Addition Theorem of Probability

A frequent business question involves the probability that one event OR another occurs — will a customer buy product A or product B, will a machine fail due to fault X or fault Y. The addition theorem answers exactly this, for two events $A$ and $B$ defined on the same sample space:

The subtraction of $P(A\cap B)$ exists precisely because, when $A$ and $B$ overlap (share some common outcomes), simply adding $P(A)$ and $P(B)$ would double-count every outcome that lies in both.

Mutually exclusive events are events that cannot occur together — they share no common outcome at all, so $P(A\cap B)=0$, and the formula simplifies:

A classic example of mutually exclusive events: rolling a single die and getting 'an even number' versus 'an odd number' — no outcome is both, so the two probabilities can simply be added. By contrast, 'an even number' and 'a number greater than 3' on a die do overlap (both include 4 and 6), so the general (subtract-the-overlap) form must be used instead. …