<p>A coin is tossed three times. Let \(A\): head on third toss, \(B\): heads on first two tosses. Find \(P(A/B)\).</p>
Step-by-Step Solution
Key Concept: Conditional probability P(A|B) = P(A∩B)/P(B). Identify what outcomes satisfy both B (heads on first two tosses) and A (head on third toss), then divide by outcomes satisfying only B.
<p><strong>Step 1:</strong> Identify the sample space for 3 coin tosses: Total outcomes = 8</p><p><strong>Step 2:</strong> Find outcomes satisfying B (heads on first two tosses): HH* where * can be H or T</p><p>B = {HHH, HHT}, so n(B) = 2, hence P(B) = 2/8 = 1/4</p><p><strong>Step 3:</strong> Find outcomes satisfying both A and B (heads on first two tosses AND head on third toss): A∩B = {HHH}</p><p>n(A∩B) = 1, hence P(A∩B) = 1/8</p><p><strong>Step 4:</strong> Apply conditional probability formula:</p><p>P(A|B) = P(A∩B)/P(B) = (1/8)/(1/4) = 1/8 × 4/1 = 1/2</p><p><strong>Alternative approach:</strong> Given B occurs (first two tosses are heads), the outcome is HH_. The third toss is independent and equally likely to be H or T, so P(A|B) = 1/2</p><p>∴ Answer: B (where B = 1/2)</p>
Correct Answer: B