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?
$\textbf{(A)}\ \frac{1}{8} \qquad \textbf{(B)}\ \frac{1}{7} \qquad \textbf{(C)}\ \frac{1}{6} \qquad \textbf{(D)}\ \frac{1}{4} \qquad \textbf{(E)}\ \frac{1}{3}$
By: dragonfly
We can solve this problem by listing it as an infinite geometric equation. We get that to have the same amount of tosses, they have a $\frac{1}{8}$ chance of getting all heads. Then the next probability is all of them getting tails and then on the second try, they all get heads. The probability of that happening is $\left(\frac{1}{8}\right)^2$.We then get the geometric equation
$x=\frac{1}{8}+\left(\frac{1}{8}\right)^2+\left(\frac{1}{8}\right)^3...$
And then we find that $x$ equals to $\boxed{\textbf{(B)}\ \frac{1}{7}}$ because of the formula of the sum for an infinite series, $\frac{\frac{1}{8}}{1-\frac{1}{8}} = \frac{1}{8}*\frac{8}{7} = \frac{1}{7}$.