An event which occurs with probability 1 is said to occur almost surely. Note that in an infinite probability space, this is not quite the same as saying it cannot conceivably occur! For instance, suppose I toss a fair coin until I get a head. Then the probability of tossing precisely k times is 2-k, and the probability of it never landing heads-up is . But it could conceivably happen! (It just never does).
Indeed, Kolmogorov's 0-1 law says that huge class of events in the above probability space occur almost surely.
In a finite probability space, "almost surely" isn't very interesting. But suppose we have a "natural" probability measure for every finite problem size n. An event is sometimes said to occur almost surely if the probability of it occurring tends to 1 as n increases.
For instance, define a random graph on n vertices by taking n vertices and connecting any 2 by an edge with probability p, independently of all other edges. Then any finite subgraph occurs almost surely in this space.