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

To solve this problem, let's break down the concept and apply probability principles step-by-step:

1. We have a total of 15 coupons, each numbered from 1 to 15.
2. We are selecting 7 coupons one by one with replacement, and we need to determine the probability that the largest number on any of these 7 coupons is exactly 9.
3. For the largest number to be exactly 9, every selected coupon must have a number less than or equal to 9. If any selected coupon has a number greater than 9, the largest number will be more than 9.
4. The probability of selecting a coupon with a number ≤ 9 in one draw is \(\frac{9}{15}\) because there are 9 favorable numbers (1 to 9) out of 15.
5. The probability that all 7 selected coupons have numbers ≤ 9 is therefore: 
   
   \(\left(\frac{9}{15}\right)^7\) 
   
   However, this includes the case where the largest number could be any of the numbers from 1 to 9, not just exactly 9.
6. To specifically have the largest number be exactly 9, at least one coupon must be 9, and all others must be ≤ 8:
7. The probability of selecting a number ≤ 8 in one draw is \(\frac{8}{15}\).
8. The event we need to compute is as follows: - Exactly 6 out of 7 draws result in numbers ≤ 8, and - At least one draw results in a 9.
9. For the 7 selected coupons, consider that the last selected number is 9, and the first 6 can be ≤ 8. The probability is: 
   
   \(7 \times \left(\frac{8}{15}\right)^6 \times \left(\frac{1}{15}\right)\) 
   
   Here, 7 is the number of ways to position the number 9 among 7 draws.
10. Let's calculate this: 
    
    \(= 7 \times \left(\frac{8}{15}\right)^6 \times \left(\frac{1}{15}\right)\) 
    \(= 7 \times \left(\frac{8}{15}\right)^6 \times \left(\frac{1}{15}\right)\)

Thus, the probability that the largest number appearing on the selected coupons is 9 does not correspond to any of the given options. The correct answer is "None of these".