Almost surely

Definition

Let $(\Omega, \mathcal{F}, P)$ be a probability space. An event $E \in \mathcal{F}$ happens almost surely if $P(E) = 1$. More generally, a property of outcomes holds almost surely (a.s.) if an event of probability $0$ contains all outcomes where it fails.

Remarks

The property need not hold for all outcomes. In this sense, "almost surely" is weaker than "surely". Two random variables $X$ and $Y$ are equal almost surely if $P(X = Y) = 1$. They then have the same law. A countable intersection of events that happen almost surely also happens almost surely. It is the probability form of "almost everywhere" in measure theory.

Explanations

L1 human

If you choose a uniform random number in $[0, 1]$, it is almost surely irrational. Rational outcomes are possible, but together they have probability $0$. If you toss a fair coin forever, you get heads at least once almost surely.