Solving a Problem using Bayes' Theorem

Solving a Problem using Bayes' Theorem

Table of Contents:

1. Introduction
2. Problem Statement
3. Tree-Based Approach 3.1 Creating the Tree Diagram 3.2 Calculating the Probability using the Tree-Based Approach
4. Formula-Based Approach 4.1 Understanding the Base Theorem 4.2 Applying the Base Theorem to the Problem 4.3 Calculating the Probability using the Formula-Based Approach
5. Conclusion

Introduction

In this article, we will explore a problem related to the base theorem. The problem involves two taxi companies, A and B, and finding the probability that a taxi belongs to Company B if it is involved in an accident. We will discuss two approaches to solve this problem: the tree-based approach and the formula-based approach. Both approaches provide a clear understanding of how the probability is calculated.

Problem Statement

The problem states that Company A has 60% of the taxis in the city, while Company B has the remaining 40%. It is Mentioned that 3% of Company A's taxis are involved in accidents, while 6% of Company B's taxis are involved in accidents. We need to find the probability that a taxi, involved in an accident, belongs to Company B. The given options for this problem are analyzed using two approaches.

Tree-Based Approach

To solve the problem using the tree-based approach, let's consider a hypothetical example. Suppose there are 1000 taxis in the city. Based on the given probabilities, 60% of the taxis (600) belong to Company A, while 40% (400) belong to Company B. Additionally, 3% of taxis from Company A (18) and 6% of taxis from Company B (24) are involved in accidents. Drawing a tree Diagram helps Visualize the problem.

Creating the Tree Diagram

By creating the tree diagram, we can depict the number of taxis involved in accidents for each company. From Company A, 18 taxis are involved in accidents, while 582 taxis are not. From Company B, 24 taxis are involved in accidents, while 376 taxis are not.

Calculating the Probability using the Tree-Based Approach

To find the probability that an accident-involved taxi belongs to Company B, we divide the number of taxis from Company B involved in accidents (24) by the total number of accident-involved taxis (42). This calculation gives us a probability value of 0.57, which aligns with option B.

Formula-Based Approach

The formula-based approach involves using the Bayes' theorem to calculate the required probability.

Understanding the Base Theorem

The formula for Bayes' theorem is as follows: the probability of A given B is equal to the probability of B given A multiplied by the probability of A, divided by the probability of B.

Applying the Base Theorem to the Problem

Using the given probabilities, we can Apply Bayes' theorem to find the probability that a taxi belongs to Company B, given that it is involved in an accident. By substituting the provided values into the formula, including the probability of an accident given that it belongs to Company B (0.06), the probability of B (0.4), and the overall probability of an accident (0.42), we determine the probability to be 0.57.

Conclusion

In this article, we discussed a problem related to the base theorem, which involved calculating the probability that a taxi belongs to Company B if it is involved in an accident. We explored two approaches: the tree-based approach and the formula-based approach. Both methods led to the same conclusion - a probability value of 0.57, supporting option B. Understanding such problems and applying the appropriate approach helps analyze and solve similar scenarios effectively.