<p>Two players toss 4 coins each. The probability that they both obtain the same number of heads is</p>
Step-by-Step Solution
Key Concept: Use binomial probability to find P(X=k) for each player independently, then sum the products P(both get k heads) for k=0,1,2,3,4. Since each player's outcome is independent, multiply their individual probabilities.
<p><strong>Step 1:</strong> For each player tossing 4 coins, the probability of getting exactly k heads is:</p><p>P(k heads) = C(4,k)/2^4 = C(4,k)/16</p><p><strong>Step 2:</strong> Since both players' tosses are independent, P(both get k heads) = P(k heads)² = [C(4,k)/16]²</p><p><strong>Step 3:</strong> Sum over all possible values k = 0, 1, 2, 3, 4:</p><p>P(same heads) = Σ[C(4,k)]²/256</p><p>= [C(4,0)² + C(4,1)² + C(4,2)² + C(4,3)² + C(4,4)²]/256</p><p>= [1 + 16 + 36 + 16 + 1]/256</p><p>= 70/256 = 35/128</p><p><strong>Note:</strong> The sum Σ[C(4,k)]² = C(8,4) = 70 by Vandermonde's identity.</p><p>∴ Answer: <strong>35/128</strong></p>
Correct Answer: C