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

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.

[todo: examples, futher points]