Index: The Book of Statistical ProofsGeneral Theorems ▷ Probability theory ▷ Probability axioms ▷ Axioms of probability

Definition: Let there be a sample space $\Omega$, an event space $\mathcal{E}$ and a probability measure $P$, such that $P(E)$ is the probability of some event $E \in \mathcal{E}$. Then, we introduce three axioms of probability:

  • First axiom: The probability of an event is a non-negative real number:
\[\label{eq:prob-ax1} P(E) \in \mathbb{R}, \; P(E) \geq 0, \; \text{for all } E \in \mathcal{E} \; .\]
  • Second axiom: The probability that at least one elementary event in the sample space will occur is one:
\[\label{eq:prob-ax2} P(\Omega) = 1 \; .\]
  • Third axiom: The probability of any countable sequence of disjoint (i.e. mutually exclusive) events $E_1, E_2, E_3, \ldots$ is equal to the sum of the probabilities of the individual events:
\[\label{eq:prob-ax3} P\left(\bigcup_{i=1}^\infty E_i \right) = \sum_{i=1}^\infty P(E_i) \; .\]

Metadata: ID: D158 | shortcut: prob-ax | author: JoramSoch | date: 2021-07-30, 11:11.