StATS: Geometric distribution (May 16, 2005)

Someone asked me about a game where A, B, and C toss a coin in order until someone gets a heads on their coin flip. What are the probabilities that A will win? B will win? C will win?

This is a simple application of the geometric distribution with P=1/2. A good reference for this distribution is

You can also take advantage of the symmetry of the situation. We don't know what the probability of winning for A, B, or C is, so set them to unknown values, X, Y, and Z. We can infer that if A's first flip of the coin is a tail, then B's probability of winning is now the same as what A's was before the coin flip. So that implies that Y=0.5X. By a similar argument, Z=0.5Y=0.25X. Since X, Y, and Z have to add up to 1, that implies that X +0.5X+0.25X=1. Solve for X to get your answer.

This page was written by Steve Simon while working at Children's Mercy Hospital. Although I do not hold the copyright for this material, I am reproducing it here as a service, as it is no longer available on the Children's Mercy Hospital website. Need more information? I have a page with general help resources. You can also browse for pages similar to this one at Category: Probability concepts.