Probability
Classical Probability
Grade 12
Question:
<p>Fifteen coupons are numbered from 1 to 15. Seven coupons are selected at random one at a time with replacement. The probability that the largest number appearing on a selected coupon is 9, is</p>
<p>(a) \(\left(\frac{9}{15}\right)^6\)</p>
<p>(b) \(\left(\frac{8}{15}\right)^7\)</p>
<p>(c) \(\left(\frac{3}{5}\right)^7\)</p>
<p>(d) none of these</p>
Step-by-Step Solution
Key Concept: For the largest number to be exactly 9, all 7 selected coupons must be from {1,2,...,9} AND at least one must be 9. Use the formula: P(max=9) = P(all ≤ 9) - P(all ≤ 8).
<p><strong>Step 1:</strong> With replacement, total outcomes = 15^7. We need P(max = 9).</p><p><strong>Step 2:</strong> For max to be exactly 9: all coupons must be from {1,2,...,9} AND at least one must be 9.</p><p><strong>Step 3:</strong> P(max = 9) = P(all ≤ 9) - P(all ≤ 8)</p><p>P(all ≤ 9) = (9/15)^7 = (3/5)^7</p><p>P(all ≤ 8) = (8/15)^7</p><p><strong>Step 4:</strong> P(max = 9) = (9/15)^7 - (8/15)^7 = (3/5)^7 - (8/15)^7</p><p>= (1/15^7)[9^7 - 8^7]</p><p><strong>Step 5:</strong> Simplifying: = (9^7 - 8^7)/15^7</p><p>∴ Answer: D</p>
Correct Answer: D