In some games such as tennis and ping pong you can reach a s

In some games, such as tennis and ping pong, you can reach a state called \"deuce\". This means that the score is tied and either player wins the game when he or she gets two points ahead of the other player. Suppose you are at deuce. Suppose the probability that you win a single point is p and this is true independently for all points. As a function of p, what is the probability that you win the game? (Hint: from a given score, condition on the outcome of the next point.)

Solution

For game to be won you need to won first point so its probability is p
SO now the possibility is you win the point and win the game or lose the point and come to earlier stage
probability of winning a game = p*p +p*(1-p)*p*p+p*(1-p)*p*(1-p)p*p...........
So this is GP with a =p*p and r= p(1-p)
so sum is a/1-r = p*p/(1-p+p^2)