<p>Let gain/loss = <br>\(W \times 100 + LW(-50 + 100) + L^2W(-50 - 50 + 100) + L^3(-150)\)<br>where \(W\) is probability that outcome is 5 or 6 and \(L\) is probability that outcome is 1, 2, 3, 4. The expected gain/loss is:</p>
Step-by-Step Solution
Key Concept: Expected value is calculated by summing each outcome multiplied by its probability. Here, we need to identify what W and L represent (W = P(5 or 6) = 1/3, L = P(1,2,3,4) = 2/3), then expand the given expression and find E[Gain/Loss] by recognizing it as a binomial expansion where each term represents a specific sequence of outcomes.
<p><strong>Step 1:</strong> Identify probabilities. W = P(rolling 5 or 6) = 2/6 = 1/3, and L = P(rolling 1,2,3,4) = 4/6 = 2/3. Verify: W + L = 1.</p><p><strong>Step 2:</strong> Recognize the structure. The expression represents: one success (W×100) + one loss (WL×50) + two losses (L²×0) + three losses (L³×(-150)), accounting for different game sequences.</p><p><strong>Step 3:</strong> Calculate expected value: E = W(100) + L(W)(50) + L²(W)(0) + L³(-150). Expanding: E = (1/3)(100) + (2/3)(1/3)(50) + (2/3)²(1/3)(0) + (2/3)³(-150).</p><p><strong>Step 4:</strong> Compute: E = 100/3 + (2/9)(50) + 0 - (8/27)(150) = 100/3 + 100/9 - 1200/27 = 900/27 + 300/27 - 1200/27 = 0.</p><p>∴ Answer: A (Expected gain/loss = 0)</p>
Correct Answer: A