Impossible Event - Asymptotically Almost Surely

In asymptotic analysis, one says that a property holds asymptotically almost surely (a.a.s.) if, over a sequence of sets, the probability converges to 1. For instance, a large number is asymptotically almost surely composite, by the prime number theorem; and in random graph theory, the statement "G(n,p_n) is connected" (where G(n,p) denotes the graphs on n vertices with edge probability p) is true a.a.s when p_n > for any ε > 0.

In number theory this is referred to as "almost all", as in "almost all numbers are composite". Similarly, in graph theory, this is sometimes referred to as "almost surely".