Let's analyze and solve both parts of this problem:

Given Information:

• The access code consists of three digits.
• Each digit can be any number from 0 through 5.
• Digits can be repeated.

Step-by-Step Solution:

(a) Probability of randomly selecting the correct access code on the first try:

1. Total possible codes: Each digit can be 0, 1, 2, 3, 4, or 5 — giving 6 possible choices per digit. Since there are 3 digits, the total number of possible codes is: $6 \times 6 \times 6 = 216$

2. Number of correct codes: There is only 1 correct code.

3. Probability: The probability $P$ of selecting the correct code on the first try is: $P = \frac{\text{Number of correct codes}}{\text{Total possible codes}} = \frac{1}{216}$

(b) Probability of not selecting the correct access code on the first try:

1. Number of incorrect codes: Since there are 216 possible codes and only 1 is correct, the number of incorrect codes is: $216 - 1 = 215$

2. Probability: The probability $P$ of not selecting the correct code on the first try is: $P = \frac{\text{Number of incorrect codes}}{\text{Total possible codes}} = \frac{215}{216}$

Final Answers:

• (a) Probability of selecting the correct code: $\frac{1}{216} \approx 0.00463$ (approximately 0.463%)
• (b) Probability of not selecting the correct code: $\frac{215}{216} \approx 0.99537$ (approximately 99.537%)

Would you like a deeper explanation or have any other questions?

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Here are 5 related questions that expand on this topic:

1. What would the probability be if there were 4 digits instead of 3, with each digit ranging from 0 to 9?
2. How does the probability change if digits cannot be repeated in the access code?
3. What is the probability of randomly selecting a code with all three digits being the same (e.g., 000, 111)?
4. If you try 5 different codes at random, what is the probability that one of them is correct?
5. What is the probability of selecting a code with no zeros (only using digits 1 through 5)?

Tip: When calculating probabilities, always ensure the total possible outcomes and the favorable outcomes are correctly counted. This helps in setting up the problem correctly from the start.