# Axiom:Axioms of Uncertainty/Axiom 3

## Axiom

Let $Z$ be a random variable.

Let $Z$ take the values $a_i$ with probability $p_i$, where $i \in \set {1, 2, \ldots, n}$.

Let $H_n: Z^n \to \R$ be a mapping which is to be defined as the uncertainty of $Z$.

$H$ fulfils the following axiom:

$\map {H_n} {p_1, p_2, \ldots, p_n} \ge 0$

and:

$\map {H_n} {p_1, p_2, \ldots, p_n} = 0$ if and only if $\exists i \in \set {1, 2, \ldots, n}: p_i = 1$

That is, uncertainty is always a positive quantity, and only reaches zero when one of the arguments is $1$.