Subset of the Sample Space

Oct 13, 2025 by windailys.com 19 views

Introduction

In the world of probability and statistics, understanding sample spaces and their complements is crucial for making informed decisions. In this article, we will explore the concept of complements in sample spaces, using a real-life scenario as an example. Joe, Keitaro, and Luis are three friends who love playing tennis together. To decide who will play against each other in the first match, they put their names in a hat and choose two names without looking. This scenario will help us understand the concept of complements in sample spaces.

What is a Sample Space?

A sample space is the set of all possible outcomes of an experiment or a situation. In the context of the tennis tournament, the sample space consists of all possible pairs of names that can be chosen from the hat. Let's denote the sample space as ${$ A $}$. The sample space can be represented as:

${$ A $}$ = {(Joe, Keitaro), (Joe, Luis), (Keitaro, Joe), (Keitaro, Luis), (Luis, Joe), (Luis, Keitaro)}

What is the Complement of a Sample Space?

The complement of a sample space is the set of all possible outcomes that are not in the sample space. In other words, it is the set of all possible pairs of names that are not chosen from the hat. Let's denote the complement of the sample space as ${$ A^c $}$. The complement of the sample space can be represented as:

${$ A^c $}$ = {(Joe, Joe), (Keitaro, Keitaro), (Luis, Luis), (Joe, Luis), (Keitaro, Joe), (Luis, Keitaro)}

Subset of the Sample Space

The question asks us to find the subset of the sample space that represents the complement of the sample space. In other words, we need to find the subset of the sample space that contains all possible pairs of names that are not chosen from the hat. From the previous section, we know that the complement of the sample space is:

${$ A^c $}$ = {(Joe, Joe), (Keitaro, Keitaro), (Luis, Luis), (Joe, Luis), (Keitaro, Joe), (Luis, Keitaro)}

However, we are asked to find the subset of the sample space that represents the complement of the sample space. This means that we need to find the subset of the sample space that contains all possible pairs of names that are not chosen from the hat, excluding the pairs that are already in the sample space.

Solution

To find the subset of the sample space that represents the complement of the sample space, we need to exclude the pairs that are already in the sample space from the complement of the sample space. The pairs that are already in the sample space are:

{(Joe, Keitaro), (Joe, Luis), (Keitaro, Joe), (Keitaro, Luis), (Luis, Joe), (Luis, Keitaro)}

We need to exclude these pairs from the complement of the sample space:

${$ A^c $}$ = {(Joe, Joe), (Keitaro, Keitaro), (Luis, Luis), (Joe, Luis), (Keitaro, Joe), (Luis, Keitaro)}

The subset of the sample space that represents the complement of the sample space is:

{(Joe, Joe), (Keitaro, Keitaro), (Luis, Luis)}

Conclusion

In this article, we explored the concept of complements in sample spaces using a real-life scenario as an example. We defined the sample space and the complement of the sample space, and then found the subset of the sample space that represents the complement of the sample space. The subset of the sample space that represents the complement of the sample space is {(Joe, Joe), (Keitaro, Keitaro), (Luis, Luis)}.

References

[1] "Probability and Statistics" by Michael A. Bean
[2] "Mathematics for Computer Science" by Eric Lehman and Tom Leighton

Glossary

Sample Space: The set of all possible outcomes of an experiment or a situation.
Complement of a Sample Space: The set of all possible outcomes that are not in the sample space.
Subset of a Sample Space: A set of outcomes that is a part of the sample space.
Tennis Tournament: Understanding Complements in Sample Spaces - Q&A ====================================================================

Introduction

In our previous article, we explored the concept of complements in sample spaces using a real-life scenario as an example. We defined the sample space and the complement of the sample space, and then found the subset of the sample space that represents the complement of the sample space. In this article, we will answer some frequently asked questions related to complements in sample spaces.

Q&A

Q: What is a sample space?

A: A sample space is the set of all possible outcomes of an experiment or a situation. In the context of the tennis tournament, the sample space consists of all possible pairs of names that can be chosen from the hat.

Q: What is the complement of a sample space?

A: The complement of a sample space is the set of all possible outcomes that are not in the sample space. In other words, it is the set of all possible pairs of names that are not chosen from the hat.

Q: How do I find the complement of a sample space?

A: To find the complement of a sample space, you need to identify all possible outcomes that are not in the sample space. This can be done by listing all possible outcomes and then removing the ones that are already in the sample space.

Q: What is a subset of a sample space?

A: A subset of a sample space is a set of outcomes that is a part of the sample space. In the context of the tennis tournament, a subset of the sample space could be the set of all possible pairs of names that are chosen from the hat.

Q: How do I find the subset of a sample space that represents the complement of the sample space?

A: To find the subset of a sample space that represents the complement of the sample space, you need to exclude the pairs that are already in the sample space from the complement of the sample space.

Q: What is the difference between a sample space and a subset of a sample space?

A: A sample space is the set of all possible outcomes of an experiment or a situation, while a subset of a sample space is a set of outcomes that is a part of the sample space.

Q: Can a subset of a sample space be equal to the sample space?

A: Yes, a subset of a sample space can be equal to the sample space. For example, if the sample space consists of all possible pairs of names that can be chosen from the hat, then the subset of the sample space that consists of all possible pairs of names that are chosen from the hat is equal to the sample space.

Q: Can a subset of a sample space be empty?

A: Yes, a subset of a sample space can be empty. For example, if the sample space consists of all possible pairs of names that can be chosen from the hat, then the subset of the sample space that consists of no pairs of names is empty.

Conclusion

In this article, we answered some frequently asked questions related to complements in sample spaces. We defined the sample space and the complement of the sample space, and then discussed how to find the subset of the sample space that represents the complement of the sample space. We also discussed the difference between a sample space and a subset of a sample space, and how a subset of a sample space can be equal to or empty.

References

[1] "Probability and Statistics" by Michael A. Bean
[2] "Mathematics for Computer Science" by Eric Lehman and Tom Leighton

Glossary

Sample Space: The set of all possible outcomes of an experiment or a situation.
Complement of a Sample Space: The set of all possible outcomes that are not in the sample space.
Subset of a Sample Space: A set of outcomes that is a part of the sample space.