Math Problem Statement
Bank of America customers select their own three-digit personal identification number (PIN) for use at ATMs.
a-1. Assuming that the first digit cannot be a zero, compute the number of possible outcomes when the digits can be repeated.
a-2. Assuming that the first digit cannot be a zero, compute the number of possible outcomes when the digits cannot be repeated.
If zero can be the first digit and the numbers can be repeated, what is the probability Mr. Jones and Mrs. Smith select the same PIN? (Round your answer to 6 decimal places.)
Which concept of probability did you use to answer part b?
multiple choice Classical Correct Empirical Subjective
Solution
Part a-1: Number of possible outcomes when digits can be repeated
We are given a three-digit PIN where:
- The first digit cannot be zero.
- The digits can be repeated.
Let's break it down:
- The first digit can be any number from 1 to 9 (since zero is excluded), so there are 9 possible choices for the first digit.
- The second digit can be any number from 0 to 9 (since repetition is allowed), so there are 10 possible choices for the second digit.
- The third digit can also be any number from 0 to 9, so there are 10 possible choices for the third digit.
Therefore, the total number of possible outcomes when digits can be repeated is:
Part a-2: Number of possible outcomes when digits cannot be repeated
Now, let's calculate the number of possible outcomes when digits cannot be repeated:
- The first digit can still be any number from 1 to 9, so there are 9 choices.
- The second digit can be any number from 0 to 9, except the number already chosen for the first digit. So, there are 9 choices for the second digit.
- The third digit can be any number from 0 to 9, except the two numbers already chosen for the first and second digits. So, there are 8 choices for the third digit.
Therefore, the total number of possible outcomes when digits cannot be repeated is:
Part b: Probability that Mr. Jones and Mrs. Smith select the same PIN
To find the probability that both Mr. Jones and Mrs. Smith select the same PIN:
- The total number of possible PINs is 900 (from part a-1).
- If Mr. Jones selects a PIN, Mrs. Smith must select exactly the same PIN for them to match. So, the number of successful outcomes is 1.
The probability is the ratio of successful outcomes to the total outcomes. Therefore, the probability is:
Rounded to six decimal places, the probability is 0.001111.
Concept of probability used:
The concept of probability used here is Classical Probability. This is based on the assumption that each possible outcome is equally likely, which is the case in this problem where each PIN has an equal chance of being selected.
Summary:
- a-1: The number of possible outcomes when digits can be repeated: 900.
- a-2: The number of possible outcomes when digits cannot be repeated: 648.
- b: The probability that Mr. Jones and Mrs. Smith select the same PIN: 0.001111.
- The concept of probability used: Classical Probability.
Would you like any further explanation or have any questions?
Here are some related questions to expand on this:
- What is the probability of selecting a PIN where all digits are even?
- How would the probability change if the first digit could be zero?
- How many possible PINs are there if the digits are allowed to be letters as well as numbers?
- What is the expected number of attempts it would take to match a specific PIN, assuming random selection?
- What is the probability that Mr. Jones and Mrs. Smith select different PINs?
Tip: When calculating probabilities, always ensure that the outcomes are equally likely, as that forms the basis for classical probability calculations.
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Math Problem Analysis
Mathematical Concepts
Probability
Combinatorics
Permutations
Classical Probability
Formulas
Total Outcomes = Number of Choices for Digit 1 × Number of Choices for Digit 2 × Number of Choices for Digit 3
Probability = Successful Outcomes / Total Outcomes
Theorems
Classical Probability Theorem
Suitable Grade Level
Grades 9-11
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