Question

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. \(0.75\)
• C. \(0.25\)
• D. \(0.375\)

Hint:

Write both probabilities with C(6,r) p^r q^(6-r) and compare.

Answer 1

The Correct Option is C

Step 1: Take the ratio:
$\frac{P(X=4)}{P(X=2)} = \frac{\binom64p^4q^2}{\binom62p^2q^4} = \frac{p^2}{q^2}$, because the binomial coefficients are equal.

Step 2: Use the condition:
$9P(X=4) = P(X=2)$ means this ratio is $\frac19$, so $\frac pq = \frac13$.

Step 3: Solve:
$q = 3p$ and $p + q = 1$, so $4p = 1$, $p = 0.25$.

Final Answer:
p equals 0.25, option (C). \[ \boxed{0.25} \]