2016 AMC 12B Problem 19

Problem 19 of 25HarderCounting & Probability

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?

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Solution

A player’s first head comes on flip

n

with probability

\left(\tfrac12\right)^n.

All three stopping on the same flip

n

has probability

\left(\left(\tfrac12\right)^n\right)^3=\left(\tfrac18\right)^n.

Summing over

n\ge1,

\displaystyle\sum_{n=1}^\infty\left(\tfrac18\right)^n =\frac{\frac{1}{8}}{1-\frac{1}{8}}=\frac17.

Thus, the correct answer is B.

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