Just to make one more observation about "almost impossible events". Consider the experiment of picking a random real number in the interval I := [0, 1) (say, but it's not really important, using an uniform random distribution) and the question "How probable is it that the picked number is rational"?
The set of non-rational numbers in I is uncountable, whereas the set of rational numbers in the same interval is countable. It means that the probability of picking a non-rational number is infinitely larger than that of picking a rational number and, consequently, the probability of picking a rational number is 0. However, nothing prevents you from picking 0.5, which is a perfectly possible result of your experiment.
As Dror points out, some events can have probability 0 and this does not mean that there is no possibility for them to occur.
What is the probability of an almost impossible event?
Mathematically speaking, it is 0.
"Almost" has a well-defined mathematical meaning. A function can be almost everywhere continuous, meaning that it is continuous everywhere but in a countable set of places. The Dirac Delta function is almost everywhere zero.
no, the explanation of "almost everywhere continuous" is not the one given in the last paragraph. The truth is more complex than this, and the linked Wikipedia article gives a correct explanation.