Q80 / 160 · AP EAMCET 2023 (16 May Shift 2) · Mathematics

If the difference between the mean and variance of a binomial variate is $\frac{5}{9}$, then the probability of getting 2 successes for an event when the experiment is conducted 5 times, is

  • Correct answer
    A
    $\frac{80}{243}$
  • B
    $\frac{18}{234}$
  • C
    $\frac{12}{241}$
  • D
    $\frac{80}{432}$

Solution

Mean $=n p$, variance $=n p q$ $\begin{aligned} & \Rightarrow n p-n p q=\frac{5}{9} \\ & \Rightarrow 5(p-p q)=\frac{5}{9}=p(1-q)=\frac{1}{9} \\ & \text { since, } p+q=1, \Rightarrow p=1-q \\ & \Rightarrow p^2=\frac{1}{q} \Rightarrow p=\frac{1}{3}, q=1-\frac{1}{3}=\frac{2}{3} \end{aligned}$ So probability of getting 2 successes $={ }^5 \mathrm{C}_2 \times\left(\frac{1}{3}\right)^2 \times\left(\frac{2}{3}\right)^3=10 \times \frac{1}{9} \times \frac{8}{27}=\frac{80}{243}$

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

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