A Student Surveyed 200 Students To Determine The Number Who Have A Dog And Those Who Have A Cat.Let \[$ A \$\] Be The Event That A Student Has A Dog And \[$ B \$\] Be The Event That A Student Has A Cat. The Student Finds That \[$

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Introduction

In probability theory, events are often denoted by letters such as A and B. These events can be related to each other in various ways, including being mutually exclusive, independent, or complementary. In this article, we will explore the concept of events and their relationships using a real-world example of a student surveying 200 students to determine the number who have a dog and those who have a cat.

The Survey

A student surveyed 200 students to determine the number who have a dog and those who have a cat. Let { A $}$ be the event that a student has a dog and { B $}$ be the event that a student has a cat. The student finds that 80 students have a dog and 120 students have a cat.

The Probability of Events A and B

The probability of an event is a measure of the likelihood of that event occurring. In this case, we can calculate the probability of events A and B as follows:

  • The probability of event A (having a dog) is 80/200 = 0.4.
  • The probability of event B (having a cat) is 120/200 = 0.6.

The Relationship Between Events A and B

Events A and B are not mutually exclusive, meaning that a student can have both a dog and a cat. In fact, 20 students have both a dog and a cat.

The Probability of the Union of Events A and B

The probability of the union of events A and B is the probability that a student has either a dog or a cat. This can be calculated using the formula:

P(A or B) = P(A) + P(B) - P(A and B)

Substituting the values we have calculated, we get:

P(A or B) = 0.4 + 0.6 - 0.2 = 0.8

The Probability of the Intersection of Events A and B

The probability of the intersection of events A and B is the probability that a student has both a dog and a cat. This is given by the value of P(A and B), which is 0.2.

The Probability of the Complement of Event A

The probability of the complement of event A is the probability that a student does not have a dog. This can be calculated as follows:

P(not A) = 1 - P(A) = 1 - 0.4 = 0.6

The Probability of the Complement of Event B

The probability of the complement of event B is the probability that a student does not have a cat. This can be calculated as follows:

P(not B) = 1 - P(B) = 1 - 0.6 = 0.4

Conclusion

In this article, we have explored the concept of events and their relationships using a real-world example of a student surveying 200 students to determine the number who have a dog and those who have a cat. We have calculated the probability of events A and B, the probability of the union of events A and B, the probability of the intersection of events A and B, and the probability of the complement of events A and B. These calculations demonstrate the importance of understanding the relationships between events in probability theory.

Discussion

The concept of events and their relationships is a fundamental aspect of probability theory. In this article, we have used a real-world example to illustrate the importance of understanding these relationships. The calculations we have performed demonstrate the power of probability theory in analyzing complex situations.

References

  • [1] Probability Theory by E.T. Jaynes
  • [2] Probability and Statistics by James E. Gentle

Further Reading

  • [1] Probability Theory and Statistics by William Feller
  • [2] Probability and Statistics for Engineers and Scientists by Ronald E. Walpole

Glossary

  • Event: A set of outcomes in a sample space.
  • Probability: A measure of the likelihood of an event occurring.
  • Mutually Exclusive: Events that cannot occur at the same time.
  • Independent: Events that do not affect each other's probability.
  • Complementary: Events that are not mutually exclusive and have a probability of 1.

Tags

  • Probability Theory
  • Events
  • Relationships
  • Calculations
  • Real-World Example
  • Student Survey
  • Dogs
  • Cats
  • Probability
  • Union
  • Intersection
  • Complement

Introduction

In our previous article, we explored the concept of events and their relationships using a real-world example of a student surveying 200 students to determine the number who have a dog and those who have a cat. In this article, we will answer some frequently asked questions related to the topic.

Q&A

Q1: What is the probability of a student having a dog?

A1: The probability of a student having a dog is 0.4, which is calculated as 80/200.

Q2: What is the probability of a student having a cat?

A2: The probability of a student having a cat is 0.6, which is calculated as 120/200.

Q3: What is the relationship between events A and B?

A3: Events A and B are not mutually exclusive, meaning that a student can have both a dog and a cat.

Q4: What is the probability of the union of events A and B?

A4: The probability of the union of events A and B is 0.8, which is calculated using the formula P(A or B) = P(A) + P(B) - P(A and B).

Q5: What is the probability of the intersection of events A and B?

A5: The probability of the intersection of events A and B is 0.2, which is the value of P(A and B).

Q6: What is the probability of the complement of event A?

A6: The probability of the complement of event A is 0.6, which is calculated as 1 - P(A).

Q7: What is the probability of the complement of event B?

A7: The probability of the complement of event B is 0.4, which is calculated as 1 - P(B).

Q8: Can a student have both a dog and a cat?

A8: Yes, a student can have both a dog and a cat.

Q9: How many students have both a dog and a cat?

A9: 20 students have both a dog and a cat.

Q10: What is the probability of a student not having a dog?

A10: The probability of a student not having a dog is 0.6, which is calculated as 1 - P(A).

Q11: What is the probability of a student not having a cat?

A11: The probability of a student not having a cat is 0.4, which is calculated as 1 - P(B).

Q12: What is the relationship between the probability of an event and its complement?

A12: The probability of an event and its complement are complementary, meaning that they add up to 1.

Conclusion

In this article, we have answered some frequently asked questions related to the topic of a student surveying 200 students to determine the number who have a dog and those who have a cat. We hope that this Q&A article has provided valuable information and insights into the concept of events and their relationships.

Discussion

The concept of events and their relationships is a fundamental aspect of probability theory. In this article, we have used a real-world example to illustrate the importance of understanding these relationships. The calculations we have performed demonstrate the power of probability theory in analyzing complex situations.

References

  • [1] Probability Theory by E.T. Jaynes
  • [2] Probability and Statistics by James E. Gentle

Further Reading

  • [1] Probability Theory and Statistics by William Feller
  • [2] Probability and Statistics for Engineers and Scientists by Ronald E. Walpole

Glossary

  • Event: A set of outcomes in a sample space.
  • Probability: A measure of the likelihood of an event occurring.
  • Mutually Exclusive: Events that cannot occur at the same time.
  • Independent: Events that do not affect each other's probability.
  • Complementary: Events that are not mutually exclusive and have a probability of 1.

Tags

  • Probability Theory
  • Events
  • Relationships
  • Calculations
  • Real-World Example
  • Student Survey
  • Dogs
  • Cats
  • Probability
  • Union
  • Intersection
  • Complement