Probability
Independent Events
Grade 12
Question:
<p>In a match, the probability that team <em>W</em> wins is <span>\(\frac{1}{2}\)</span> and the probability that team <em>L</em> wins is <span>\(\frac{1}{2}\)</span>. Two matches are played. Find the probability that the results are <span>\(W_1 L_2\)</span> or <span>\(L_1 W_2\)</span> (i.e., the teams split the wins).</p>
<p>\(\frac{1}{2}\)</p>
<p>\(\frac{1}{4}\)</p>
<p>\(\frac{1}{3}\)</p>
<p>\(\frac{3}{4}\)</p>
Step-by-Step Solution
Key Concept: The events 'W₁L₂' and 'L₁W₂' are mutually exclusive, so you add their individual probabilities. Each match is independent, so multiply probabilities within each event.
<p><strong>Step 1:</strong> Identify the favorable outcomes. We need either:<br>• Match 1: W wins, Match 2: L wins (W₁L₂), OR<br>• Match 1: L wins, Match 2: W wins (L₁W₂)</p><p><strong>Step 2:</strong> Calculate P(W₁L₂):<br>Since matches are independent: P(W₁L₂) = P(W₁) × P(L₂) = ½ × ½ = ¼</p><p><strong>Step 3:</strong> Calculate P(L₁W₂):<br>P(L₁W₂) = P(L₁) × P(W₂) = ½ × ½ = ¼</p><p><strong>Step 4:</strong> Add the mutually exclusive events:<br>P(teams split wins) = P(W₁L₂) + P(L₁W₂) = ¼ + ¼ = ½</p><p><strong>∴ Answer: B (or ½)</strong></p>
Correct Answer: B