Coin problems are trading-interview bread and butter: minimal setup, instant probabilistic depth, endless variants. Nearly all of them fall into five families.
The five families
- Waiting times. Expected flips to see HH = 6, but HT = 4. The asymmetry (a failed HH attempt wastes progress, a failed HT attempt doesn't) is the single most-asked coin fact. General tool: the ABRACADABRA martingale or pattern Markov chains.
- Pattern races. Which appears first, HHT or THH? Penney's game: THH beats HHT 3:1, and for any pattern the opponent can pick a better one — the game is non-transitive, a fact interviewers love to spring. Worked here.
- Biased coins. Von Neumann's trick extracts a fair coin from an unfair one: flip twice, HT means heads, TH means tails, otherwise repeat. Worked here. Reverse direction: detecting bias from data is a statistics question — how many flips to distinguish 0.5 from 0.51? (Order of $1/0.01^2 = 10{,}000$.)
- Counting and parity. Probability of exactly k heads, more heads than tails, even number of heads (a slick generating-function one-liner: $\frac{1 + (1-2p)^n}{2}$).
- Game/strategy wrappers. Stop-when-ahead games, betting on streaks, "flip until you choose to stop, paid the fraction of heads" — optimal-stopping questions in coin costume.
Worked example
Why 6 flips for HH but 4 for HT? Let $E$ be expected flips for HT: after a head, you wait only for a tail (expected 2 more); getting that first head takes expected 2 — total 4. For HH, a tail during the attempt resets you completely: $E = 2 + \frac{1}{2}\cdot 0 + \frac{1}{2}(E)$... solving the two-state recursion gives 6. The structural reason: HT's progress is never destroyed, HH's is. Explaining WHY, not just computing, is what earns the follow-up-free pass.
Practice with solutions
- HHT before THH — Penney's game machinery.
- Fair outcome from a biased coin — Von Neumann's trick.
- Eight coins, two weighings — the information-theoretic cousin.
- The probability bank — 113 free coin/dice/probability problems with solutions.
More topic guides
- Bayes' Theorem in Quant Interviews
- Dice Questions in Quant Interviews
- Gambler's Ruin in Quant Interviews
- The Kelly Criterion in Quant Interviews
- Markov Chains in Quant Interviews
- Martingales in Quant Interviews
- The Monty Hall Problem — and the Variants Interviews Actually Ask
- Optimal Stopping in Quant Interviews
- Random Walks in Quant Interviews
- All guides & explainers
Frequently asked questions
Why does HH take 6 expected flips but HT only 4?
A tail during an HH attempt destroys all progress, while an HT attempt keeps its progress (once you have the H, any flip either finishes or leaves you still holding an H). Non-overlapping progress makes HT cheaper — the general rule is quantified by the ABRACADABRA martingale.
What is Penney's game?
A pattern race where players choose 3-flip patterns and watch the sequence. It is non-transitive: for every pattern there is another that beats it (e.g. THH beats HHT 3:1), so the second player always has an advantage.
How do you get a fair coin flip from a biased coin?
Von Neumann's trick: flip twice; HT = heads, TH = tails, HH/TT = repeat. The two accepted outcomes have equal probability p(1−p) regardless of the bias.
How many flips to detect a small bias?
Distinguishing p = 0.5 from p = 0.5 + ε needs on the order of 1/ε² flips — about 10,000 for a 1% bias. The √n statistics of the binomial set the scale.
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