Question

In a Binomial distribution, if \(p=q\) and \(n\geq 4\), then \[ 2^nP(X=5)= \]

• A. \(5\)
• B. \({}^{n}C_{2}\)
• C. \(10\)
• D. \({}^{n}C_{5}\)

Hint:

Whenever \(p=q=\frac12\), the binomial probability simplifies to \[ P(X=r)=\frac{{}^{n}C_r}{2^n}. \]

Answer 1

The Correct Option is D

Step 1: Deduce p and q from the given condition.
For a binomial distribution, p + q = 1. With p = q, we obtain p = q = 1/2.

Step 2: Write the binomial probability mass function.
P(X = 5) = ⁿC₅ p⁵ qⁿ⁻⁵ = ⁿC₅ (1/2)⁵ (1/2)ⁿ⁻⁵ = ⁿC₅ (1/2)ⁿ.

Step 3: Multiply both sides by 2ⁿ.
2ⁿ P(X = 5) = 2ⁿ · ⁿC₅ · (1/2)ⁿ = ⁿC₅.

Step 4: Final conclusion.
The expression simplifies to ⁿC₅.