Abstract: An evolutionarily stable strategy (ESS), a concept introduced by Maynard-Smith and Price (1973), is a strategy that is immune to invasions by rare alternative ("mutant") strategies. Does an ESS always exist? No: there are finite games with no ESS (unlike Nash equilibria, which always exist in this case). How robust is this non-existence phenomenon? In particular, does an ESS exist for almost every game?Next, consider the convex hull of n random points in the plane. How many vertices are there? This is a classical problem, originally studied by Renyi and Sulanke (1963).We analyze these two problems and get some surprising results, having to do with the underlying "random" distribution.