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