Q120 / 180 · KCET 2024 · Mathematics

If a random variable $X$ follows the binomial distribution with parameters $n=5, p$ and $P(X=2)=9 P(X=3)$, then $p$ is equal to

  • A
    $10$
  • Correct answer
    B
    $1 / 10$
  • C
    $5$
  • D
    $1 / 5$

Solution

$\because p(X=2)=9 \times p(X=3)$ (where, $n=5$ and $q=1-p)$ $\Rightarrow \quad{ }^5 C_2 p^2(1-p)^3=9 \cdot{ }^5 C_3 p^3(1-p)^2$ $\Rightarrow \frac{5!}{2!3!} p^2(1-p)^3=9 \cdot \frac{5!}{3!2!} p^3(1-p)^2$ $\Rightarrow \quad \frac{p^2(1-p)^3}{p^3(1-p)^2}=9 \Rightarrow \frac{1-p}{p}=9$ $\Rightarrow \quad 9 p+p=1 \quad \Rightarrow \quad 10 p=1$ $\therefore \quad p=\frac{1}{10}$

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

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