Understanding the Relationship between Not Independent and Mutually Exclusive Events

It is a common misconception that if two events are not independent, they must be mutually exclusive. However, this is not necessarily true. Understanding the definitions and relationships between these concepts is crucial for a clear grasp of probability theory.

Definitions

Independent Events: Two events A and B are considered independent if the occurrence of one does not affect the probability of the other occurring. Mathematically, this is expressed as:

PA ∩ B PA · PB

Mutually Exclusive Events: Two events A and B are mutually exclusive or disjoint if they cannot occur at the same time. This means:

PA ∩ B 0

Not Independent Events

Not independent events imply that the occurrence of one event affects the probability of the other. However, this does not necessarily mean that they cannot occur together. For instance, if event A occurs, it might increase the probability of event B occurring, but both can still happen simultaneously. This kind of relationship indicates that the events are dependent, but the form of this dependence is not mutually exclusive.

Mutual Exclusivity

Mutually exclusive events cannot occur together. If one event happens, the other cannot. Therefore, mutually exclusive events are inherently dependent since the occurrence of one excludes the possibility of the other.

Conclusion

Thus, while mutually exclusive events are definitely dependent, because the occurrence of one excludes the possibility of the other, not independent events can still occur together. This means that not independent events do not imply mutual exclusivity. The dependency between not independent events can range from exclusive to non-exclusive, and understanding this distinction is essential for accurate probability analysis.

Further Clarification

Being not independent signifies that there exists some form of relationship between these events, but this relationship is not fully known without additional information. This relationship could be mutual exclusivity, where the dependency of one event on the other means that if one exists, the other cannot. Alternatively, it could be the case that one event is a subset of another, meaning the occurrence of the first event necessarily implies the occurrence of the other.

While theory may provide a more mathematical and rigorous approach, the aim of this explanation is to make these concepts accessible and understandable to everyone. Thank you to all, including Amit Goyal and others, who contributed valuable insights and their time.

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