Suppose you’re at a “deuce” in the game of tennis. This means that to win the match, you need to be two points ahead of the other player (i.e. you need to score two points in a row).
The probability that you win a point at any given time is 0.25. What is the probability that you win the match?
Find the error in the following solution:
Let p be the probability that we win the match. Then we can write:
$ p = 0.25cdot[0.25 + 0.75cdot p] + 0.75cdot [0.25cdot p + 0.75cdot 0]$
p = 1/12
(this is incorrect)