firstname.lastname@example.org wrote: > > I overheard a conversation today (among my undergrad peers, not grads or > professors or anything) where some guy was explaining to another that if > you throw darts at the real number line, the probability of hitting a > rational is 0 because there are only aleph_0 rationals and there are > 2^aleph_0 irrationals. When I first heard this I kinda thought it was > nonsense. The probability of hitting a rational _CAN'T_ be 0, because, > well, they exist and you _COULD_ actually hit one! Then I started > thinking that it kinda makes sense in a way to say that if you throw a > dart at the integers, you have a 20% chance of hitting one divisible by > 5, even though there are aleph_0 integers, and aleph_0 integers > divisible by 5. (Well, maybe it doesn't after all....) > > I guess I want to know how to state this "throwing darts" business more > precisely. I don't have any sort of formal or rigorous understanding of > what is meant by "probability", but that is probably the key to > resolving this matter. I'm guessing that "probability" is not well > defined or applicable to infinite sets in this way.
I think the crux is that for any definition of probability that applies to infinite sets in this way, you cannot also define 'a probability of zero' as meaning the same as 'cannot happen'.