Sequential Coin Toss Win Chance

This interview question considers a two-player coin-tossing game where the winner is determined by a specific short pattern appearing in the running sequence of flips. The players alternate turns and the same fair coin is used throughout, so neither has an obvious mechanical advantage, making the asymmetry come purely from turn order and the stopping rule. Candidates must think about how the game can end, how many flips it might take, and how the current partial sequence of heads and tails affects who is currently favored to win. This kind of sequential pattern-stopping game is common in quant interviews for trading and research roles, where reasoning about path-dependent events and turn-based advantages is important.

The solution leans on conditional probability, recursive equations, and a careful definition of game states. A strong answer identifies a minimal set of states that capture all relevant history for the Markov property, then writes down relationships between the winning probabilities from each state. The interviewer watches for the ability to structure the problem cleanly, avoid double counting of paths, and manipulate the resulting algebra without getting lost in enumerating long sequences of flips.

What it tests

Problems of this class are governed by recursive conditioning and the principle of state equivalence: after each event, the game's future can often be described by the same probability as the original, but possibly with swapped roles or altered starting conditions. The key is to recognize that the process can be broken down into a set of mutually exclusive, exhaustive cases (like first flip H or T), and that after each case, the game's structure either resets or transitions to a similar subgame. This recursive structure allows us to set up equations relating the unknown probabilities to themselves, which can then be solved algebraically. The reason this works is that the Markov property holds: the future evolution of the game depends only on the current state, not on the path taken to reach it. By exploiting this, we can reduce complex-seeming sequential games to solvable systems of equations.