  By adopting a Multilevel Monte Carlo (MLMC) framework, we show that only a
handful of costly fine scale computations are needed to accurately estimate
statistics of the failure of a composite structure, as opposed to the thousands
typically needed in classical Monte Carlo analyses. We introduce the MLMC
method, compare its theoretical complexity with classical Monte Carlo, and give
a simple-to-implement algorithm which includes a simple extension called MLMC
with selective refinement to efficiently calculated structural failure
probabilities. To demonstrate the huge computational gains we present two
benchmark problems in composites: (1) the effects of fibre waviness on the
compressive strength of a composite material, (2) uncertain buckling
performance of a composite panel with uncertain ply orientations. For our most
challenging test case, estimating a rare ($\sim 1/150$) probability of buckling
failure of a composite panel, we see a speed-up factor $> 1000$. Our approach
distributed over $1024$ processors reduces the computation time from $218$ days
to just $4.5$ hours. This level of speed up makes stochastic simulations that
would otherwise be unthinkable now possible.?
