Probability & Statistics

P(A ∪ B) = P(A) + P(B) − P(A ∩ B); subtract the overlap so it is not double-counted.

For independent events, P(A ∩ B) = P(A)·P(B). In general, P(A ∩ B) = P(A)·P(B|A).

P(A|B) = P(A ∩ B) / P(B), valid when P(B) > 0.

They are independent when P(A ∩ B) = P(A)·P(B), equivalently P(A|B) = P(A).

It reverses conditioning: P(A|B) = P(B|A)·P(A) / P(B), where P(B) may be expanded by total probability.

P(B) = Σ P(B|Ai)·P(Ai), summing over the mutually exclusive, exhaustive events Ai.