Question

A die is tossed twice. A 'success' is getting an even number on a toss. Find the variance of number of successes. [NCERT EXEMPLAR]

Answer 1

$$\begin{gathered} \mathrm{We}\mathrm{have}, \\ p=\mathrm{probability}\mathrm{of}\mathrm{getting}\mathrm{an}\mathrm{even}\mathrm{number}\mathrm{on}\mathrm{a}\mathrm{toss}=\frac{3}{6}=\frac{1}{2}\mathrm{and} \\ q=1-p=1-\frac{1}{2}=\frac{1}{2} \\ \mathrm{Let}X\mathrm{denote}\mathrm{a}\mathrm{success}\mathrm{of}\mathrm{getting}\mathrm{an}\mathrm{even}\mathrm{number}\mathrm{on}\mathrm{a}\mathrm{toss}.\mathrm{Then}, \\ X\mathrm{follows}\mathrm{binomial}\mathrm{distribution}\mathrm{with}\mathrm{parameters}n=2\mathrm{and}p=\frac{1}{2} \\ \therefore \mathrm{Var}\left(X\right)=npq=2\times \frac{1}{2}\times \frac{1}{2}=\frac{1}{2} \end{gathered}$$