<p><strong>For Problems 16–18:</strong> Two fair dice are rolled. Let \(P(A_i) > 0\) denote the event that the sum of the faces of the dice is divisible by \(i\).</p><p>Which one of the following events is most probable?</p>
Step-by-Step Solution
Key Concept: Compare probabilities of events A₂, A₃, A₄, A₆ by counting favorable outcomes where the sum is divisible by each value. The sum ranges from 2 to 12 with 36 total outcomes, so calculate how many sums are divisible by i for each i.
<p><strong>Step 1: Find outcomes for each event</strong></p><p>Total outcomes when rolling two dice = 36</p><p><strong>Step 2: Count sums divisible by 2 (even sums)</strong></p><p>Even sums: 2,4,6,8,10,12 → outcomes: 1+35 = 18 outcomes</p><p>P(A₂) = 18/36 = 1/2</p><p><strong>Step 3: Count sums divisible by 3</strong></p><p>Sums divisible by 3: 3,6,9,12 → outcomes: 2+5+4+1 = 12 outcomes</p><p>P(A₃) = 12/36 = 1/3</p><p><strong>Step 4: Count sums divisible by 4</strong></p><p>Sums divisible by 4: 4,8,12 → outcomes: 3+5+1 = 9 outcomes</p><p>P(A₄) = 9/36 = 1/4</p><p><strong>Step 5: Count sums divisible by 6</strong></p><p>Sums divisible by 6: 6,12 → outcomes: 5+1 = 6 outcomes</p><p>P(A₆) = 6/36 = 1/6</p><p><strong>Step 6: Compare probabilities</strong></p><p>P(A₂) = 1/2 > P(A₃) = 1/3 > P(A₄) = 1/4 > P(A₆) = 1/6</p><p>∴ Answer: A (Event A₂ is most probable)</p>
Correct Answer: A