Probability of Complementary Event

Theorem

Given an event \(A\) in the probability space \((\Omega, \mathcal{F}, P)\), the probability of the complementary event is given by:

\[ P(A^{c}) = P(\Omega - A) = 1 - P(A).\]

Proof

Let \(A \in \mathcal{F}\), we have that:

\[\begin{align*} 1 &= P(S) \\ &= P(A \cup A^{c}) \\ &= P(A) + P(A^{c}) \\ \end{align*}\]

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

Hence \(P(A^{c}) = 1 - P(A)\).

Corollary

Given events \(A\) and \(B\) in a probability space:

\[ P(A^{c} \mid B) = 1 - P(A \mid B)\]

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