If two events are mutually exclusive, we know that the...

If two events are mutually exclusive, we know that the probability of their union is the sum of their individual probabilities. However, this is not the case for events that are not mutually exclusive. Verify this assertion by considering the events $A$ and $B$ of Exercise 3.2.

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Key Concepts

Addition Rule for Probabilities

This rule provides a method for calculating the probability of the union of two events. When events are not mutually exclusive, the probability of their union is given by the sum of the probabilities of each event minus the probability of their intersection, ensuring that the overlapping outcomes are not counted twice.

Inclusion-Exclusion Principle

A fundamental counting principle in probability that correctly accounts for overlapping events. It ensures that when combining the probabilities of events that are not mutually exclusive, the intersection (shared outcomes) is subtracted from the total to avoid overestimation.

Mutually Exclusive Events

These are events that cannot occur at the same time, meaning the occurrence of one event excludes the possibility of the other. In probability theory, when two events are mutually exclusive, the probability of their union is simply the sum of their individual probabilities because they do not overlap.

Non-Mutually Exclusive Events

These are events that may occur simultaneously, meaning they have some overlap. The fact that one event can occur with another necessitates an adjustment in calculating their combined probability to account for the overlapping outcomes.

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