Question 11
Alfred and Bonnie play a game in which they take turns tossing a fair coin. The winner of a game is the first person to obtain a head. Alfred and Bonnie play this game several times with the stipulation that the loser of a game goes first in the next game. Suppose that Alfred goes first in the first game, and that the probability that he wins the sixth game is $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. What are the last three digits of $m+n?$