Dayshawn Can Choose Two Of His Four T-shirts To Take On A Weekend Trip. If The T-shirts Are Labeled \[$A, B, C,\$\] And \[$D\$\], Which Choice Represents The Sample Space, \[$S\$\], For The Event?A. \[$S = \{ABCD\}\$\]B.

by ADMIN 221 views

Introduction

In mathematics, the concept of sample space is a fundamental idea in probability theory. It represents the set of all possible outcomes of an experiment or event. In this article, we will explore the concept of sample space and how it applies to a real-world scenario involving Dayshawn's t-shirt choices.

What is Sample Space?

The sample space, denoted by S, is the set of all possible outcomes of an experiment or event. It is a collection of all the possible results that can occur when an experiment is conducted. In other words, it is the set of all possible values that a random variable can take.

Example: Dayshawn's T-Shirt Choices

Let's consider the scenario where Dayshawn has four t-shirts labeled A, B, C, and D, and he needs to choose two of them to take on a weekend trip. We want to find the sample space, S, for this event.

Choosing Two T-Shirts

To choose two t-shirts from the four available, Dayshawn has several options. He can choose any two t-shirts from the set {A, B, C, D}. The order in which he chooses the t-shirts does not matter, as long as he selects two of them.

Calculating the Sample Space

To calculate the sample space, we need to list all the possible combinations of two t-shirts that Dayshawn can choose. We can use the combination formula to find the number of possible combinations:

nCk = n! / (k!(n-k)!)

where n is the total number of items, k is the number of items to choose, and ! denotes the factorial function.

In this case, n = 4 (the total number of t-shirts) and k = 2 (the number of t-shirts to choose). Plugging these values into the formula, we get:

4C2 = 4! / (2!(4-2)!) = 24 / (2 * 2) = 6

So, there are 6 possible combinations of two t-shirts that Dayshawn can choose.

Listing the Sample Space

Now that we know there are 6 possible combinations, we can list them out:

S = {AB, AC, AD, BC, BD, CD}

Conclusion

In conclusion, the sample space, S, for the event of Dayshawn choosing two t-shirts from the set {A, B, C, D} is {AB, AC, AD, BC, BD, CD}. This represents all the possible outcomes of the experiment, and it is the set of all possible values that a random variable can take.

Answer

The correct answer is:

B. S = {AB, AC, AD, BC, BD, CD}

Discussion

This problem illustrates the concept of sample space in mathematics. It shows how to calculate the sample space for a given event and how to list out all the possible outcomes. The sample space is a fundamental idea in probability theory, and it has many applications in real-world scenarios.

Related Concepts

  • Probability: The study of chance events and their likelihood of occurring.
  • Random Variables: A variable that takes on a value randomly from a probability distribution.
  • Experiments: A procedure that is repeated multiple times to collect data.

Real-World Applications

  • Statistics: The study of the collection, analysis, interpretation, presentation, and organization of data.
  • Data Science: The study of extracting insights and knowledge from data.
  • Machine Learning: The study of algorithms that enable machines to learn from data.

Conclusion

Q: What is the sample space in probability theory?

A: The sample space, denoted by S, is the set of all possible outcomes of an experiment or event. It is a collection of all the possible results that can occur when an experiment is conducted.

Q: How do I calculate the sample space for a given event?

A: To calculate the sample space, you need to list out all the possible combinations of the event. You can use the combination formula to find the number of possible combinations:

nCk = n! / (k!(n-k)!)

where n is the total number of items, k is the number of items to choose, and ! denotes the factorial function.

Q: What is the difference between a sample space and a probability distribution?

A: A sample space is the set of all possible outcomes of an experiment or event, while a probability distribution is a function that assigns a probability to each outcome in the sample space.

Q: Can a sample space have more than one outcome?

A: Yes, a sample space can have more than one outcome. In fact, the sample space is a collection of all possible outcomes of an experiment or event.

Q: How do I determine the sample space for a given event?

A: To determine the sample space for a given event, you need to identify all the possible outcomes of the event. You can use the combination formula to find the number of possible combinations.

Q: What is the significance of the sample space in probability theory?

A: The sample space is a fundamental concept in probability theory, as it represents the set of all possible outcomes of an experiment or event. It is used to calculate probabilities and make predictions about the outcome of an event.

Q: Can a sample space be empty?

A: Yes, a sample space can be empty. This occurs when there are no possible outcomes for an event.

Q: How do I represent a sample space mathematically?

A: A sample space can be represented mathematically using set notation. For example, the sample space for the event of rolling a die can be represented as:

S = {1, 2, 3, 4, 5, 6}

Q: What is the relationship between a sample space and a random variable?

A: A sample space is the set of all possible values that a random variable can take.

Q: Can a sample space be infinite?

A: Yes, a sample space can be infinite. This occurs when there are an infinite number of possible outcomes for an event.

Q: How do I calculate the probability of an event given a sample space?

A: To calculate the probability of an event given a sample space, you need to identify the number of outcomes in the sample space that satisfy the event, and then divide that number by the total number of outcomes in the sample space.

Q: What is the significance of the sample space in real-world applications?

A: The sample space is a fundamental concept in many real-world applications, including statistics, data science, and machine learning. It is used to calculate probabilities and make predictions about the outcome of an event.

Conclusion

In conclusion, the sample space is a fundamental concept in probability theory that represents the set of all possible outcomes of an experiment or event. It is used to calculate probabilities and make predictions about the outcome of an event. The sample space can be represented mathematically using set notation, and it can be infinite. The significance of the sample space in real-world applications cannot be overstated, as it is used in many fields, including statistics, data science, and machine learning.