What is Uniform distribution?

$X$ follows the uniform distribution $\mathcal U\intcc ab$ if its density is constant, $f(t) = \frac{1}{b-a}$ on $\intcc{a}{b}$. Then for $\intcc{c}{d} \subseteq \intcc ab$, $\P(c \leq X \leq d) = \frac{d - c}{b - a}$: probability proportional to length.