Math Problem Statement
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?
Solution
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:
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:
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:
Substituting the value from part (a):
Final answers:
(a) The probability of selecting the correct access code on the first try is .
(b) The probability of not selecting the correct access code on the first try is .
Would you like further details or clarifications on any part of this?
Here are 5 related questions to expand your understanding:
- What is the probability of guessing the correct code if the range of digits is from 0 to 9?
- How would the probability change if the code consisted of 4 digits instead of 3?
- What is the probability of guessing a code with unique digits (no repetitions)?
- If you are given two attempts, what is the probability of guessing the code correctly at least once?
- 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.
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Math Problem Analysis
Mathematical Concepts
Probability
Combinatorics
Complementary Events
Formulas
Total possible codes = 9^3 = 729
P(correct) = 1 / 729
P(not correct) = 1 - P(correct) = 728 / 729
Theorems
Complementary Rule
Suitable Grade Level
Grades 9-12