The access code for a gym locker consists of three digits. Each digit can be any number from 0 through 8​, and each digit can be repeated. Complete parts​ (a) and​ (b).
​(a) What is the probability of randomly selecting the correct access code on the first​ try?
​(b) What is the probability of not selecting the correct access code on the first​ try?

Let's go through this step by step.

### Part (a) - Probability of selecting the correct access code on the first try

The access code consists of 3 digits, and each digit can be any number from 0 through 8. This means there are 9 possible choices (0 to 8) for each digit.

The total number of possible access codes is the product of the number of choices for each digit:
\[
\text{Total possible codes} = 9 \times 9 \times 9 = 9^3 = 729
\]

Since there is only 1 correct code out of the 729 possible codes, the probability of randomly selecting the correct code on the first try is:
\[
P(\text{correct}) = \frac{1}{729}
\]

### Part (b) - Probability of not selecting the correct access code on the first try

The probability of **not** selecting the correct code is the complement of the probability of selecting the correct one. The complement rule states:
\[
P(\text{not correct}) = 1 - P(\text{correct})
\]

Substituting the value from part (a):
\[
P(\text{not correct}) = 1 - \frac{1}{729} = \frac{728}{729}
\]

### Final answers:
(a) The probability of selecting the correct access code on the first try is \(\frac{1}{729}\).

(b) The probability of not selecting the correct access code on the first try is \(\frac{728}{729}\).

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Would you like further details or clarifications on any part of this?

Here are 5 related questions to expand your understanding:
1. What is the probability of guessing the correct code if the range of digits is from 0 to 9?
2. How would the probability change if the code consisted of 4 digits instead of 3?
3. What is the probability of guessing a code with unique digits (no repetitions)?
4. If you are given two attempts, what is the probability of guessing the code correctly at least once?
5. How would the total number of possible codes change if one digit can be repeated only twice?

**Tip:** The complement rule is a helpful strategy when calculating the probability of "not" an event occurring. It simplifies calculations in many probability problems.

The access code for a gym locker consists of three digits. Each digit can be any number from 0 through 8​, and each digit can be repeated. Complete parts​ (a) and​ (b).
(a) What is the probability of randomly selecting the correct access code on the first​ try?
(b) What is the probability of not selecting the correct access code on the first​ try?

Calculate the probability of selecting the correct 3-digit gym locker access code on the first try, where each digit ranges from 0 to 8. Explore concepts of combinatorics and the complementary rule, with a step-by-step solution for Grades 9-12.

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