# Probability distribution (countable sample space)

# Definition

A probability distribution on a countable Sample space $\Omega$ is a Function \(\mathbb{P}: \Omega \to [0,1]\) such that \(\sum_{\omega \in \Omega} \mathbb{P}(\omega) = 1\).

# Intuition

We express a belief that “$x\in \Omega$ happens with probability \(r\)” by setting \(\mathbb{P}(x) = r\). So a probability distribution divides up our anticipation of what will happen, out of the set \(\Omega\) of things that might possibly happen.

examples, futher points

Parents:

- Probability theory
The logic of science; coherence relations on quantitative degrees of belief.