# Axioms of probability

## Partitions

**Definition:** \(E_1, E_2, \dots\) is a **partition** of \(E\) if:

- the \(E_i\)’s are disjoint, i.e., \[E_i \cap E_j = \emptyset \text{ when } i\neq j,\]
- and their union is \(E\), i.e., \(\cup_i E_i = E\).

## Axioms of probability

- A
**probability**is a function \(\mathbb{P}\) that satisfy the following constraints:- \(\mathbb{P}\) should take events as input and return a number between zero and one: \[\mathbb{P}(E) \in [0, 1].\]
*Additivity axiom*: if \(E_1, E_2, \dots\) is a partition of \(E\), then \[\mathbb{P}(E) = \sum_i \mathbb{P}(E_i).\]- \(\mathbb{P}(S) = 1\)

- Thanks to the constraints, even if I only specify a few known probabilities I can recover many other ones mathematically/computationally.