Question

A dice is thrown ten times. If getting an even number is considered a success, then the probability of four successes is

• A. ${}^{10}C_4{\left(\frac{1}{2}\right)}^4$
• B. ${}^{10}C_4{\left(\frac{1}{2}\right)}^6$
• C. ${}^{10}C_4(\frac{1}{2}{)}^8$
• D. ${}^{10}C_4(\frac{1}{2}{)}^{10}$

Answer 1

The correct option is D

${}^{10}C_4(\frac{1}{2}{)}^{10}$

An explanation for the correct option:

Step 1: Find the probability of success and failure for a given event.

We have been given that, dice is thrown ten times.

We need to find the probability of getting an even number is considered a success, then the getting of four successes.

Probability of getting an even number $\left(p\right)=\frac{3}{6}$

Probability of getting an odd number$\left(q\right)=\frac{3}{6}$

Let, $x$ represents a number of successes.

Step 2: Using the formula of Binomial distribution, find the required probability.

here, $n=10,x=4,p=\frac{1}{2},q=\frac{1}{2}$

the required probability would be,

$$\begin{gathered} \Rightarrow P={}^{10}C_4\times {\left(\frac{1}{2}\right)}^4\times {\left(\frac{1}{2}\right)}^{10-4} \\ \Rightarrow P={}^{10}C_4\times {\left(\frac{1}{2}\right)}^4\times {\left(\frac{1}{2}\right)}^6 \\ \Rightarrow \mathit{P}\mathbf{=}{}^{\mathbf{10}}\mathit{C}_{\mathbf{4}}\mathbf{\times }{\mathbf{\left(}\frac{\mathbf{1}}{\mathbf{2}}\mathbf{\right)}}^{\mathbf{10}} \end{gathered}$$

Therefore, option (D) is the correct answer.