One over infinity is not a zero at all, because it's an ill-defined operation. You need to say exactly what you mean, and which number system you're working in. Are you in the surreals? Non-standard analysis? The hyper-reals?
If you work with standard analysis, your comment seems to be a non-sequiteur. If not, you need to say which system you're in. Personally, I think the other systems are largely unhelpful, although independently interesting.
And of course an infinitesimal is not zero (nil being a different technical term).
If I've missed the point(s) you were tying to make, you'll need to be more explicit. Probably further "discussion" here is not going to be fruitful, but I'll be interested to read anything you care to add.
I'm sorry, I'm not a statistician (or any kind of mathematician) so I don't know the proper terminology. I'm just noting that the only way your previous example seems work from my perspective is if you take one over infinity to be zero.
I'll explain: The set of all of the points between two points is an infinite set, and the probability of selecting any single member of a set is approximately one over the size of the set, so it would seem that the probability of selecting any individual member of an infinite set would be one over 'infinity' that is, an undefined value.
Granted, the limit of one over x as x approaches infinity is zero, but the probability of selecting any single member of an infinite set isn't zero in the same sense that the probability of selecting a member outside the set is zero. That is, if I pick a point between 0 and 1, and I select point x, there was obviously not impossible to select point x. So while it may be 'mathematically convenient'* to say that the probability of selecting any given point is zero I think that it might be better to say that it might be better to say that the chance is infinitesimally small, rather than zero per se.
So while I may certainly be wrong, I think that you are univocating when you say that a range is an example of an infinite number of zeros being non-zero. When I was talking about the chance of a individual monkey typing out Shakespeare was zero, I meant that, if it was impossible for any monkey to type (imagine that we are using exclusively dead monkeys) no monkey could ever type out anything, much less Shakespeare.
The problem is that if you assign a probably to a point, then the probability you assign has to be strictly smaller that every positive number, so it has to be zero. The system that works with finite sets of taking the number of things you're interested in and dividing by the number of things that might happen doesn't apply. Contradictions and/or inconsistencies arise from that sort of thinking, seductive though it is.
My example about the ranges was an attempt to show that the otherwise excellent ideas from the finite world simply don't apply, so you have to do something else. You suggested that ...
> If the probability of producing the desired result
> in any given attempt is zero, then the probability
> of producing the desired result given infinite
> attempts is still zero.
... and that's what I was refuting.
And perhaps the things to take away from this is that just because the probability of an event is zero, that doesn't mean the event can't happen. Talking about the probability being "infinitesimally small" requires that you make a whole pile of other definitions precise, and that way lies madness. Or at least, non-standard analysis, which is pretty close to the same thing.
In my opinion it doesn't help the understanding of the "interested layman."
All this, of course, is only valid in the theoretical musings of mathematicians (or those doing math at the time) In the "real world" it's all different again.
Doing this sort of thing correctly and consistently is hard.
If you work with standard analysis, your comment seems to be a non-sequiteur. If not, you need to say which system you're in. Personally, I think the other systems are largely unhelpful, although independently interesting.
And of course an infinitesimal is not zero (nil being a different technical term).
If I've missed the point(s) you were tying to make, you'll need to be more explicit. Probably further "discussion" here is not going to be fruitful, but I'll be interested to read anything you care to add.