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A proper definition would probably (hah) use the [[Kolmogorov axioms]].
Intuition: probability is "chance". However: something like 50% is not exact. If
something has the probability 50% we don't expect it to happen exactly 50 times
out of 100 tries. Rather, we expect
$$\lim_{n \rightarrow \infty} \mathrm{N}(\mathrm{heads}) = 50\% \cdot \mathrm{N}(n \ \mathrm{throws})$$
# Some rules
$$P(S) = 1$$
$$P(\phi) = 0$$
$$P(A') = 1 - P(A)$$
$$P(A_1 \cup A_2 \cup ...) = P(A_1) + P(A_2) + ...$$
$$P(A \cup B) = P(A) + P(B) - P(A \cap B)$$
$$P(A \cup B \cup C) = P(A) + P(B) + P(C) - P(A \cap B) - P(B \cap C) - P(A \cap C) + P(A \cap B \cap C)$$
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