  This is the fourth paper in a series. We prove a conjecture made
independently by Boston et al and Shalev. The conjecture asserts that there is
an absolute positive constant delta such that if G is a finite simple group
acting transitively on a set of size n > 1, then the proportion of derangements
in G is greater than delta. We show that with possibly finitely many
exceptions, one can take delta = .016. Indeed, we prove much stronger results
showing that for many actions, the proportion of derangements goes to 1 as n
increases and prove similar results for families of permutation
representations.
