Data Science Interview Prep: Q70

First-Move Advantage in a Biased Coin Toss Game. (Category: Probability)

Players A and B take turns flipping a biased coin with the probability of heads being p. The first player to flip a heads wins. If Player A starts, what is Player A’s chance of winning?

Solution:

To begin, let’s map out the possible outcomes when Player A flips first, using a tree diagram.

Here, p represents the probability of flipping a head, and (1 - p) represents the probability of flipping a tail.

Let X be the probability that Player A wins.

Two cases arise:

• Player A flips a head on the first try:

  • Player A wins immediately with probability p.

• Player A flips a tail on the first try:

  • Player B then flips. If Player B also flips a tail, the game resets to the original situation, with Player A’s chance of winning again equal to X.

    This occurs with probability (1−p)·(1−p)·X = (1-p)² · X.

As the two outcomes are mutually exclusive, the probability of Player A winning is given by the sum of their probabilities, from which we solve for X:

Therefore, the probability of Player A winning is 1/(2 - p), where 0 < p < 1.

Accounting for the bias in the coin:

• If the coin is heavily biased toward heads, i.e., p → 1, then given that Player A flips first, X → 1. Player A almost always wins.

• If the coin is heavily biased toward tails, i.e., p → 0, then given that Player A flips first, X → 1/2. In this case, the game may continue for many flips, but Player A’s first-mover advantage disappears, leaving both players equally likely to win.

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Reference:

• Biased coin problem: https://math.stackexchange.com/questions/4236624/general-solution-to-two-players-alternately-flip-a-coin-which-may-be-biased-wh