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In probability theory, mutually exclusive events are events that cannot happen at the same time. If one event occurs, the other cannot. For example, when flipping a coin, the outcomes "heads" and "tails" are mutually exclusive because both cannot appear in a single flip.
The concept of mutually exclusive events dates back to the early development of probability theory in the 17th century. Mathematicians like Blaise Pascal and Pierre de Fermat laid the groundwork for the formal study of probability through their correspondence in the 1650s. The distinction between mutually exclusive and non-mutually exclusive events became clearer as the study of probability evolved, especially with the work of Andrey Kolmogorov in the 20th century, who formalized probability theory.
When dealing with mutually exclusive events, the probability of either event happening is simply the sum of their individual probabilities. This is based on the idea that the occurrence of one event excludes the possibility of the other, meaning they cannot overlap.
For example, if Event A has a probability of P(A)P(A) and Event B has a probability of P(B)P(B), and A and B are mutually exclusive, the probability of either event A or event B happening is:
P(A∪B)=P(A)+P(B)
Here, P(A∪B) represents the probability of either event A or event B occurring.
For two mutually exclusive events A and B:
P(A∪B)=P(A)+P(B)
P(A) = Probability of event A
P(B) = Probability of event B
P(A∪B) = Probability of event A or event B happening
For more than two events that are mutually exclusive, the formula can be extended:
P(A1∪A2∪⋯∪An)=P(A1)+P(A2)+⋯+P(An)
where A1, A2, …, An are mutually exclusive events.
Mutually exclusive events cannot happen at the same time.
The probability of either event occurring is the sum of their individual probabilities.
Non-mutually exclusive events, on the other hand, can occur together, and their combined probability requires subtracting the intersection (overlap) of events to avoid double-counting.