Tennis Deuce probability to win 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 = 0.25\cdot[0.25 + 0.75\cdot p] + 0.75\cdot [0.25\cdot p + 0.75\cdot 0]$ p = 1/10 (this is incorrect, correct answer is 1/12)

Your formula is fine. You arithmetic is off.

$p=0.25[0.25+0.75\cdot p]+0.75[0.25\cdot p+0.75\cdot 0]$

multiply it out:

$p = 0.0625 + 0.375 p\\\ 0.625 p = 0.0625\\\ p = 0.1$

infinite series approach

$p = (\frac 14)^2\sum_\limits{n=0}^\infty (2\cdot \frac 14\cdot \frac 34)^n\\\ \dfrac {\frac 1{16}}{1-\frac {6}{16}} = \frac 1{10}$