A probability space is Omega, an event space, and P: E -> [0,+\infty)

Why do we use infinite sums (A_1...A_\infty) instead of just finite sum?

Ex. relevant question: I have a fair coin and I flip it until I get a
tails. Are you sure you'll eventually get tails?

Yes.

Lemma: What are the odds that n coin flips result in no tails?
Complement: What are the odds that n coin flips result in only heads?

Each flip is independent and fair, so:

P(A_1 \cap A_2 \cap \ldots \cap A_n) = 1/2^n
P("""^C) = 1 - 1/2^n

Using an infinite union A_1...A_\omega = lim_n->\infty P(A_1\cap...A_n)

This axiom defines this relation.

Properties of probabilities (can be derived from original axioms):
    For all A \in E, P(A^C) = 1-P(A)
    For all A,B \in E, if A \subset B, P(A) \leq P(B)
    For all A,B \in E, P(A \cup B) = P(A)+P(B)-P(A\cap B)
        Can construct Poincare lemma or inclusion/exclusion.