Complement Rule - Probability

Let A be an event, A' be its complement, and S be the sample space. By the definition of the complement, the union of an event and its complement is equal to the entire sample space. A⋃ A' = S Because A⋃ A' and S represent the same event, their probabilities are equal. P(A⋃ A') = P(S) Since the complement of A consists of the outcomes that are not in A, events A and A' are disjoint. By the Addition Rule of Probability, the probability of the union is the sum of the individual probabilities of each of the events. P(A⋃ A') = P(A) + P(A') Now the Transitive Property of Equality can be applied to the equalities. P(A⋃ A') = P(A) + P(A') P(A⋃ A') = P(S) ⇓ P(A) + P(A') = P(S) Additionally, the probability of the entire sample space P(S) is equal to 1. By applying the Transitive Property of Equality once more, the rule is proven.

Using the Subtraction Property of Equality, the formula for the probability of A is obtained. P(A) + P(A') = 1 ⇕ P(A) = 1 - P(A')