<p><b>For Problems 10 and 11:</b> Football teams \(T_1\) and \(T_2\) have to play two games against each other. It is assumed that the outcomes of the two games are independent. The probabilities of \(T_1\) winning, drawing and losing a game against \(T_2\) are \(\dfrac{1}{2}\), \(\dfrac{1}{6}\) and \(\dfrac{1}{3}\), respectively. Each team gets 3 points for a win, 1 point for a draw and 0 point for a loss in a game. Let \(X\) and \(Y\) denote the total points scored by teams \(T_1\) and \(T_2\), respectively, after two games.</p><p><b>Problem 10:</b> \(P(X > Y)\) is</p>
Step-by-Step Solution
Key Concept: Calculate P(X > Y) by identifying all game outcomes where T₁ scores more total points than T₂, then sum probabilities using independence of the two games. The key is recognizing that X > Y occurs when T₁ wins at least one game, or when both games are draws (impossible for X > Y), or specific win-loss combinations.
<p><strong>Step 1:</strong> Identify all outcomes where X > Y after 2 games.</p><p>Per game: P(T₁ wins) = 1/2, P(Draw) = 1/6, P(T₁ loses) = 1/3</p><p><strong>Step 2:</strong> List favorable cases:</p><p>• <strong>T₁ wins both games:</strong> X = 6, Y = 0 → X > Y ✓<br/>P = (1/2)(1/2) = 1/4</p><p>• <strong>T₁ wins first, draws second:</strong> X = 3+1 = 4, Y = 0+1 = 1 → X > Y ✓<br/>P = (1/2)(1/6) = 1/12</p><p>• <strong>T₁ draws first, wins second:</strong> X = 1+3 = 4, Y = 1+0 = 1 → X > Y ✓<br/>P = (1/6)(1/2) = 1/12</p><p>• <strong>T₁ wins first, loses second:</strong> X = 3+0 = 3, Y = 0+3 = 3 → X = Y ✗</p><p>• <strong>T₁ loses first, wins second:</strong> X = 0+3 = 3, Y = 3+0 = 3 → X = Y ✗</p><p>• <strong>T₁ draws both:</strong> X = 1+1 = 2, Y = 1+1 = 2 → X = Y ✗</p><p>• <strong>T₁ draws first, loses second:</strong> X = 1+0 = 1, Y = 1+3 = 4 → X < Y ✗</p><p>• <strong>T₁ loses first, draws second:</strong> X = 0+1 = 1, Y = 3+1 = 4 → X < Y ✗</p><p>• <strong>T₁ loses both:</strong> X = 0, Y = 6 → X < Y ✗</p><p><strong>Step 3:</strong> Sum all favorable probabilities:</p><p>P(X > Y) = 1/4 + 1/12 + 1/12 = 3/12 + 1/12 + 1/12 = <strong>5/12</strong></p><p>∴ Answer: B</p>
Correct Answer: B