Q80 / 160 · AP EAMCET 2024 (21 May Shift 2) · Mathematics

If $X \sim B(5, p)$ is a binomial variate such that $p(X=3)=$ $p(X=4)$, then $p(|X-3| \lt 2)=$

  • A
    $\frac{242}{243}$
  • B
    $\frac{201}{243}$
  • Correct answer
    C
    $\frac{200}{243}$
  • D
    $\frac{121}{243}$

Solution

$\begin{aligned} & \text { } X \sim B(5, p) \\ & p(X=3)=p(X=4) \\ & \Rightarrow{ }^5 C_3 p^3(1-p)^2={ }^5 C_4 p^4(1-p) \\ & \Rightarrow 10(1-p)=5 p \Rightarrow p=\frac{10}{15}=\frac{2}{3} \\ & p(|X-3| \lt 2)=p(-2 \lt X-3 \lt 2)=p(1 \lt X \lt 5) \\ & =p(X=2)+p(X=3)+p(X=4)=p(X=2)+2 p(X=3) \\ & ={ }^5 C_2\left(\frac{2}{3}\right)^2\left(\frac{1}{3}\right)^3+2^5 C_3\left(\frac{2}{3}\right)^3\left(\frac{1}{3}\right)^2=\frac{200}{243} .\end{aligned}$

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

Relevant Questions