Question

A die is thrown $100$ times, and getting an even number is considered a success.

The variance of the number of successes is

• A. $10$
• B. $25$
• C. $18$
• D. $11$

Answer 1

The correct option is B

$25$

The explanation for the correct option:

Step1. Calculate the probability of even numbers:

Total number of times a die is thrown $\mathrm{n}=100$

Total number of sample space $=6$

Possibilities of getting an even number$=3$

Probability of getting an even number

$\mathrm{P}\left(\mathrm{E}\right)=\frac{3}{6}$ $\mathbf{[}\mathbf{\because }\mathbf{P}\mathbf{r}\mathbf{o}\mathbf{b}\mathbf{a}\mathbf{b}\mathbf{i}\mathbf{l}\mathbf{i}\mathbf{t}\mathbf{y}\mathbf{}\mathbf{=}\mathbf{}\frac{\mathbf{T}\mathbf{o}\mathbf{t}\mathbf{a}\mathbf{l}\mathbf{}\mathbf{n}\mathbf{u}\mathbf{m}\mathbf{b}\mathbf{e}\mathbf{r}\mathbf{}\mathbf{o}\mathbf{f}\mathbf{}\mathbf{f}\mathbf{a}\mathbf{v}\mathbf{o}\mathbf{u}\mathbf{r}\mathbf{a}\mathbf{b}\mathbf{l}\mathbf{e}\mathbf{}\mathbf{o}\mathbf{u}\mathbf{t}\mathbf{c}\mathbf{o}\mathbf{m}\mathbf{e}}{\mathbf{T}\mathbf{o}\mathbf{t}\mathbf{a}\mathbf{l}\mathbf{}\mathbf{n}\mathbf{u}\mathbf{m}\mathbf{b}\mathbf{e}\mathbf{r}\mathbf{}\mathbf{o}\mathbf{f}\mathbf{}\mathbf{o}\mathbf{u}\mathbf{t}\mathbf{c}\mathbf{o}\mathbf{m}\mathbf{e}\mathbf{s}}\mathbf{]}\mathbf{}$

$\Rightarrow P\left(E\right)=\frac{1}{2}$

Getting an even number is a success $\text{'}\mathrm{p}\text{'}=\frac{1}{2}$

Probability of not getting an even number $\begin{array}{rcl}\text{'}\mathrm{q}\text{'}& =& 1-\frac{1}{2}\mathbf{[}\mathbf{\because }\mathbf{}\mathbf{p}\mathbf{+}\mathbf{}\mathbf{q}\mathbf{}\mathbf{=}\mathbf{1}\mathbf{]}\end{array}$

$\begin{array}{rcl}& =& \frac{1}{2}\end{array}$

Step 2. Calculate variance:

Variance $\mathbf{=}\mathbf{}\mathit{n}\mathit{p}\mathit{q}$

Substitute the value of $\mathrm{n},\mathrm{p}\mathrm{and}\mathrm{q}$ in the formula, and we get

Variance $=100\times \frac{1}{2}\times \frac{1}{2}$

$=\mathbf{25}$

Hence, Option (B) is the correct answer.