Question

A fair coin is tossed $9$ times. Find the probability that it shows head exactly $5$ times.

Answer 1

Let $X=$number of head shows

$n=9,p=\frac{1}{2},q=\frac{1}{2}$

We know that, $P(X=x)={}^n{C}_x{p}^n\cdot q^{n-x}wherex=0,1,...,n$

$\therefore P(X=5)={}^9{C}_5{\left(\frac{1}{2}\right)}^5\cdot {\left(\frac{1}{2}\right)}^{9-5}$

$={}^9{C}_5{\left(\frac{1}{2}\right)}^5\cdot {\left(\frac{1}{2}\right)}^4$

$={}^9{C}_5{\left(\frac{1}{2}\right)}^9$

$=\frac{9\times 8\times 7\times 6}{4\times 3\times 2\times 1}\times \frac{1}{2^9}$

$=\frac{126}{512}$

$\therefore P(X=5)=0.2460$