Question

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?

$\dfrac18$

$\dfrac17$

$\dfrac16$

$\dfrac14$

$\dfrac13$

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 $n\ge1,$ $\displaystyle\sum_{n=1}^\infty\left(\tfrac18\right)^n =\frac{1/8}{1-1/8}=\frac17.$

Thus, the correct answer is B.

2016 AMC 12B Problem 19

Solution:

Problem 19 in Other Years