(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.

Log in or register to write something here or to contact authors.