Question

A die is thrown 5 times. Find the probability that an odd number will come up exactly three times.

Answer 1

Here, $n=5,p=(\frac{1}{6}+\frac{1}{6}+\frac{1}{6})=\frac{1}{2}$ and $q=1-p=1-\frac{1}{2}=\frac{1}{2}$
Also, $r=3$$\therefore P(X=r){=}^n{C}_r\left(p{)}^r\right(q{)}^{n-r}{=}^5{C}_3{\left(\frac{1}{2}\right)}^3{\left(\frac{1}{2}\right)}^{5-3}$
$=\frac{5!}{3!2!}\cdot \frac{1}{8}\cdot \frac{1}{4}=\frac{10}{32}=\frac{5}{16}$