Probability
Binomial Distribution
Grade 12
Question:
<p>A fair coin is tossed \(n\) times. \(X\) = number of heads, \(X \sim \text{Bin}(n, 1/2)\). Which are TRUE?</p>
<p>\(E(X) = \dfrac{n}{2}\)</p>
<p>\(\text{Var}(X) = \dfrac{n}{4}\)</p>
<p>\(P(X = k) = P(X = n-k)\) for all \(k\)</p>
<p>\(P(X \geq n/2) = P(X \leq n/2)\) when \(n\) is even</p>
Step-by-Step Solution
Key Concept: For Bin(n, 1/2): mean = n/2, variance = n(1/2)(1/2) = n/4. Symmetric distribution gives P(X=k) = P(X=n-k).
<p><strong>A:</strong> \(E(X)=np=\frac{n}{2}\) ✓.</p><p><strong>B:</strong> \(\text{Var}(X)=np(1-p)=\frac{n}{4}\) ✓.</p><p><strong>C:</strong> \(P(X=k)=\binom{n}{k}(\frac{1}{2})^n = \binom{n}{n-k}(\frac{1}{2})^n = P(X=n-k)\) ✓ (symmetric).</p><p><strong>D:</strong> Symmetric about n/2 → ✓.</p>
Correct Answer: ABCD