Consider a tennis game in which the server wins each point with probability $p$, independent of the score.
Now I need to find the expected duration of the game. I know i have to condition on the #points but I'm not sure exactly how. To find the expected duration, given the game never goes to deuce i found:
En = E(#points|Game never goes to deuce)
= P(lose game)*E(#points|Game never goes to deuce,lose game) + P(win game)*E(#points|Game never goes to deuce,win game)
But to find the expected number of points all together is giving me some trouble.I started the conditioning like this:
En = E(#points)
=E(#points | Server wins the game)*P(server wins the game) + E(#points | server loses the game)*P(server loses the game) (1)
But now I'm not sure how to continue. I think the answer is going to be some binomial that goes to infinity since technically the server can win or lose after infinitely many points.
Am I on the right track with equation (1)?