Question

An unbiased die is thrown twice. A success is getting a number greater than 4. Find the probability distribution of the number of successes.

Answer 1

Let X denote getting a number greater than 4 .

Then, X follows a binomial distribution with n=2

$$\begin{gathered} p=P(X>4)=P(X=5\mathrm{or}6) \\ =\frac{1}{6}+\frac{1}{6} \\ =\frac{1}{3} \\ q\mathit{}=1-p=\frac{2}{3} \\ \mathrm{P}(X=r)={}^2\mathrm{C}_r{\left(\frac{1}{3}\right)}^r{\left(\frac{2}{3}\right)}^{2-r},\mathit{}r=0,1,2 \\ \mathrm{Substituting}\mathrm{for}\mathit{}r\mathrm{we}\mathrm{get}\mathrm{probability}\mathrm{distribution}\mathrm{of}X\mathrm{as}\mathrm{follows}. \end{gathered}$$

$$\begin{gathered} X\mathit{}\mathit{}\mathit{}012 \\ P\left(X\right)\frac{4}{9}\frac{4}{9}\frac{1}{9} \end{gathered}$$