We give a new method for analysing the mixing time of a Markov chain using
path coupling with stopping times. We apply this approach to two hypergraph
problems. We show that the Glauber dynamics for independent sets in a
hypergraph mixes rapidly as long as the maximum degree \Delta of a vertex
and the minimum size m of an edge satisfy m\geq 2\Delta+1. We also show
that the Glauber dynamics for proper q-colourings of a hypergraph mixes
rapidly if m\geq 4 and q > \Delta, and if m=3 and q\geq1.65\Delta.
We give related results on the hardness of exact and approximate counting
for both problems.