Describing Events as Combinations — And, Or, Not

And, or, and not describe how events combine, and each word changes the count differently. And asks for outcomes in both events at once, a smaller, more exclusive group. Or asks for outcomes in either event, a bigger group, and that bigger group is where double-counting sneaks in. Not just takes everything outside the event, found by subtracting the event's probability from 1.

Picture two overlapping circles in a Venn diagram, one for event A and one for event B. If you add P(A) and P(B), the outcomes sitting in the overlap, where both events happen, get counted once inside each circle, so they are counted twice total. The addition rule fixes this: P(A or B) = P(A) + P(B) - P(A and B). Subtracting the overlap once leaves it counted exactly once, matching reality. When two events cannot happen together, the overlap is empty, P(A and B) = 0, and the rule simplifies to plain addition: P(A or B) = P(A) + P(B).

This idea sits right next to conditional probability and independence, since P(A and B) often has to be worked out first using those tools before the addition rule can even be applied. It also connects to counting problems, where overlaps correspond to outcomes satisfying two conditions at once, and to expected value problems that sum probabilities across combined events.