Answer :
To solve the problem, we need to determine the probability that the third digit in the locker combination is 7, given that the first two digits are 9 and 8, and that digits cannot be repeated. Let's break this down step-by-step:
1. Identify the given information:
- The first digit is 9.
- The second digit is 8.
- We need to find the probability that the third digit is 7.
- Digits cannot be repeated, and only nonzero digits (1 through 9) can be used.
2. Determine the total nonzero digits available:
- The nonzero digits in this context are 1 through 9, totaling 9 digits.
3. Calculate the digits already chosen:
- We have already used the digits 9 and 8 for the first two positions.
4. Find the remaining digits available for the third position:
- Since 9 and 8 are already used, they cannot be chosen again.
- This leaves us with the digits: 1, 2, 3, 4, 5, 6, 7.
5. Count the remaining available digits:
- There are 7 remaining digits (1 through 7).
6. Determine the favorable outcome:
- The favorable outcome is choosing the digit 7 for the third position.
- There is only one favorable outcome (choosing the digit 7).
7. Calculate the probability:
- Probability is given by the number of favorable outcomes divided by the total number of possible outcomes.
- In this scenario, there is 1 favorable outcome (choosing 7) out of 7 remaining possible digits.
The probability can be expressed as:
[tex]\[ \text{Probability} = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \frac{1}{7} \][/tex]
Therefore, the probability that the third digit is 7 is \(\frac{1}{7}\).
Thus, the correct answer is [tex]\( \boxed{\frac{1}{7}} \)[/tex], which corresponds to option D.
1. Identify the given information:
- The first digit is 9.
- The second digit is 8.
- We need to find the probability that the third digit is 7.
- Digits cannot be repeated, and only nonzero digits (1 through 9) can be used.
2. Determine the total nonzero digits available:
- The nonzero digits in this context are 1 through 9, totaling 9 digits.
3. Calculate the digits already chosen:
- We have already used the digits 9 and 8 for the first two positions.
4. Find the remaining digits available for the third position:
- Since 9 and 8 are already used, they cannot be chosen again.
- This leaves us with the digits: 1, 2, 3, 4, 5, 6, 7.
5. Count the remaining available digits:
- There are 7 remaining digits (1 through 7).
6. Determine the favorable outcome:
- The favorable outcome is choosing the digit 7 for the third position.
- There is only one favorable outcome (choosing the digit 7).
7. Calculate the probability:
- Probability is given by the number of favorable outcomes divided by the total number of possible outcomes.
- In this scenario, there is 1 favorable outcome (choosing 7) out of 7 remaining possible digits.
The probability can be expressed as:
[tex]\[ \text{Probability} = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \frac{1}{7} \][/tex]
Therefore, the probability that the third digit is 7 is \(\frac{1}{7}\).
Thus, the correct answer is [tex]\( \boxed{\frac{1}{7}} \)[/tex], which corresponds to option D.
