Probability Interview Questions
Probability is the single most-tested topic in quant interviews, and the best answers almost never come from brute-force computation. This playlist trains the reflexes interviewers actually probe: Bayesian updating and the base-rate trap, conditional probability and the law of total probability, sym
How to think about probability questions
Probability puzzles feel like a grab-bag of tricks until you notice they're all bookkeeping over the same sample space. Define the outcomes cleanly, decide what's independent and what isn't, and most “paradoxes” turn into one careful count.
CONDITION ON WHAT YOU KNOW
The engine behind nearly every hard problem is updating: a new piece of information reshapes the odds. Bayes' rule is just the careful version of “reweight the worlds consistent with what I observed” — and the classic traps (the boy-girl problem, the two-envelope swap) are all failures to pin down exactly which worlds the evidence rules out.
SPLIT, DON'T MULTIPLY BLINDLY
Independence is a privilege, not a default. When events truly don't interact you may multiply; when they do, partition the event into disjoint cases and add. Getting this boundary right — product when independent, sum when mutually exclusive — is what separates a clean two-line answer from a wrong one.
After a handful of these you stop reaching for formulas and start drawing the sample space — once the worlds are laid out cleanly, the arithmetic is the easy part.
Probability questions (94)
- Positive Test With Low Prevalence
- Strictly Increasing Dice Rolls
- Normal Tail Probability via Standardization
- Bayesian Posterior Calculation
- Memorylessness of the Geometric Distribution
- Maximum of Multiple Dice Rolls
- Posterior Fair Value from Noisy Signal
- The Gaussian Integral
- Uncorrelated but Not Independent
- Bayes' Theorem: Condition Prevalence Among Smokers
- Minimum of Three Uniform Random Variables in an Interval
- Hypergeometric Probability: Energy Names in a Hedge Basket
- Bertrand's Box Paradox
- Probability That the Median Exceeds a Threshold
- Probability Two Friends Sit Adjacent in a Row
- The Other Ball in the Box
- Inverse Transform Sampling
- Ant Collision on an Equilateral Triangle
- Bayesian Disease Testing with Two Independent Tests
- Probability of an ETF Up Day
- Ten of Spades Before All Face Cards
- Two Children and a Boy
- Monty Hall With 100 Doors
- Probability of Rolling a Sum of 4 Before a Sum of 2
- Two Trips with At Least One International in December
- Conditional Probability of Two Heads Given at Least One
- Witness Reliability and Bayes' Theorem
- Poisson Limit of Lottery Winners
- Gambler's Ruin on a Fair Coin
- Probability a Random Triangle Contains the Center of a Circle
- CLT Approximation for Coin Flips
- Probability That You Have the Bad Coin
- Breaking a Stick into a Triangle
- Probability of Negative Return Given a Sharpe Ratio of 1
- Bayesian Coin Identification
- Random Chord Length on a Unit Circle
- Markov Bound on Total Cheese Weight
- Race Between Two Patterns on a Fair Die
- Probability of a Straight in a 5-Card Poker Hand
- Even vs Odd Heads and Coin Fairness
- Monte Carlo Estimation of Pi and e
- Monty Hall with an Imperfect Host
- Pattern Race: HTH vs HHT
- Probability of Meeting at Office Hours
- Bayesian Coin: Fair vs. Double-Headed
- Winning From Deuce With a Weaker Serve
- Birthday Problem
- Probability of Truth Given Corroboration
- HHT Before THH in Coin Flips
- Probability No One Gets Their Own Hat
- Monty Hall With Envelopes
- Competing Poisson Processes
- Probability of a Void Suit in a Card Hand
- Generate Uniform 1-7 from Uniform 1-5
- Biased-Host Monty Hall
- Biased Random Walk Reaching Position 1
- Probability the Bowl is Empty When the Last Orange is Drawn
- Probability That One Exponential Beats Another
- At Least One Low Dice Sum in Five Rolls
- Probability of No Consecutive Heads in N Coin Tosses
- Truth Through a Chain of Liars
- Aces Before Kings: Probability via Exchangeability
- Non-Transitive Dice
- Sum of 8 Given At Least One Die Shows 3
- Optimal Switching in the Two Envelope Problem
- Race to Three Heads
- Even Before Odd
- Poisson Thinning for Toxic Quote Hits
- Bayesian Updating with Sequential Trade Direction
- Probability That One Interval Contains All Others
- Parity of Heads with One Fair and Many Biased Coins
- Posterior Update from a Trade Followed by Silence
- Order Probabilities and Expectations for I.I.D. Variables
- Convergence in Probability vs. Almost Sure Convergence
- Bayesian Update with Three Binary Signals
- Acute Triangle from Random Points on a Circle
- One-Sided Chebyshev Bound vs. Markov's Inequality
- Simulating a 2:3 Odds Ratio with Fair Coins
- Random Walk Absorption on a Graph
- Marginal Distribution of a Beta-Binomial Mixture
- Bookmaker Mispricing and Static Arbitrage
- Pattern Race: HHT vs HTH
- Meeting Window with Uniform Arrivals
- Weekend Rain Bounds via Frechet-Hoeffding
- 100 Prisoners and Drawers
- Limiting Probability of the Last Die Face
- Last Guest Gets Their Own Sandwich
- Probability a Random Quadratic Has Two Real Roots
- Probability a Broken Stick Triangle Is Acute
- Random Walk to the Boundary of a Square
- Simpson's Paradox with Poisson Scoring
- Five Points on a Hemisphere
- Must the Success Rate Cross 80%?
- Probability the Last Die Roll Is Two When Summing Past 100
Probability interview questions FAQ
What kind of probability questions show up in quant interviews?
This page collects 94 probability 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 probability interview questions?
The set spans 31 easy, 46 medium and 17 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 probability for quant interviews?
Work through them by difficulty, starting just below your level, and write the solution out before checking. 36 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 probability 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.