# If on an average $1$ ship out of $10$ do not arrive safely to ports. Find the mean and the standard deviation of ship returning safely out of a total of $500$ ships

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 numbers of ships out of 500 that return safely to the port.
The probability of a ship not returning safely=$\large\frac{1}{10}$$=q$
$\therefore p=1-\large\frac{1}{10}=\frac{9}{10}$
$X\sim B(500,\large\frac{9}{10})$
Step 2:
The mean=np
$\qquad\quad\;\;=500\times \large\frac{9}{10}$
$\qquad\quad\;\;=450$
The standard deviation =$\sqrt{npq}$
$\qquad\qquad\qquad\qquad=\sqrt{500\times \large\frac{9}{10}\times \frac{1}{10}}$
$\qquad\qquad\qquad\qquad=\sqrt{45}$
$\qquad\qquad\qquad\qquad=3\sqrt{5}$