1.6.3 Inclusion-Exclusion Principle

The axiom of additivity allows us to find the probability of mutually exclusive events by simply adding their individual probabilities. However, when events are not mutually exclusive, we can not simply add the probabilities as doing so double counts the probability of any outcome which is in both events. For instance, if the probability of rain is 0.6 and the probability of wind is 0.8, the probability of rain or wind will be 1.4 which is incorrect as the probability can’t exceed 1. Since rain and wind can occur simultaneously, they are not mutually exclusive events, and therefore, their probabilities can’t be added directly.

This is where the inclusion-exclusion principle comes into play. This principle allows us to modify the additivity axiom by subtracting the probability of the intersection of the two events, ensuring an accurate calculation of the overall probability in such cases. The formula for the principle is:

Fig 1.15 depicts the union of two non-mutually exclusive events. The intersection area,   , is included in both circles (highlighted by the brown region). To ensure accurate probability calculations, it is essential to subtract this overlapping area once, as it would otherwise be counted twice.