The following can be proved using only set theory and the three axioms of probability.

Probability of an event not happening

P(not A)   =   1 − P(A)

Probability that one of two events happens

For any two events, A and B

P(A or B)   =   P(A)   +  P(B)  −  P(A and B)

(Both proved in full version)

Note that the second result applies to events that may not be mutually exclusive.

Illustration with Venn diagrams

Many results can also be explained using Venn diagrams that represent the whole sample space by a rectangle and events by shapes drawn in it.

Formulae involving addition and subtraction of probabilities correspond to the same formulae about areas in the Venn diagram. For example, the two results above correspond to:

and