# 1988 AHSME Problems/Problem 28

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## Problem

An unfair coin has probability $p$ of coming up heads on a single toss. Let $w$ be the probability that, in $5$ independent toss of this coin, heads come up exactly $3$ times. If $w = 144 / 625$, then $\textbf{(A)}\ p\text{ must be }\tfrac{2}{5}\qquad \textbf{(B)}\ p\text{ must be }\tfrac{3}{5}\qquad\\ \textbf{(C)}\ p\text{ must be greater than }\tfrac{3}{5}\qquad \textbf{(D)}\ p\text{ is not uniquely determined}\qquad\\ \textbf{(E)}\ \text{there is no value of } p \text{ for which }w =\tfrac{144}{625}$

## Solution

We have $w = {5\choose3}p^{3}(1-p)^{2} = 10p^{3}(1-p)^{2}$, so we need to solve $10p^{3}(1-p)^{2} = \frac{144}{625} \implies p^{3}(1-p)^{2} = \frac{72}{3125}$. Now observe that when $p=0$, the left-hand side evaluates to $0$, which is less than $\frac{72}{3125}$; when $p=\frac{1}{2}$, it evaluates to $\frac{1}{32}$, which is more than $\frac{72}{3125}$, and when $p=1$, it evaluates to $0$ again. Thus, since $p^{3}(1-p)^{2}$ is continuous, the Intermediate Value Theorem tells us there is a solution between $0$ and $\frac{1}{2}$ and another solution between $\frac{1}{2}$ and $1$, meaning that there is not a unique value of $p$, so the answer is $\boxed{\text{D}}$.

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