Axioms of probability

Axiom 1

For any event A, 0 <= P(A) <= 1

Axiom 2

P(S) = 1

Something in the sample space will occur.

Axiom 3

If A1, A2, …, An are a collection of n mutually exclusive events, then:

P(union of all events) = Sum(1, n)P(Ak)

In other words, the sum of all probabilities of these events is exactly equal to the probability of the event: union of all events.

More generally, if A1, A2, … is an infinite collection of mutually exclusive events, then the probability of the union of those events is the same as the sum of all probabilities of each event.

Consequences

Ac = "A complement"

A ∩ Ac = {} and A ∪ Ac = S, so
1 = P(S) = P(A ∪ Ac) = P(A) + P(Ac) which implies
P(Ac) = 1 - P(A)

If A ∩ B = {}, then P(A ∩ B) = 0 = P({})

P(A ∪ B) = P(A) + P(B) - P(A ∩ B)

You can use these to calculate many probabilities.