TITLE: Why do we say "almost surely" in Probability Theory?
QUESTION [29 upvotes]: I recently asked a question and got a great answer that involved proving that "X is almost surely one of the roots of P".
I know (now) that "almost surely" means "with probability 1", but I've never understood why that phrase exists.
When something has a probability of 1, it's going to happen, no almost about it. What's the story there?  I've tried looking this up online, but I just got more definitions, not meaningful explanations.

REPLY [31 votes]: In terms of the sample space of events $Ω$, an event $E$ happens almost surely if $P(E) = 1$, whereas an event happens surely if $E=Ω $. 
An example: suppose we are (independently) flipping a (fair) coin infinitely many times. The event
$$\{ \text{I will get heads infinitely often}\}$$ is an almost sure event (because it is possible get only a finite number of heads...but how likely is that? Rigorous proof uses Borel Cantelli, if you are interested)
In contrast, $$\{\text{ I will get heads or tails on my 16th flip} \}$$ must happen. This is a sure event.