Question

If \(X\sim B(n,\frac12)\), minimum \(n\) such that \[ P(X\ge2)\ge0.6 \] is

• A. 6
• B. 3
• C. 5
• D. 4 \bigskip

Hint:

For binomial inequalities, always use complement \(1-P(0)-P(1)\).

Answer 1

The Correct Option is A

Concept: Use complement: \[ P(X\ge2)=1-P(0)-P(1) \]

Step 1: Write probabilities.
\[ P(0)=\left(\frac12\right)^n \quad P(1)=n\left(\frac12\right)^n \]

Step 2: Inequality.
\[ 1-\frac{n+1}{2^n}\ge0.6 \] \[ \frac{n+1}{2^n}\le0.4 \]

Step 3: Test values.
For \(n=6\): \[ \frac{7}{64}=0.109 \le 0.4 \] For \(n=5\): \[ \frac{6}{32}=0.1875 \le 0.4 \] Minimum satisfying condition is: \[ \boxed{6} \] \[ \boxed{(A)} \]