<p>A fair coin is tossed 100 times. The probability of getting tails 1, 3, ..., 49 times is</p>
Step-by-Step Solution
Key Concept: Use symmetry of the binomial distribution: P(odd number of tails) = P(even number of tails) = 1/2 for a fair coin, which follows from the binomial expansion of (1/2 + 1/2)^100 and (1/2 - 1/2)^100.
<p><strong>Step 1:</strong> Consider the binomial expansion:</p><p>(p + q)^n = Σ C(n,k)·p^k·q^(n-k)</p><p><strong>Step 2:</strong> For a fair coin with p = q = 1/2, setting p = 1/2 and q = 1/2:</p><p>(1/2 + 1/2)^100 = Σ C(100,k)·(1/2)^100 = 1</p><p><strong>Step 3:</strong> Now consider (p - q)^n with p = q = 1/2:</p><p>(1/2 - 1/2)^100 = Σ C(100,k)·(1/2)^k·(1/2)^(100-k)·(-1)^(100-k) = 0</p><p><strong>Step 4:</strong> This expands to: Σ(k even) C(100,k)·(1/2)^100 - Σ(k odd) C(100,k)·(1/2)^100 = 0</p><p><strong>Step 5:</strong> Therefore: P(even tails) = P(odd tails) = 1/2</p><p>∴ Answer: <strong>1/2</strong></p>
Correct Answer: B