AIME 2009 I · 第 3 题

AIME 2009 I — Problem 3

专题: Contest Math
难度: L4
来源: AIME

题目详情

Problem

A coin that comes up heads with probability $p > 0$ and tails with probability $1 - p > 0$ independently on each flip is flipped $8$ times. Suppose that 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 $m + n$ .

解析

Solution 1

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 $\binom {8}{3}p^5(1-p)^3$ .

$\begin{aligned} 25\binom {8}{3}p^3(1-p)^5&=\binom {8}{3}p^5(1-p)^3 \\ 25(1-p)^2&=p^2 \\ 25p^2-50p+25&=p^2 \\ 24p^2-50p+25&=0 \\ p&=\frac {5}{6}\end{aligned}$ Therefore, the answer is $5+6=\boxed{011}$ .

Solution 2

We start as shown above. However, when we get to $25(1-p)^2=p^2$ , we square root both sides to get $5(1-p)=p$ . We can do this because we know that both $p$ and $1-p$ are between $0$ and $1$ , so they are both positive. Now, we have:

$\begin{aligned} 5(1-p)&=p \\ 5-5p&=p \\ 5&=6p \\ p&=\frac {5}{6}\end{aligned}$ Now, we get $5+6=\boxed{011}$ .

~Jerry_Guo

Solution 3

Rewrite it as : $(P)^3$ $(1-P)^5=\frac {1}{25}$ $(P)^5$ $(1-P)^3$

This can be simplified as $24P^2 -50P + 25 = 0$

This can be factored into $(4P-5)(6P-5)$

This yields two solutions: $\frac54$ (ignored because it would result in $1-p<0$ ) or $\frac56$

Therefore, the answer is $5+6$ = $\boxed {011}$

Video Solution

https://youtu.be/NL79UexadzE

~IceMatrix

Video Solution 2

https://www.youtube.com/watch?v=P00iOJdQiL4

~Shreyas S