(probability:)

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.

(combinatorics:)

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.