  Let $K_{n,n}$ be the complete bipartite graph with $n$ vertices in each side.
For each vertex draw uniformly at random a list of size $k$ from a base set $S$
of size $s=s(n)$. In this paper we estimate the asymptotic probability of the
existence of a proper colouring from the random lists for all fixed values of
$k$ and growing $n$. We show that this property exhibits a sharp threshold for
$k\geq 2$ and the location of the threshold is precisely $s(n)=2n$ for $k=2$,
and approximately $s(n)=\frac{n}{2^{k-1}\ln 2}$ for $k\geq 3$.
