Q100 / 150 · MHT CET 2026 (20 April Shift 1) · Mathematics
A random variable $X \sim B(n, p)$ follows a binomial distribution with $n = 6$. If $9P(X = 4) = P(X = 2)$, then the probability of success $p$ is..
- A$0.125$
- B$0.75$
- Correct answerC$0.25$
- D$0.375$
Solution
Given $X \sim B(n, p)$ with $n = 6$.
The probability mass function is $P(X = k) = ^{n}C_{k} p^k (1-p)^{n-k}$.
Substituting $k = 4$ and $k = 2$ in the given condition $9P(X = 4) = P(X = 2)$:
$9 \times ^{6}C_{4} p^4 (1-p)^2 = ^{6}C_{2} p^2 (1-p)^4$
Since $^{6}C_{4} = ^{6}C_{2} = 15$, dividing both sides by $15 p^2 (1-p)^2$ gives:
$9p^2 = (1-p)^2$
Taking the square root on both sides, since $p > 0$ and $1-p > 0$:
$3p = 1 - p$
$4p = 1$
$p = 0.25$
Answer: $0.25$
The probability mass function is $P(X = k) = ^{n}C_{k} p^k (1-p)^{n-k}$.
Substituting $k = 4$ and $k = 2$ in the given condition $9P(X = 4) = P(X = 2)$:
$9 \times ^{6}C_{4} p^4 (1-p)^2 = ^{6}C_{2} p^2 (1-p)^4$
Since $^{6}C_{4} = ^{6}C_{2} = 15$, dividing both sides by $15 p^2 (1-p)^2$ gives:
$9p^2 = (1-p)^2$
Taking the square root on both sides, since $p > 0$ and $1-p > 0$:
$3p = 1 - p$
$4p = 1$
$p = 0.25$
Answer: $0.25$
Hence, the correct answer is option: 'C'.