Question

A die is tossed thrice. If getting a four is considered a success, then the mean and variance of the probability distribution of the number of successes are

• A. $\frac{1}{2},\frac{1}{12}$
• B. $\frac{1}{6},\frac{5}{12}$
• C. $\frac{1}{12},\frac{1}{2}$
• D. $\frac{1}{2},\frac{5}{12}$

Answer 1

The correct option is D

$\frac{1}{2},\frac{5}{12}$

Explanation for the correct option:

Step1. Find the possibilities of success and failure.

Possible outcomes $=\left\{1,2,3,4,5,6\right\}$

Number of times dice thrown$\text{'}\mathrm{n}\text{'}=3$

Getting four is success

Probability of success $\text{'}p\text{'}=\frac{1}{6}$

Probability of failure $\text{'}q\text{'}=1-p$

$$\begin{gathered} =1-\frac{1}{6} \\ =\frac{5}{6} \end{gathered}$$

Step2. Calculate the mean of the probability distribution:

$\mathrm{Mean}=3\times \frac{1}{6}$ $\mathit{M}\mathit{e}\mathit{a}\mathit{n}\mathbf{=}\mathbf{}\mathit{n}\mathit{p}$

$=\frac{1}{2}$

Step3. Calculate the variance of the probability distribution:

$\mathrm{Variance}=3\times \frac{1}{6}\times \frac{5}{6}$$\mathbf{}\mathbf{}\mathit{V}\mathit{a}\mathit{r}\mathit{i}\mathit{a}\mathit{n}\mathit{c}\mathit{e}\mathbf{}\mathbf{=}\mathbf{}\mathit{n}\mathit{p}\mathit{q}$

$=\frac{5}{12}$

Hence, Option (D) is the correct answer.