Question

A coin comes up with heads with probability $p>0$ and tails with probability $1–p>0$ . When it is flipped eight times, the probability of five heads and three tails is $25$ times the probability of three heads and five tails. Let $p=\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Then $n–m$ is equal to

Answer 1

We have, ${}^8{C}_5{p}^5{\left(1p\right)}^3=25{}^8{C}_3{p}^3{\left(1p\right)}^5$

$\Rightarrow {\left(\frac{p}{1p}\right)}^2=25$

$\Rightarrow p=\frac{5}{6}$

So $(n-m)=1$