Probability
Classical Probability
Grade 12
Question:
<p>An unbiased ordinary dice is rolled four times. Out of the four face-values obtained, the probability that the minimum face-value is not less than 2 and the maximum face-value is not greater than 5, is</p>
<p>(a) \(\frac{16}{81}\)</p>
<p>(b) \(\frac{1}{6}\)</p>
<p>(c) \(\frac{80}{81}\)</p>
<p>(d) \(\frac{65}{81}\)</p>
Step-by-Step Solution
Key Concept: The minimum is ≥2 AND maximum is ≤5 means all four rolls must be in {2,3,4,5}. Use the favorable outcomes divided by total outcomes.
<p><strong>Step 1:</strong> Interpret the conditions. 'Minimum face-value not less than 2' means min ≥ 2. 'Maximum face-value not greater than 5' means max ≤ 5.</p><p><strong>Step 2:</strong> For BOTH conditions to hold simultaneously, all four rolls must come from the set {2, 3, 4, 5}. No roll can be 1 (violates min ≥ 2) and no roll can be 6 (violates max ≤ 5).</p><p><strong>Step 3:</strong> Total possible outcomes when rolling a dice 4 times = 6⁴ = 1296.</p><p><strong>Step 4:</strong> Favorable outcomes (all four rolls from {2,3,4,5}) = 4⁴ = 256.</p><p><strong>Step 5:</strong> Probability = 256/1296 = 16/81.</p><p>∴ Answer: <strong>16/81</strong></p>
Correct Answer: A