Neighbor Distance in the Unit Ball
- Nested Uniform Random Variables
- Umbrella Markov Chain: Expected Rainy Commutes Until Getting Wet
- Expected Number and Variance of Records in a Random Permutation
- Expected Seated Count with No-Adjacent Rule
- Expected Payoff of Max Difference of Dice
- Optimal Reroll Strategy for a Die Game
- Expected Adjacent Cat-Dog Pairs
- Expected Time to First Non-Record in Uniform Draws
- Expected Value and Variance of a Rademacher Determinant
- Linear MMSE Estimator for a Conditional Exponential
- Mean and Variance of Nested Uniform Random Variables
- Pattern Waiting Time
- Comparing Three Dice Expectations
- Conditional Expectation of Brownian Motion Given Terminal Value
- Expectation of a Ratio of IID Variables
- Maximum Ones in a Nilpotent Binary Matrix
- Bayesian Coin Inference and Kelly Betting with Beta Prior
- Mean and Variance of a Mixture of Normals
- Expected Length of a Zigzag Prefix
- Expected Longest Piece From Two Random Cuts
- Time Until the Second Ace
- Variance of the Determinant of a Random Sign Matrix
- Expected Tosses for Three Consecutive Heads
Two Sigma probability questions (26)
- Breaking a Stick into a Triangle
- Fair Outcome from an Extremely Biased Coin
- Five Points on a Hemisphere
- Jointly Normal and Uncorrelated Implies Independent
- Optimal Sequential Rank Guessing
- Correlation of Nested Binomial Counts Under a Random Parameter
- Leading Digit of Powers of Two
- Meeting Probability for N Uniform Arrivals
- Mixture of Normals and Random Sign Products
- Biased-Host Monty Hall
- Meeting Window with Uniform Arrivals
- Probability a Random Triangle Contains the Center of a Circle
- Bayesian Coin Selection and Sequential Betting
- Distinct Outcomes in Dice Rolls
- Conditional Probability of a Normal Linear Combination
- Cumulative Dice Sum Threshold Distribution
- Coin Flip Probabilities with Four Fair Coins
- Ascending Order Probability for Three Cards
- Random Walk Absorption on a Graph
- Three Six-Sided Dice vs One Twenty-Sided Die
- Chebyshev Bound for Bivariate Normal Difference
- Birthday Problem Generalization via Poisson Approximation
- Bounds on P(B)
- Parity of Heads with One Fair and Many Biased Coins
- Tournament Survival Probability with Deterministic Outcomes
- Posterior for a Double-Headed Coin
Two Sigma coding questions (10)
- Knuth-Morris-Pratt Pattern Search
- Kruskal's MST with Union-Find
- Reservoir Sampling from an Unknown-Length Stream
- Wildcard Pattern Matching
- Median from a Data Stream
- Random Number Generator Without Replacement
- Counting Friend Circles
- Dijkstra's Algorithm with Binary Heap
- Hash Table Implementation and Collision Resolution
- Merging Overlapping Intervals
Two Sigma game theory questions (9)
- Optimal Stopping in a Blue-Red Ball Draw
- Optimal Stopping in the Red-Black Card Game
- Urn Game Optimal Stopping Strategy
- Pirate Hat Puzzle: Cooperative Parity Strategy
- Nim-Style Counting Game to 100
- Optimal Stopping with Absorbing Perfect Squares
- Paying for a Better Die
- Uber vs Taxi Driver Strategy Comparison
- Choosing a Lottery Ticket by Expected Value
Two Sigma optimization questions (4)
Two Sigma random variables questions (3)
Two Sigma combinatorics questions (3)
Two Sigma regression questions (2)
Two Sigma statistics questions (1)
Two Sigma linear algebra questions (1)
Two Sigma market microstructure questions (1)
Two Sigma finance questions (1)
Two Sigma brain teasers questions (1)
Two Sigma interview FAQ
What kind of questions does Two Sigma ask in quant interviews?
Candidates most often report expected value, probability and coding questions. This page collects 100 of them, 100 stamped with the month they were last reported — each with a full worked solution.
How hard are Two Sigma interview questions?
The set spans 13 easy, 56 medium and 31 hard problems. Most sit at medium difficulty — solvable in a few minutes with clean reasoning — with a harder tail that rewards knowing the canonical tricks.
How do I prepare for the Two Sigma quant interview?
Work through this set by topic (use the sidebar), starting from your weakest area. 26 problems are free to open with their full solution, so you can judge the quality before anything else. Then walk the full Two Sigma interview guide for the round-by-round funnel and the online assessment.
Are these the actual Two Sigma interview questions?
They are built from candidate-reported Two Sigma questions. We rewrite each prompt for clarity and author the worked solutions ourselves — we don't claim the wording is verbatim, and we never invent questions or recycle generic lists. 100 of 100 carry the month they were last reported.