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.
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$\begingroup$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.
$\endgroup$ 8 $\begingroup$There is a difference between "almost surely" and "surely."
Consider choosing a real number uniformly at random from the interval $[0,1]$. The event "$1/2$ will not be chosen" has probability $1$, but is not impossible.
I recommend reading the relevant Wikipedia article, which I found very clarifying when I was learning probability.
$\endgroup$ 1 $\begingroup$Subtle difference is present.
If some event occurs, then it has probability measure $1$. But if some event occurs with probability $1$, then the event need not occur, and the event that the event does not occur has probability measure $0$.
Note that probability is not about what happenED, for if so then we do not need probability theory. Probability is about what may happen.
So it is not that a random walk must go back to the origin, but that it will go back to the origin with probability $1$.
You may be referred to the original, axiomatic treatment of probability by A. Kolmogorov
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