# Probability

This article is all about the basics of probability. There are two interpretations of a probability, but the difference only matters when we will consider inference.

• Frequency
• The degree of belief

### Axioms of Probability

A function $$P$$ which assigns a value $$P(A)$$ to every event $$A$$ is a probability measure or probability distribution if it satisfies the following three axioms.

1. $$P(A) \geq 0 \text{ } \forall \text{ } A$$
2. $$P(\Omega) = 1$$
3. If $$A_1, A_2, …$$ are disjoint then $$P(\bigcup_{i=1}^{\infty} A_i) = \sum_{i=1}^{\infty} P(A_i)$$

These axioms give rise to the following five properties.

1. $$P(\emptyset) = 0$$
2. $$A \subset B \Rightarrow P(A) \leq P(B)$$
3. $$0 \leq P(A) \leq 1$$
4. $$P(A^\mathsf{c}) = 1 – P(A)$$
5. $$A \cap B = \emptyset \Rightarrow P(A \cup B) = P(A) + P(B)$$