Question

A dice is thrown $5$ times then the probability that an even number will come up exactly $3$ times is

• A. $\frac{5}{16}$
• B. $\frac{1}{2}$
• C. $\frac{3}{16}$
• D. $\frac{3}{2}$

Answer 1

The correct option is A

$\frac{5}{16}$

An explanation for the correct option:

Step 1: Find the probability of success and failure.

We have been given a dice is thrown $5$ times.

We need to find the probability that an even number will come up exactly $3$ times.

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

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

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

the probability that an even number will come up exactly $3$ times is:

here, $n=5,x=3,p=\frac{1}{2},q=\frac{1}{2}$

$$\begin{gathered} \Rightarrow P={}^{5}C_3\times {\left(\frac{1}{2}\right)}^3\times {\left(\frac{1}{2}\right)}^{5-3} \\ \Rightarrow P=\frac{5!}{3!(5-3)!}\times \left(\frac{1}{8}\right)\times {\left(\frac{1}{2}\right)}^2 \\ \Rightarrow P=\frac{5}{16} \end{gathered}$$

Therefore, option (A) is the correct answer.