Almost surely
In mathematics, the phrase almost surely is a subtle, precise way to say that something is certain except for cases that almost never happen, though still possible. The concept comes into play only in problems that involve infinite time or infinitesimal space.
The locution almost surely is to probability theory as almost everywhere is to measure theory. Formally it is equivalent to "with probability 1". For example, imagine throwing a dart at the unit square; the probability that the dart lands in any subregion of the square is the area of that subregion. The area of the diagonal of the square is zero, so the probability that the dart lands exactly on the diagonal is zero. But the diagonal is not the empty set; a point on the diagonal is no less probable than is any point at which the dart could land. One says that the dart will almost surely not land on the diagonal. In other words an event is "almost sure" if the probability of its complement is zero, even though the complement may not be empty.
Instead of infinitesimal space, one may consider infinite time.
Suppose that a coin is flipped again and again.
A sequence heads, heads, heads, ..., ad infinitum,
without ever coming up tails, is possible in some sense -- it does not violate any laws of physics to suppose that tails never appears -- but it is very, very improbable.
In fact,
such a sequence has probability zero.
However, any real sequence must have a finite length,
and any possible realization has probability greater than zero.
The difficulty comes when the sequence is hypothetically extended to infinite length. It is common to talk about almost sure convergence or divergence of random variables.
An example of the fine distinction between 'sure' and 'almost sure' can be found in the difference between constant and almost surely constant random variables. In measure theoretic probability theory these two types of random variable are not identical, but for practical purposes they are equivalent, since if a constant random variable X and an almost surely constant random variable Y represent ths same constant c, then they share the same distribution functions.
