Probability of Complementary Event

Theorem

Given an event A in the probability space (Ω,F,P), the probability of the complementary event is given by:

P(Ac)=P(Ω−A)=1−P(A).
Proof

Let A∈F, we have that:

1=P(S)=P(A∪Ac)=P(A)+P(Ac)

where 1=P(S) follows from the definition of a probability space and the additivity of probabilities on the last line follows from the fact that A and Ac are disjoint by definition, and the property of the measure that allows countable unions of disjoint sets to be measured individually.

Hence P(Ac)=1−P(A).

Corollary

Given events A and B in a probability space:

P(Ac∣B)=1−P(A∣B)

Intuitively this is just because these probabilities are much the same as in the first case, treating B as the underlying probability space itself.