Q268 / 270 · NDA 2021 (Phase 1) · Mathematics

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

  • A
    $\frac{5}{36}$
  • B
    $\frac{25}{36}$
  • C
    $\frac{25}{54}$
  • Correct answer
    D
    $\frac{25}{216}$

Solution

Mean $(n p)=\frac{2}{3}$ $\operatorname{Variance}(n p q)=\frac{5}{9}$ $\therefore \quad \frac{\text { Variance }(n p q)}{\operatorname{mean}(n p)}=q=\frac{\frac{5}{9}}{\frac{2}{3}}=\frac{5}{6}$ $p=1-q=1-\frac{5}{6}=\frac{1}{6}$ Here number of trial $n \times \frac{1}{6}=\frac{2}{3}$ Random Variable $X=2$ $\therefore \quad$ Probability $={ }^4 \mathrm{C}_2(\mathrm{P}){ }^{4-2} \mathrm{Q}^2={ }^4 C_2 \times\left(\frac{1}{6}\right)^2 \times\left(\frac{5}{6}\right)^2=$ $\frac{25}{216}$

Hence, the correct answer is option: 'D'.

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