Step 1:

Let $X$ be the random variable denoting the number of times 1 or 5 turns up when a die is thrown .

The probability of getting 1 or 5 in one throw is $p=\large\frac{2}{6}=\frac{1}{3}$

$q=1-\large\frac{1}{3}=\frac{2}{3}$ (not getting 1 or 6)

The number of throws(Bernoulli’s Trials )=n=120

$\therefore X$ follows a binomial distribution with parameters $p=\large\frac{1}{3},$$n=120$

Step 2:

The mean =np

$\qquad\quad\;\;= 120\times \large\frac{1}{3}$$=40$

The variance =$npq$

$\qquad\qquad\;\;=120\times \large\frac{1}{3}\times \frac{2}{3}$

$\qquad\qquad\;\;=\large\frac{80}{3}$