Probability
Classical Probability
Grade None
Question:
<p>Fifteen coupons are numbered 1, 2, …, 15 respectively. Seven coupons are selected at random one at a time with replacement. The probability that the largest number appearing on a selected coupon as 9, is</p>
<p>(a) \(\left(\dfrac{9}{19}\right)^6\)</p>
<p>(b) \(\left(\dfrac{8}{15}\right)^7\)</p>
<p>(c) \(\left(\dfrac{3}{5}\right)^7\)</p>
<p>(d) None of these</p>
Step-by-Step Solution
Key Concept: For the maximum of selected coupons to be exactly 9, all selections must be from {1,2,...,9} AND at least one must be 9. Use inclusion-exclusion: P(max=9) = P(all ≤9) - P(all ≤8).
<p><strong>Step 1:</strong> We select 7 coupons with replacement from {1,2,...,15}. We need P(maximum selected = 9).</p><p><strong>Step 2:</strong> For max to equal 9: (i) all selections must be from {1,2,...,9}, AND (ii) at least one selection must be 9.</p><p><strong>Step 3:</strong> Using complementary counting:<br>P(max = 9) = P(all ≤ 9) - P(all ≤ 8)</p><p><strong>Step 4:</strong> Calculate each probability:<br>• P(all 7 selections ≤ 9) = (9/15)^7 = (3/5)^7<br>• P(all 7 selections ≤ 8) = (8/15)^7</p><p><strong>Step 5:</strong> Therefore:<br>P(max = 9) = (9/15)^7 - (8/15)^7 = (3/5)^7 - (8/15)^7</p><p><strong>Step 6:</strong> Simplify: (3/5)^7 - (8/15)^7 = (3/5)^7 - (8/15)^7<br>= 3^7/5^7 - 8^7/15^7<br>Finding common denominator and factoring gives the answer in option D.</p><p>∴ <strong>Answer: D</strong></p>
Correct Answer: D