Probability
Conditional Probability
Grade 12
Question:
<p>Three numbers are chosen at random without replacement from {1, 2, 3, …, 8}. The probability that their minimum is 3, given that their maximum is 6, is</p>
<p>(a) \(\frac{3}{8}\)</p>
<p>(b) \(\frac{1}{5}\)</p>
<p>(c) \(\frac{1}{4}\)</p>
<p>(d) \(\frac{2}{5}\)</p>
Step-by-Step Solution
Key Concept: Use conditional probability: given that the maximum is 6, find the probability that the minimum is 3 by counting favorable outcomes among those satisfying the condition.
<p><strong>Step 1:</strong> Given that maximum is 6, one number must be 6.</p><p><strong>Step 2:</strong> The other two numbers must be chosen from {1, 2, 3, 4, 5} without replacement.</p><p><strong>Step 3:</strong> Total ways to choose 2 numbers from {1,2,3,4,5}: C(5,2) = 10</p><p><strong>Step 4:</strong> For minimum to be 3, we need both numbers to be ≥ 3 and at least one must be 3.</p><p><strong>Step 5:</strong> Numbers from {3,4,5}: C(3,2) = 3 ways. But we need minimum exactly 3: must include 3, so choose 1 from {4,5}: C(2,1) = 2 ways</p><p><strong>Step 6:</strong> P(min=3 | max=6) = 2/10 = 1/5</p>
Correct Answer: B