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

Options

A. `((9)/(10))^(6)`
B. `((8)/(15))^(7)`
C. `((3)/(5))^(7)`
D. none

Answer

To solve the problem, we need to find the probability that the largest number appearing on a selected coupon is 9 when we select 7 coupons from a set of 15 numbered coupons (1 to 15) with replacement.

### Step-by-Step Solution:

1. **Understanding the Problem**:
We are selecting 7 coupons, and we want the largest number among those selected coupons to be 9. This means that all selected numbers must be from the set {1, 2, 3, 4, 5, 6, 7, 8, 9}.

2. **Identifying Favorable Outcomes**:
The favorable outcomes are the numbers from 1 to 9. Therefore, when we select a coupon, it can only be one of these 9 numbers.

3. **Calculating Total Outcomes**:
Since there are 15 coupons, the total number of outcomes for selecting one coupon is 15. When selecting 7 coupons, the total number of possible outcomes is:
\[ 15^7 \]

4. **Calculating Favorable Outcomes**:
The number of favorable outcomes (where the selected numbers are from 1 to 9) for each selection is 9. Therefore, the total number of favorable outcomes when selecting 7 coupons is:
\[ 9^7 \]

5. **Finding the Probability**:
The probability \( P \) that the largest number appearing on a selected coupon is 9 can be calculated using the formula:
\[ P = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{9^7}{15^7} \]
This simplifies to:
\[ P = \left(\frac{9}{15}\right)^7 = \left(\frac{3}{5}\right)^7 \]

6. **Final Result**:
Thus, the probability that the largest number appearing on a selected coupon is 9 is:
\[ P = \left(\frac{3}{5}\right)^7 \]