Do "almost surely" means probability 1 or approaching 1? When mathematicians say that something is almost surely to happens what does it means? Does it means that has probability 1 of happening or a value approaching 1?
 A: "Almost surely" means that the probability is exactly equal to $1$. I think I know why such a question arises, so let me elaborate over an example.
Suppose we are tossing a coin $n$ times. What is the probability that we get at least one head? Well, the probability of this not happening is $\frac{1}{2^n}$, so the probability of there being some heads is $1-\frac{1}{2^n}$. This probability tends to $1$ as $n\rightarrow\infty$. So does this mean that if we toss a coin infinitely many times we are going to have at least one head almost surely?
The implication is not quite immediate. In certain way, throwing coin an infinite number of times is a "limit" of throwing coins $n$ times, and because of that in this "limit" the probability of getting at least one head is equal to $1$. So it is true that in infinite tosses we will get a head almost surely, because that event has probability $1$, but this is implied by the fact that the probabilities for finite number of tosses tend to $1$.
This distinction is important because in mathematics very often some processes cannot be thought of as "limits" of some finite processes (for example, randomly choosing a real number). In such contexts, speaking of the probability "approaching" $1$ simply doesn't make any sense. But an event has its own probability nevertheless, and it might be equal to $1$, in which case we say it happens almost surely.
