Axioms of Probability — Section 1: Foundations of Probability

Kolmogorov's three axioms are the foundation that all of probability theory builds on. A function $P$ defined on subsets of a sample space $\Omega$ is a probability if and only if it satisfies:

Non-negativity: $P(A) \ge 0$ for every event $A$ . You can't have a negative probability.

Normalization: $P(\Omega) = 1$ . The total probability across all possible outcomes is $1$ .

Countable additivity: for any countable collection of pairwise disjoint events $A_1, A_2, \dots$ ,

P\!\left(\bigcup_i A_i\right) = \sum_i P(A_i)

These three axioms are minimal — but every classical result in probability follows from them. Some immediate consequences:

$P(\emptyset) = 0$
$P(A^c) = 1 - P(A)$
For any two events, $P(A \cup B) = P(A) + P(B) - P(A \cap B)$ (inclusion–exclusion)

The last one matters because $A$ and $B$ might overlap. If they do, naively adding $P(A) + P(B)$ would double-count the intersection, so we subtract it back.

Kolmogorov's three axioms are the foundation that all of probability theory builds on. A function $P$ defined on subsets of a sample space $\Omega$ is a probability if and only if it satisfies:

Non-negativity: $P(A) \ge 0$ for every event $A$ . You can't have a negative probability.

Normalization: $P(\Omega) = 1$ . The total probability across all possible outcomes is $1$ .

Countable additivity: for any countable collection of pairwise disjoint events $A_1, A_2, \dots$ ,

P\!\left(\bigcup_i A_i\right) = \sum_i P(A_i)

These three axioms are minimal — but every classical result in probability follows from them. Some immediate consequences:

$P(\emptyset) = 0$
$P(A^c) = 1 - P(A)$
For any two events, $P(A \cup B) = P(A) + P(B) - P(A \cap B)$ (inclusion–exclusion)

The last one matters because $A$ and $B$ might overlap. If they do, naively adding $P(A) + P(B)$ would double-count the intersection, so we subtract it back.