[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]