Skip to main content

2016 AMC 12B Problem 19

Problem 19 of 25HarderProbability & Statistics

Tom, Dick, and Harry are playing a game. Starting at the same time, each of them flips a fair coin repeatedly until he gets his first head, at which point he stops. What is the probability that all three flip their coins the same number of times?

Answer choices

Show solution

Solution

A player’s first head comes on flip nn with probability (12)n.\left(\tfrac12\right)^n. All three stopping on the same flip nn has probability ((12)n)3=(18)n.\left(\left(\tfrac12\right)^n\right)^3=\left(\tfrac18\right)^n. Summing over n≥1,n\ge1, ∑n=1∞(18)n=181−18=17.\displaystyle\sum_{n=1}^\infty\left(\tfrac18\right)^n =\frac{\frac{1}{8}}{1-\frac{1}{8}}=\frac17. Thus, the correct answer is B.
AoPS wiki

Tagged: geometric distribution · independent events

More practice