2017 Gauss 8 Problem 23

Mike and Alain play a game in which each player is equally likely to win. The first player to win three games becomes the champion, and no further games are played. If Mike has won the first game, what is the probability that Mike becomes the champion?

\textbf{(A)}\ \frac{1}{4}\quad \textbf{(B)}\ \frac{5}{8}\quad \textbf{(C)}\ \frac{11}{16}\quad \textbf{(D)}\ \frac{3}{5}\quad \textbf{(E)}\ \frac{3}{4}

If there are no answer choices shown, enter a numerical answer.

Full credit to this problem is given to the CEMC, you may view all Gauss contests here.