What is the formula for calculating the probability when events are not mutually exclusive?

When calculating the probability of two events that are not mutually exclusive, it is important to consider the possibility that both events can occur simultaneously. The correct formula accounts for this overlap by subtracting the probability of both events occurring together from the sum of the probabilities of each individual event.

The formula P(a or b) = P(a) + P(b) - P(ab) effectively avoids double-counting the intersection of events a and b. When a and b can occur at the same time, just adding their probabilities would inflate the result because it counts the overlap twice. By subtracting P(ab), which represents the probability of both events occurring together, the calculation accurately reflects the true likelihood of either event occurring.

This method is essential for assessing risk and making informed decisions in industrial hygiene, where overlapping exposures could impact worker safety. Therefore, recognizing the relationships between events and applying the correct formula ensures that probability assessments are precise and meaningful.