<p><b>For Problems 1–3:</b> A fair die is tossed repeatedly until a 6 is obtained. Let \(X\) denote the number of tosses required.</p><p><b>Problem 1:</b> The probability that \(X = 3\) equals</p>
Step-by-Step Solution
Key Concept: X = 3 means we get non-6 results on tosses 1 and 2, then a 6 on toss 3. Each toss is independent with P(6) = 1/6 and P(not 6) = 5/6.
<p><strong>Step 1:</strong> Identify the event. X = 3 means we fail (don't get 6) on tosses 1 and 2, then succeed (get 6) on toss 3.</p><p><strong>Step 2:</strong> Calculate the probability using independence.</p><p>P(X = 3) = P(not 6 on toss 1) × P(not 6 on toss 2) × P(6 on toss 3)</p><p>P(X = 3) = (5/6) × (5/6) × (1/6) = 25/216</p><p><strong>Step 3:</strong> Verify this matches the geometric distribution formula: P(X = n) = (1 - p)^(n-1) × p, where p = 1/6.</p><p>P(X = 3) = (5/6)² × (1/6) = 25/216</p><p>∴ Answer: A (25/216)</p>
Correct Answer: A