Probability Of Event A Given Union And Conditional Probabilities

by ADMIN 65 views

Introduction

In probability theory, understanding the relationships between events is crucial for making informed decisions and predictions. This article delves into a specific problem involving two events, A and B, defined within a sample space S. We are given the probability of their union, P(A ∪ B) = 0.84, and the conditional probability P(B|A) = 0.41, with the added condition that A and B are independent events. Our goal is to determine the probability of event A, P(A). This problem showcases the interplay between fundamental probability concepts such as union, conditional probability, and independence.

The solution requires careful application of the definitions and formulas associated with these concepts. We will begin by outlining the necessary theoretical background, including the formula for the probability of a union of events and the definition of conditional probability. The condition of independence will also play a key role, as it simplifies the relationship between P(A ∩ B) and the individual probabilities P(A) and P(B). By combining these elements, we can construct an equation that allows us to solve for the unknown probability P(A). The final answer will be a numerical value rounded to five decimal places, adhering to the precision required in many practical applications of probability theory. Through a step-by-step approach, this article aims to provide a clear and comprehensive understanding of the problem and its solution, making it accessible to both students and practitioners of probability.

Theoretical Background

Before diving into the solution, it is essential to review the core concepts and formulas that will be used. The probability of the union of two events, P(A ∪ B), is given by the formula:

P(A ∪ B) = P(A) + P(B) - P(A ∩ B)

This formula states that the probability of either event A or event B occurring (or both) is the sum of their individual probabilities minus the probability of both occurring simultaneously (A ∩ B). This subtraction is necessary to avoid double-counting the outcomes that belong to both A and B. Understanding this formula is crucial for solving problems involving the probabilities of combined events.

Next, let's consider the definition of conditional probability. The conditional probability of event B given event A, denoted as P(B|A), is defined as:

P(B|A) = P(A ∩ B) / P(A)

This formula expresses the probability of event B occurring, given that event A has already occurred. It is a fundamental concept in probability theory and is widely used in various fields such as statistics, finance, and engineering. The conditional probability allows us to update our beliefs about the likelihood of an event based on new information. In the context of this problem, P(B|A) = 0.41 provides valuable information about the relationship between events A and B.

The final key concept is the independence of events. Two events A and B are said to be independent if the occurrence of one does not affect the probability of the other. Mathematically, this is expressed as:

P(A ∩ B) = P(A) * P(B)

This simple yet powerful equation is a cornerstone of probability theory. It allows us to simplify calculations when dealing with independent events, as the joint probability (P(A ∩ B)) can be expressed as the product of the individual probabilities. In this problem, the condition that A and B are independent is crucial for linking the given information and solving for the unknown P(A). Understanding and applying these theoretical concepts is essential for tackling complex probability problems.

Problem Setup

In this specific problem, we are given the following information:

  • P(A ∪ B) = 0.84: The probability of the union of events A and B.
  • P(B|A) = 0.41: The conditional probability of event B given event A.
  • A and B are independent events: This condition simplifies the relationship between P(A ∩ B) and the individual probabilities P(A) and P(B).

Our goal is to determine the probability of event A, denoted as P(A). To achieve this, we will leverage the theoretical background discussed earlier, combining the formulas for the union of events, conditional probability, and independence. The problem setup provides a clear roadmap for the solution, as it highlights the knowns and the unknown, and it sets the stage for the subsequent steps.

The key to solving this problem lies in recognizing how these pieces of information fit together. The union probability provides a relationship between P(A), P(B), and P(A ∩ B). The conditional probability provides another relationship between P(A ∩ B) and P(A). The independence condition further simplifies the expression for P(A ∩ B). By strategically combining these relationships, we can create an equation that allows us to isolate and solve for P(A). This problem setup exemplifies a common approach in probability problem-solving: identifying the key relationships and using them to construct a solvable equation. The next section will delve into the step-by-step solution, illustrating how these concepts are applied in practice.

Step-by-Step Solution

To determine the probability of event A, we will follow a step-by-step approach, utilizing the given information and the theoretical concepts discussed earlier.

  1. Express P(A ∩ B) using the conditional probability formula:

    We know that P(B|A) = P(A ∩ B) / P(A). Therefore, we can rewrite P(A ∩ B) as:

    P(A ∩ B) = P(B|A) * P(A)

    Substituting the given value of P(B|A) = 0.41, we get:

    P(A ∩ B) = 0.41 * P(A)

    This step is crucial as it expresses the intersection probability in terms of the unknown P(A), setting the stage for further substitutions and simplifications.

  2. Express P(A ∩ B) using the independence condition:

    Since A and B are independent events, we know that P(A ∩ B) = P(A) * P(B). This provides an alternative expression for the intersection probability, which will be valuable in linking the given union probability.

  3. Substitute into the union probability formula:

    The formula for the probability of the union of two events is P(A ∪ B) = P(A) + P(B) - P(A ∩ B). Substituting the given value P(A ∪ B) = 0.84, we have:

    0. 84 = P(A) + P(B) - P(A ∩ B)

    Now, we substitute P(A ∩ B) = 0.41 * P(A) from Step 1 into this equation:

    0. 84 = P(A) + P(B) - 0.41 * P(A)

    Simplifying, we get:

    0. 84 = 0.59 * P(A) + P(B)

    This equation relates P(A) and P(B), but we still need to eliminate P(B) to solve for P(A).

  4. Eliminate P(B) using the independence condition:

    From Step 2, we know that P(A ∩ B) = P(A) * P(B). We also know from Step 1 that P(A ∩ B) = 0.41 * P(A). Equating these two expressions, we get:

    P(A) * P(B) = 0.41 * P(A)

    Assuming P(A) is not zero (otherwise, the problem becomes trivial), we can divide both sides by P(A):

    P(B) = 0.41

    This crucial step provides the value of P(B), which allows us to solve for P(A).

  5. Solve for P(A):

    Now that we have P(B) = 0.41, we can substitute it back into the equation from Step 3:

    0. 84 = 0.59 * P(A) + 0.41

    Subtracting 0.41 from both sides, we get:

    0. 43 = 0.59 * P(A)

    Dividing both sides by 0.59, we find:

    P(A) = 0.43 / 0.59 ≈ 0.72881

    Therefore, the probability of event A is approximately 0.72881.

  6. Round the answer to five decimal places:

    Rounding 0.72881 to five decimal places, we get:

    P(A) ≈ 0.72881

    This final step ensures that the answer meets the specified precision requirement.

By following this step-by-step solution, we have successfully determined the probability of event A, utilizing the given information and the fundamental concepts of probability theory. Each step builds upon the previous one, demonstrating a logical and methodical approach to problem-solving in probability.

Final Answer

Therefore, the probability of event A, rounded to five decimal places, is 0.72881.

Conclusion

In this article, we addressed a probability problem involving the determination of the probability of an event A, given the probability of the union of two events (A and B), the conditional probability P(B|A), and the condition that A and B are independent. The problem required a careful application of fundamental probability concepts such as the formula for the union of events, the definition of conditional probability, and the condition of independence.

The solution involved a step-by-step approach, starting with expressing P(A ∩ B) in terms of P(A) using the conditional probability formula and the independence condition. We then substituted these expressions into the union probability formula, which allowed us to create an equation relating P(A) and P(B). By further utilizing the independence condition, we were able to eliminate P(B) and solve for P(A). The final answer was obtained by rounding the result to five decimal places, as required.

This problem highlights the importance of understanding the relationships between different probability concepts and how they can be combined to solve complex problems. The systematic approach used in this article serves as a valuable framework for tackling similar problems in probability theory. Moreover, the solution demonstrates the practical application of theoretical concepts, reinforcing the understanding of probability principles. By mastering these concepts and techniques, one can confidently address a wide range of probability problems in various fields, including statistics, finance, and engineering. The ability to analyze and solve probability problems is a valuable skill in today's data-driven world, and this article provides a solid foundation for developing that skill.