Abby's Fair Coin Puzzle

This quant interview puzzle is about random events occurring over repeated independent trials with unequal success probabilities. It lives at the intersection of probability theory and game fairness, where one player has a biased advantage but the turn order counterbalances it. In quant prep terms, it forces you to model a simple yet nontrivial stochastic game and express its long-run outcomes cleanly.

It trains your ability to formalize an evolving random process, recognize when the state of the system effectively "resets," and translate that into compact probability expressions. You practice working with absorbing Markov-style structures, infinite-horizon reasoning, geometric patterns, and conditioning on partial histories in a clean algebraic way.

This matters for quant interviews because many derivatives pricing, risk, and trading problems reduce to similar recursive probability structures. Interviewers use it to see whether you can turn an informal game description into a rigorous probabilistic model, reason about fairness, and handle infinite processes under time pressure.

What it tests

This problem class is governed by the structure of absorbing Markov processes with alternating turns and independent success probabilities. The key is that the process resets after each pair of failures, so the probability of eventual success for the first player can be written recursively or as an infinite geometric series. The symmetry or fairness condition (equal chances for both players) often leads to an equation where the unknown success probability is balanced against the advantage of going first. The underlying reason this works is that each round is memoryless: after both fail, the game is statistically identical to the starting position, except the roles may be swapped. This recursive self-similarity is what allows us to sum over all possible rounds or use a fixed-point equation.