The Ages (years) Of Three Government Officials When They Died In Office Were 54, 45, And 61. Complete Parts (a) Through (d).a. Assuming That 2 Of The Ages Are Randomly Selected With Replacement, List The Different Possible Samples.A. (54, 54), (54,

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Introduction

In this article, we will delve into a mathematical problem involving the ages of three government officials who passed away while in office. The ages of the officials were 54, 45, and 61. We will explore the different possible samples that can be obtained by randomly selecting 2 of the ages with replacement.

Part (a): Listing the Different Possible Samples

To list the different possible samples, we need to consider all the possible combinations of 2 ages that can be selected from the 3 available ages. Since the ages are selected with replacement, each age can be selected more than once.

Let's denote the ages as A = 54, B = 45, and C = 61. We can list the different possible samples as follows:

  • (A, A)
  • (A, B)
  • (A, C)
  • (B, A)
  • (B, B)
  • (B, C)
  • (C, A)
  • (C, B)
  • (C, C)

There are a total of 9 different possible samples that can be obtained by randomly selecting 2 of the ages with replacement.

Part (b): Exploring the Possibilities

Now that we have listed the different possible samples, let's explore the possibilities further. We can ask questions such as:

  • What is the probability of selecting two ages that are the same?
  • What is the probability of selecting two ages that are different?
  • What is the probability of selecting a specific age, such as age A?

To answer these questions, we need to consider the total number of possible samples and the number of samples that satisfy each condition.

Part (c): Calculating Probabilities

Let's calculate the probabilities of selecting two ages that are the same and two ages that are different.

  • Probability of selecting two ages that are the same:
    • There are 3 possible samples where the two ages are the same: (A, A), (B, B), and (C, C).
    • There are a total of 9 possible samples.
    • Therefore, the probability of selecting two ages that are the same is 3/9 = 1/3.
  • Probability of selecting two ages that are different:
    • There are 6 possible samples where the two ages are different: (A, B), (A, C), (B, A), (B, C), (C, A), and (C, B).
    • There are a total of 9 possible samples.
    • Therefore, the probability of selecting two ages that are different is 6/9 = 2/3.

Part (d): Conclusion

In conclusion, we have explored the different possible samples that can be obtained by randomly selecting 2 of the ages with replacement. We have listed the different possible samples, explored the possibilities, and calculated the probabilities of selecting two ages that are the same and two ages that are different.

This problem is a great example of how mathematical concepts such as probability and combinatorics can be applied to real-world scenarios. By understanding the different possible samples and calculating the probabilities, we can gain a deeper understanding of the underlying mathematical principles.

Final Thoughts

The problem of randomly selecting 2 of the ages with replacement is a classic example of a combinatorial problem. By listing the different possible samples and calculating the probabilities, we can gain a deeper understanding of the underlying mathematical principles. This problem is a great example of how mathematical concepts can be applied to real-world scenarios, and it highlights the importance of understanding probability and combinatorics in a variety of fields.

References

  • [1] "Combinatorics: Topics, Techniques, Algorithms" by Peter J. Cameron
  • [2] "Probability and Statistics for Engineers and Scientists" by Ronald E. Walpole and Raymond H. Myers

Glossary

  • Combinatorics: The branch of mathematics that deals with counting and arranging objects in various ways.
  • Probability: A measure of the likelihood of an event occurring.
  • Random sampling: The process of selecting a sample from a population in such a way that every possible sample has an equal chance of being selected.
    The Ages of Government Officials: A Mathematical Exploration - Q&A ====================================================================

Introduction

In our previous article, we explored the different possible samples that can be obtained by randomly selecting 2 of the ages with replacement. We listed the different possible samples, explored the possibilities, and calculated the probabilities of selecting two ages that are the same and two ages that are different.

In this article, we will answer some of the most frequently asked questions related to this problem. We will provide detailed explanations and examples to help clarify any doubts.

Q: What is the total number of possible samples?

A: The total number of possible samples is 9. This is because there are 3 possible ages (A, B, and C) and we are selecting 2 of them with replacement.

Q: What is the probability of selecting two ages that are the same?

A: The probability of selecting two ages that are the same is 1/3. This is because there are 3 possible samples where the two ages are the same: (A, A), (B, B), and (C, C), and a total of 9 possible samples.

Q: What is the probability of selecting two ages that are different?

A: The probability of selecting two ages that are different is 2/3. This is because there are 6 possible samples where the two ages are different: (A, B), (A, C), (B, A), (B, C), (C, A), and (C, B), and a total of 9 possible samples.

Q: What is the probability of selecting a specific age, such as age A?

A: The probability of selecting a specific age, such as age A, is 1/3. This is because there are 3 possible samples where age A is selected: (A, A), (A, B), and (A, C), and a total of 9 possible samples.

Q: What is the probability of selecting two ages that are consecutive?

A: The probability of selecting two ages that are consecutive is 1/3. This is because there are 3 possible samples where the two ages are consecutive: (A, B), (B, C), and (C, A), and a total of 9 possible samples.

Q: Can you provide an example of how to calculate the probability of selecting two ages that are the same?

A: Yes, here is an example:

Let's say we want to calculate the probability of selecting two ages that are the same. We know that there are 3 possible samples where the two ages are the same: (A, A), (B, B), and (C, C). We also know that there are a total of 9 possible samples.

To calculate the probability, we can use the following formula:

Probability = (Number of favorable outcomes) / (Total number of possible outcomes)

In this case, the number of favorable outcomes is 3 (the 3 possible samples where the two ages are the same), and the total number of possible outcomes is 9 (the total number of possible samples).

Therefore, the probability of selecting two ages that are the same is:

Probability = 3/9 = 1/3

Q: Can you provide an example of how to calculate the probability of selecting two ages that are different?

A: Yes, here is an example:

Let's say we want to calculate the probability of selecting two ages that are different. We know that there are 6 possible samples where the two ages are different: (A, B), (A, C), (B, A), (B, C), (C, A), and (C, B). We also know that there are a total of 9 possible samples.

To calculate the probability, we can use the following formula:

Probability = (Number of favorable outcomes) / (Total number of possible outcomes)

In this case, the number of favorable outcomes is 6 (the 6 possible samples where the two ages are different), and the total number of possible outcomes is 9 (the total number of possible samples).

Therefore, the probability of selecting two ages that are different is:

Probability = 6/9 = 2/3

Conclusion

In this article, we have answered some of the most frequently asked questions related to the problem of randomly selecting 2 of the ages with replacement. We have provided detailed explanations and examples to help clarify any doubts.

We hope that this article has been helpful in understanding the different possible samples and calculating the probabilities of selecting two ages that are the same and two ages that are different.

References

  • [1] "Combinatorics: Topics, Techniques, Algorithms" by Peter J. Cameron
  • [2] "Probability and Statistics for Engineers and Scientists" by Ronald E. Walpole and Raymond H. Myers

Glossary

  • Combinatorics: The branch of mathematics that deals with counting and arranging objects in various ways.
  • Probability: A measure of the likelihood of an event occurring.
  • Random sampling: The process of selecting a sample from a population in such a way that every possible sample has an equal chance of being selected.