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 $E_2 = [b,c]$

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