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# A die is thrown $120$ times and getting $1$ or $5$ is considered a success.Find the mean and variance of the number of successes.

Toolbox:
• A random variable $X$ is said to follow a binomial distribution of its probability mass function is given by
• $P(X=x)=p(x)=\left\{\begin{array}{1 1}nC_xp^xq^{n-x},&x=0,1.......n\\0,&otherwise\end{array}\right.$
• Constants of Binomial Distribution :
• Mean = np
• Variance = npq
• Standard deviation =$\sqrt{ variance }=\sqrt{ npq}$
• In a bionomial distribution Mean > Variance.
• The parameters of the distribution are $n,p\quad X\sim B(n,p)$
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 =$npq$
$\qquad\qquad\;\;=120\times \large\frac{1}{3}\times \frac{2}{3}$
$\qquad\qquad\;\;=\large\frac{80}{3}$