2009 AIME I Problems/Problem 3

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Problem

A coin that comes up heads with probability $p > 0$ and tails with probability $1 - p > 0$ independently on each flip is flipped eight times. Suppose the probability of three heads and five tails is equal to $\frac {1}{25}$ of the probability of five heads and three tails. Let $p = \frac {m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$ .

Solution

The probability of three heads and five tails is $\binom {8}{3}p^3(1-p)^5$ and the probability of five heads and three tails is $\binom {8}{3}p^5(1-p)^3$ .

$\begin{align*}25\binom {8}{3}p^3(1-p)^5&=\binom {8}{3}p^5(1-p)^3 \\25(1-p)^2&=p^2 \\5(1-p)&=p \\5-5p&=p \\5&a...$

Therefore, the answer is $5+6=\boxed{11}$ .

2009 AIME I (Problems • Resources)
Preceded by Problem 2	Followed by Problem 4
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15

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2009 AIME I Problems/Problem 3

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