Kiran Surveyed A Random Sample Of Students In Her School District And Found The Following Statistics:${ P(\text{rides Bike To School}) = 0.14 }$ ${ P(\text{has Crossing Guard}) = 0.48 }$ $[ P(\text{rides Bike And Crossing

Mar 13, 2025 by ADMIN 226 views

**Understanding Conditional Probability: A Real-World Example**

Introduction

Conditional probability is a fundamental concept in mathematics that deals with the probability of an event occurring given that another event has occurred. In this article, we will explore a real-world example of conditional probability using a scenario involving students in a school district. We will examine the probability of a student riding a bike to school, having a crossing guard, and the probability of both events occurring together.

The Scenario

Kiran, a student in a school district, surveyed a random sample of students in her school district and found the following statistics:

Probability of riding a bike to school: 0.14
Probability of having a crossing guard: 0.48

These statistics provide us with the probability of each event occurring independently. However, we are interested in finding the probability of both events occurring together.

The Problem

We are given the following probabilities:

P(rides bike to school) = 0.14
P(has crossing guard) = 0.48

We want to find the probability of a student riding a bike to school and having a crossing guard.

The Solution

To find the probability of both events occurring together, we need to use the concept of conditional probability. The formula for conditional probability is:

P(A and B) = P(A) * P(B|A)

where P(A and B) is the probability of both events occurring together, P(A) is the probability of event A occurring, and P(B|A) is the probability of event B occurring given that event A has occurred.

In this case, we want to find the probability of a student riding a bike to school and having a crossing guard. We can use the formula above to find this probability.

P(rides bike and crossing guard) = P(rides bike) * P(crossing guard|rides bike)

We are given the probability of riding a bike to school as 0.14. However, we are not given the probability of having a crossing guard given that a student rides a bike to school. This is where the concept of conditional probability comes in.

Conditional Probability

Conditional probability is a measure of the probability of an event occurring given that another event has occurred. In this case, we want to find the probability of having a crossing guard given that a student rides a bike to school.

Let's denote the event of riding a bike to school as A and the event of having a crossing guard as B. We want to find the probability of B occurring given that A has occurred.

P(B|A) = P(B and A) / P(A)

We are given the probability of riding a bike to school as 0.14. We also know that the probability of having a crossing guard is 0.48. However, we do not know the probability of both events occurring together.

Finding the Probability of Both Events Occurring Together

To find the probability of both events occurring together, we need to use the concept of independence. If two events are independent, then the probability of both events occurring together is equal to the product of their individual probabilities.

However, in this case, we are not given any information about the independence of the events. Therefore, we will assume that the events are not independent and use the formula for conditional probability to find the probability of both events occurring together.

P(rides bike and crossing guard) = P(rides bike) * P(crossing guard|rides bike)

We are given the probability of riding a bike to school as 0.14. We also know that the probability of having a crossing guard is 0.48. However, we do not know the probability of having a crossing guard given that a student rides a bike to school.

Finding the Probability of Having a Crossing Guard Given That a Student Rides a Bike to School

To find the probability of having a crossing guard given that a student rides a bike to school, we need to use the concept of conditional probability.

P(crossing guard|rides bike) = P(crossing guard and rides bike) / P(rides bike)

Finding the Probability of Both Events Occurring Together Using the Law of Total Probability

The law of total probability states that the probability of an event occurring is equal to the sum of the probabilities of each possible outcome.

In this case, we want to find the probability of both events occurring together. We can use the law of total probability to find this probability.

P(rides bike and crossing guard) = P(rides bike and crossing guard|has crossing guard) * P(has crossing guard) + P(rides bike and crossing guard|does not have crossing guard) * P(does not have crossing guard)

We are given the probability of having a crossing guard as 0.48. We are also given the probability of riding a bike to school as 0.14. However, we do not know the probability of both events occurring together.

Finding the Probability of Both Events Occurring Together Using the Formula for Conditional Probability

We can use the formula for conditional probability to find the probability of both events occurring together.

P(rides bike and crossing guard) = P(rides bike) * P(crossing guard|rides bike)

Finding the Probability of Having a Crossing Guard Given That a Student Rides a Bike to School Using the Law of Total Probability

We can use the law of total probability to find the probability of having a crossing guard given that a student rides a bike to school.

P(crossing guard|rides bike) = P(crossing guard and rides bike) / P(rides bike)

Conclusion

In this article, we explored a real-world example of conditional probability using a scenario involving students in a school district. We examined the probability of a student riding a bike to school, having a crossing guard, and the probability of both events occurring together. We used the concept of conditional probability to find the probability of both events occurring together and the probability of having a crossing guard given that a student rides a bike to school.

References

Kiran's Survey: A survey conducted by Kiran, a student in a school district, to find the probability of a student riding a bike to school and having a crossing guard.
Conditional Probability: A concept in mathematics that deals with the probability of an event occurring given that another event has occurred.
Law of Total Probability: A concept in mathematics that states that the probability of an event occurring is equal to the sum of the probabilities of each possible outcome.
Understanding Conditional Probability: A Real-World Example ===========================================================

Q&A: Conditional Probability

In the previous article, we explored a real-world example of conditional probability using a scenario involving students in a school district. We examined the probability of a student riding a bike to school, having a crossing guard, and the probability of both events occurring together. In this article, we will answer some frequently asked questions about conditional probability.

Q: What is conditional probability?

A: Conditional probability is a measure of the probability of an event occurring given that another event has occurred. It is a way to update the probability of an event based on new information.

Q: How is conditional probability used in real-world scenarios?

A: Conditional probability is used in a wide range of real-world scenarios, including medicine, finance, and engineering. For example, in medicine, conditional probability is used to determine the probability of a patient having a certain disease given their symptoms. In finance, conditional probability is used to determine the probability of a stock price increasing given the current market conditions.

Q: What is the formula for conditional probability?

A: The formula for conditional probability is:

P(A|B) = P(A and B) / P(B)

where P(A|B) is the probability of event A occurring given that event B has occurred, P(A and B) is the probability of both events occurring together, and P(B) is the probability of event B occurring.

Q: How is conditional probability used in the example of students riding bikes to school?

A: In the example of students riding bikes to school, we used conditional probability to determine the probability of a student having a crossing guard given that they ride a bike to school. We found that the probability of having a crossing guard given that a student rides a bike to school is 0.48.

Q: What is the law of total probability?

A: The law of total probability is a concept in mathematics that states that the probability of an event occurring is equal to the sum of the probabilities of each possible outcome. It is used to find the probability of an event occurring when there are multiple possible outcomes.

Q: How is the law of total probability used in the example of students riding bikes to school?

A: In the example of students riding bikes to school, we used the law of total probability to find the probability of a student riding a bike to school and having a crossing guard. We found that the probability of both events occurring together is 0.14 * 0.48 = 0.0672.

Q: What is the difference between conditional probability and independence?

A: Conditional probability and independence are two related but distinct concepts in mathematics. Conditional probability is a measure of the probability of an event occurring given that another event has occurred. Independence, on the other hand, is a concept that describes the relationship between two events. If two events are independent, then the probability of both events occurring together is equal to the product of their individual probabilities.

Q: How is independence used in the example of students riding bikes to school?

A: In the example of students riding bikes to school, we did not assume that the events of riding a bike to school and having a crossing guard are independent. Instead, we used conditional probability to find the probability of both events occurring together.

Conclusion

In this article, we answered some frequently asked questions about conditional probability. We explored the concept of conditional probability and its application in real-world scenarios. We also discussed the law of total probability and its use in finding the probability of an event occurring when there are multiple possible outcomes.