  Let $H$ and $K$ be subgroups of a finite group $G$. Pick $g \in G$ uniformly
at random. We study the distribution induced on double cosets. Three examples
are treated in detail: 1) $H = K = $ the Borel subgroup in
$GL_n(\mathbb{F}_q)$. This leads to new theorems for Mallows measure on
permutations and new insights into the LU matrix factorization. 2) The double
cosets of the hyperoctahedral group inside $S_{2n}$, which leads to new
applications of the Ewens's sampling formula of mathematical genetics. 3)
Finally, if $H$ and $K$ are parabolic subgroups of $S_n$, the double cosets are
`contingency tables', studied by statisticians for the past 100 years.
