2.3 Axioms of Probability

Probability law cannot be arbitrary, it must satisfy the Axioms of Probability:

  1. Non-negativity: \(\mathbb{P}[A] \geq 0 \text{, for any } A \subseteq \Omega\)

  2. Normalization: \(\mathbb{P}[\Omega] = 1\)

  3. Additivity: For any disjoint sets \(\{A_1, A_2, ...\}\) \[ \mathbb{P}\left[\bigcup_{i=1}^{\infty}A_i\right]=\sum_{i=1}^{\infty}\mathbb{P}[A_i] \]

Why these three axioms?

  • Axiom I (Non-negativity) ensures that probability is never negative.
  • Axiom II (Normalization) ensures that probability is never greater than 1.
  • Axiom III (Additivity) allows us to add probabilities when two events do not overlap. Natural extension from naive probability (counting)

Example

For the case where \(\Omega = [0,1]\) and \(\mathbb{P}[E] = \int_a^b f(x) dx\) we require (Axiom I) \(f(x) \geq 0\) and (Axiom II) \(\mathbb{P}[\Omega] = \int_0^1 f(x) dx =1\). Axiom III is also satisfied due to the additivity of integration. For example, consider two (non overlapping) intervals \(E_1 = [a,b]\) and \(E_2 = [b,c]\)

\[ \mathbb{P}[E_1 \cup E_2] = \int_a^c f(x) dx\\ = \int_a^b f(x) dx + \int_b^c f(x) dx\\ = \mathbb{P}[E_1] + \mathbb[E_2] \]