<p>If \(A\) and \(B\) each toss three coins. The probability that both get the same number of heads is</p>
Step-by-Step Solution
Key Concept: Use the binomial probability distribution to find P(exactly k heads) for each person, then sum the probabilities for matching outcomes. Both independent events must yield the same number of heads (0, 1, 2, or 3).
<p><strong>Step 1:</strong> Find P(exactly k heads in 3 tosses) for one person using binomial probability: P(k heads) = C(3,k) × (1/2)³</p><p>• P(0 heads) = C(3,0)/8 = 1/8<br>• P(1 head) = C(3,1)/8 = 3/8<br>• P(2 heads) = C(3,2)/8 = 3/8<br>• P(3 heads) = C(3,3)/8 = 1/8</p><p><strong>Step 2:</strong> Since A and B toss independently, P(both get k heads) = P(A gets k heads) × P(B gets k heads) = [P(k heads)]²</p><p>• P(both get 0) = (1/8)² = 1/64<br>• P(both get 1) = (3/8)² = 9/64<br>• P(both get 2) = (3/8)² = 9/64<br>• P(both get 3) = (1/8)² = 1/64</p><p><strong>Step 3:</strong> Sum all cases: P(same number) = 1/64 + 9/64 + 9/64 + 1/64 = 20/64 = <strong>5/16</strong></p><p>∴ Answer: C</p>
Correct Answer: C