Question

In a binomial distribution, the mean is $\frac{2}{3}$ and variance is $\frac{5}{9}$. What is the probability that random variable $D=2?$

• (A) $\frac{5}{36}$
• (B) $\frac{25}{36}$
• (C) $\frac{25}{54}$
• (D) $\frac{25}{216}$

Answer 1

The Correct Option is D

Explanation:
Concept:Mean = npVariance = npqWhere n is the total number of cases, p is probability of favorable cases and q is probability of unfavorable cases(1 - p)Given mean $=np=\frac{2}{3}\dots$ (i)And variance $=npq=\frac{5}{9}\dots$ (ii)On dividing equation (i) and (ii)$\frac{\text{ variance }}{\text{ mean }}=\frac{\frac{5}{9}}{\frac{2}{3}}$$q=\frac{5}{6}$$p=1-q=\frac{1}{6}$$np=\frac{2}{3}$$n=4$$\therefore$ The probability that random variable $D=2$$P={}^n{C}_2{p}^2{q}^{n-2}$$P={}^4{C}_2\times {\left(\frac{1}{6}\right)}^2\times {\left(\frac{5}{6}\right)}^2$$P=6\times \frac{1}{36}\times \frac{25}{36}$$P=\frac{25}{216}$Hence, the correct option is (D).