Question

A die is thrown three times. Let X be 'the number of twos seen'. Find the expectation of X. [NCERT EXEMPLAR]

Answer 1

$$\begin{gathered} \mathrm{We}\mathrm{have}, \\ p=\mathrm{probability}\mathrm{of}\mathrm{getting}\mathrm{a}\mathrm{number}\mathrm{two}\mathrm{in}\mathrm{a}\mathrm{throw}=\frac{1}{6}\mathrm{and} \\ q=1-p=1-\frac{1}{6}=\frac{5}{6} \\ \mathrm{As},X\mathrm{denote}\text{'}\mathrm{the}\mathrm{number}\mathrm{of}\mathrm{twos}\mathrm{seen}\text{'}. \\ \mathrm{So},X\mathrm{follows}\mathrm{binomial}\mathrm{distribution}\mathrm{with}\mathrm{parameters}n=3\mathrm{and}p=\frac{1}{6} \\ \therefore \mathrm{E}\left(X\right)=np=3\times \frac{1}{6}=\frac{1}{2} \end{gathered}$$