Question

A pair of dice is thrown 4 times. If getting a doublet is considered a success, find the probability distribution of the number of successes and, hence, find its mean.

Answer 1

Let X be the number of times a doublet is obtained in four throws.

Then, p = probability of success in one throw of a pair of dice =$\frac{6}{36}=\frac{1}{6}$
$$\begin{gathered} \mathrm{and}q=\frac{5}{6};n=4 \\ P(X=r)={}^{4}C_r{\left(\frac{\mathit{1}}{\mathit{6}}\right)}^r{\left(\frac{5}{6}\right)}^{4-r},r=0,1,2,3,4 \\ \mathrm{As}n=4\mathrm{and}p=\frac{1}{6}, \\ \text{m}\mathrm{ean}=np=\frac{4}{6}=\frac{2}{3} \end{gathered}$$
$$\begin{gathered} \mathit{\therefore }\mathit{}P(X=r)={}^{4}C_r{\left(\frac{\mathit{5}}{\mathit{6}}\right)}^r{\left(\frac{\mathit{1}}{\mathit{6}}\right)}^{n\mathit{-}r},r=0,1,2,3,4 \\ \text{The d}\mathrm{istribution}\mathrm{is}\mathrm{as}\mathrm{follows}: \\ X01234 \\ P\left(X\right){\left(\frac{5}{6}\right)}^4\frac{20}{6^4}\frac{150}{6^4}\frac{500}{6^4}\frac{1}{6^4} \end{gathered}$$