Understanding Mutually Exclusive and Exhaustive Events in Probability Theory

In probability theory, understanding the concepts of mutually exclusive and exhaustive events is fundamental. This article delves into the details of these concepts and illustrates how they can be applied to solve a specific problem involving two events, A and B. The problem states that A and B are mutually exclusive and exhaustive events, and also that the probability P(B) is twice that of P(A). This article will guide you through the step-by-step solution to find P(A).

Understanding Mutually Exclusive Events

Mutually exclusive events are those in which the occurrence of one event precludes the occurrence of the other. In other words, if event A occurs, event B cannot occur, and vice versa. This is denoted as:

P(A and B) 0

Understanding Exhaustive Events

Exhaustive events, on the other hand, are those that cover all possible outcomes in a given sample space. Therefore, the sum of the probabilities of all possible outcomes must equal 1. In other words:

P(A) P(B) ... 1

The Given Problem

The problem specifies that A and B are mutually exclusive and exhaustive events, and that the probability of B (denoted as P(B)) is twice the probability of A (denoted as P(A)).

Given Information: Pa (P(A)) x Pb (P(B)) 2x Since A and B are exhaustive, Pa Pb 1

Step-by-Step Solution

Substituting the Relationship

Given that Pb 2 * Pa, we can substitute this into the equation for exhaustive events:

Pa 2 * Pa 1

Solving for Pa

Combining the terms, we get:

3 * Pa 1

Dividing both sides by 3, we get:

Pa 1 / 3

Finding Pb

Since Pb 2 * Pa, we can now find Pb as follows:

Pb 2 * (1 / 3) 2 / 3

Conclusion

Therefore, the solution to the problem is:

Pa 1 / 3 Pb 2 / 3

This problem illustrates the application of the concepts of mutually exclusive and exhaustive events in probability theory, and it highlights the importance of understanding these concepts for solving complex probability problems.

Additional Considerations and Verification

In the context of the problem, it's important to verify all conditions:

That Pb (2 / 3) is less than or equal to 1 to meet the requirement of a probability. The sum of Pa and Pb (1 / 3 2 / 3) equals 1, confirming they are exhaustive events.

A global perspective is crucial for confirming that every possibility covered is accounted for, and that the probabilities are correctly bounded.