Maggie's Bank Has Assigned Her A Temporary 3-digit PIN To Use With Her ATM Card. Each Digit Is A Number From 1 To 5 Inclusive, And No Digit Can Be Used More Than Once In The PIN.Which Multiplication Problem Can Be Used To Determine The Probability That
Introduction
Maggie's bank has assigned her a temporary 3-digit PIN to use with her ATM card. Each digit is a number from 1 to 5 inclusive, and no digit can be used more than once in the PIN. In this problem, we will determine the probability that Maggie's PIN is a certain number.
Understanding the Problem
To solve this problem, we need to understand the concept of probability. Probability is a measure of the likelihood of an event occurring. In this case, the event is Maggie's PIN being a certain number.
Calculating the Total Number of Possible PINs
Since each digit can be a number from 1 to 5 inclusive, and no digit can be used more than once in the PIN, we need to calculate the total number of possible PINs. This can be done using the concept of permutations.
A permutation is an arrangement of objects in a specific order. In this case, we have 5 options for the first digit, 4 options for the second digit (since one digit has already been used), and 3 options for the third digit (since two digits have already been used).
Using the formula for permutations, we can calculate the total number of possible PINs as follows:
5 (options for the first digit) × 4 (options for the second digit) × 3 (options for the third digit) = 60
Therefore, there are 60 possible PINs that Maggie can have.
Calculating the Probability
To calculate the probability that Maggie's PIN is a certain number, we need to know the number of favorable outcomes (i.e., the number of PINs that match the certain number) and the total number of possible outcomes (i.e., the total number of possible PINs).
Let's assume that the certain number is 3. We need to calculate the number of PINs that contain the digit 3.
Since the digit 3 can be in any of the three positions (first, second, or third), we need to calculate the number of PINs that contain the digit 3 in each position.
Calculating the Number of PINs with 3 in the First Position
If the digit 3 is in the first position, the second digit can be any of the remaining 4 numbers (1, 2, 4, or 5), and the third digit can be any of the remaining 3 numbers (1, 2, or 4).
Using the formula for permutations, we can calculate the number of PINs with 3 in the first position as follows:
4 (options for the second digit) × 3 (options for the third digit) = 12
Therefore, there are 12 PINs with 3 in the first position.
Calculating the Number of PINs with 3 in the Second Position
If the digit 3 is in the second position, the first digit can be any of the remaining 4 numbers (1, 2, 4, or 5), and the third digit can be any of the remaining 3 numbers (1, 2, or 4).
Using the formula for permutations, we can calculate the number of PINs with 3 in the second position as follows:
4 (options for the first digit) × 3 (options for the third digit) = 12
Therefore, there are 12 PINs with 3 in the second position.
Calculating the Number of PINs with 3 in the Third Position
If the digit 3 is in the third position, the first digit can be any of the remaining 4 numbers (1, 2, 4, or 5), and the second digit can be any of the remaining 3 numbers (1, 2, or 4).
Using the formula for permutations, we can calculate the number of PINs with 3 in the third position as follows:
4 (options for the first digit) × 3 (options for the second digit) = 12
Therefore, there are 12 PINs with 3 in the third position.
Calculating the Total Number of PINs with 3
To calculate the total number of PINs with 3, we need to add the number of PINs with 3 in each position.
12 (PINs with 3 in the first position) + 12 (PINs with 3 in the second position) + 12 (PINs with 3 in the third position) = 36
Therefore, there are 36 PINs with 3.
Calculating the Probability
To calculate the probability that Maggie's PIN is 3, we need to divide the number of favorable outcomes (i.e., the number of PINs with 3) by the total number of possible outcomes (i.e., the total number of possible PINs).
Probability = Number of favorable outcomes / Total number of possible outcomes = 36 (PINs with 3) / 60 (total number of possible PINs) = 0.6
Therefore, the probability that Maggie's PIN is 3 is 0.6 or 60%.
Conclusion
In this problem, we calculated the probability that Maggie's PIN is a certain number (3). We used the concept of permutations to calculate the total number of possible PINs and the number of PINs with 3. We then calculated the probability by dividing the number of favorable outcomes by the total number of possible outcomes.
Final Answer
Introduction
In our previous article, we explored the problem of Maggie's ATM PIN and calculated the probability that her PIN is a certain number (3). In this article, we will answer some frequently asked questions related to this problem.
Q: What is the total number of possible PINs that Maggie can have?
A: The total number of possible PINs that Maggie can have is 60. This is calculated using the concept of permutations, where each digit can be a number from 1 to 5 inclusive, and no digit can be used more than once in the PIN.
Q: How many PINs contain the digit 3?
A: There are 36 PINs that contain the digit 3. This is calculated by adding the number of PINs with 3 in each position (first, second, and third).
Q: What is the probability that Maggie's PIN is 3?
A: The probability that Maggie's PIN is 3 is 0.6 or 60%. This is calculated by dividing the number of favorable outcomes (i.e., the number of PINs with 3) by the total number of possible outcomes (i.e., the total number of possible PINs).
Q: How did you calculate the number of PINs with 3 in each position?
A: We calculated the number of PINs with 3 in each position using the formula for permutations. For example, if the digit 3 is in the first position, the second digit can be any of the remaining 4 numbers (1, 2, 4, or 5), and the third digit can be any of the remaining 3 numbers (1, 2, or 4).
Q: What is the significance of the probability that Maggie's PIN is 3?
A: The probability that Maggie's PIN is 3 is significant because it gives us an idea of how likely it is that Maggie's PIN is a certain number. In this case, the probability is 0.6 or 60%, which means that there is a 60% chance that Maggie's PIN is 3.
Q: Can you explain the concept of permutations in more detail?
A: Permutations are arrangements of objects in a specific order. In the context of this problem, we used permutations to calculate the total number of possible PINs and the number of PINs with 3. Permutations are an important concept in mathematics and are used to solve a wide range of problems.
Q: How can you apply the concept of permutations to real-world problems?
A: Permutations can be applied to a wide range of real-world problems, such as:
- Calculating the number of possible combinations of a set of objects
- Determining the probability of a certain event occurring
- Solving problems involving arrangements of objects in a specific order
Conclusion
In this article, we answered some frequently asked questions related to the problem of Maggie's ATM PIN. We explained the concept of permutations and how it can be used to calculate the total number of possible PINs and the number of PINs with 3. We also discussed the significance of the probability that Maggie's PIN is 3 and how it can be applied to real-world problems.
Final Answer
The final answer is that the probability that Maggie's PIN is 3 is 0.6 or 60%.