Combining events: AND, OR, NOT

Real questions rarely involve a single event. "Rain and a delay." "A king or a heart." Two rules — one for or, one for and — combine probabilities, and each has a trap.

The OR rule (addition)

For the chance that A or B happens, add their probabilities — but subtract the overlap so you don't count it twice:

$P(A \cup B) = P(A) + P(B) - P(A \cap B)$

If the events are mutually exclusive (they can't both happen, like rolling a 2 or a 5), the overlap is zero and it simplifies to just P(A) + P(B).

The AND rule (multiplication)

For the chance that A and B both happen, multiply. When the events are independent — one doesn't affect the other, like two separate coin flips — it's clean:

$P(A \cap B) = P(A) \cdot P(B)$

So two heads in a row is 1/2 · 1/2 = 1/4. Three in a row, 1/8. Independent chances multiply, and multiplying fractions shrinks fast — which is why long winning streaks are rare.

Two pairs of words people mix up

They are not the same. Mutually exclusive events are actually highly dependent: if one happens, the other definitely didn't.

Why this matters

These two rules are the grammar of probability. Reliability engineering ("will any component fail?"), genetics ("inherit both traits?"), and everyday risk all reduce to stacking ands and ors — multiply for and, add-minus-overlap for or.