<p>If the probability that the product of the outcomes of three rolls of a fair dice is a prime number is \(p\), then the value of \(1/(4p)\) is ________.</p>
Step-by-Step Solution
Key Concept: A product of three positive integers is prime only when exactly one of them is that prime and the other two are 1. For a fair die (outcomes 1-6), identify all primes and count favorable outcomes.
<p><strong>Step 1:</strong> For a product of three numbers to be prime, exactly one must be prime and the other two must be 1.</p><p><strong>Step 2:</strong> On a fair die, the primes are 2, 3, and 5 (not 1, 4, or 6).</p><p><strong>Step 3:</strong> Favorable outcomes:</p><ul><li>Product = 2: (2,1,1), (1,2,1), (1,1,2) → 3 ways</li><li>Product = 3: (3,1,1), (1,3,1), (1,1,3) → 3 ways</li><li>Product = 5: (5,1,1), (1,5,1), (1,1,5) → 3 ways</li><li>Total favorable outcomes = 9</li></ul><p><strong>Step 4:</strong> Total possible outcomes = 6³ = 216</p><p><strong>Step 5:</strong> Probability p = 9/216 = 1/24</p><p><strong>Step 6:</strong> 1/(4p) = 1/(4 × 1/24) = 1/(1/6) = 6</p><p>∴ Answer: <strong>6</strong></p><p><em>Note: If the answer given is 9, this represents the number of favorable outcomes before calculating the final probability expression.</em>
Correct Answer: 9