Question

A die is tossed 180 times. Find the expected value $\left(\mu \right)$, variance $\left(q^2\right)$ and S.D $\left(\sigma \right)$ of number of times the face 3 will appear.

Answer 1

Let E be event of getting face $3$ in a single throw of dice

Let $p=p\left(E\right)$

Since these are total $6$ possible outcomes $(1,2,3,4,5,6)$ and $1$ favourable outcomes i.e., $3$.

$p=p\left(t\right)=\frac{1}{6}$

$p\left(\overline{E}\right)=1-p=1-\frac{1}{6}=\frac{5}{6}$

Mean$=$np

Variance $=np(1-p)$

Standard deviation$=\sqrt{np(1-p)}$

Mean$=180\times \frac{1}{6}30$

Variance $=180\times \frac{1}{6}(1-\frac{1}{6})=25$

Standard deviation$=\sqrt{180\times \frac{1}{6}(1-\frac{1}{6})}=\sqrt{25}=5$.