Definition

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

Intuition

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