Question

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 is 9, is

• A. \(\left(\frac{9}{10}\right)^6\)
• B. \(\left(\frac{8}{15}\right)^7\)
• C. \(\left(\frac{3}{5}\right)^7\)
• D. None of these

Hint:

For "largest number is exactly \(k\)" with replacement, subtract the case where all numbers \(\leq k-1\) from all numbers \(\leq k\).

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
Probability with replacement. Largest number is exactly 9 means all numbers are \(\leq 9\) and at least one is exactly 9.

Step 2: Detailed Explanation:
Total outcomes = \(15^7\).
Favorable: All numbers from 1 to 9: \(9^7\) outcomes.
All numbers from 1 to 8: \(8^7\) outcomes (these have largest \(\leq 8\)).
So favorable = \(9^7 - 8^7\).
Probability = \(\frac{9^7 - 8^7}{15^7} = \left(\frac{9}{15}\right)^7 - \left(\frac{8}{15}\right)^7 = \left(\frac{3}{5}\right)^7 - \left(\frac{8}{15}\right)^7\).
None of the given options match this exactly.

Step 3: Final Answer:
Option (D) None of these.