basic question

[BonJoviJones]

"Probability of 1 and dead certainty are two different things."

Really? What's the definition of "Probability of 1" then?

(I swear, I learn as much on the Probability forum as I have in some of my math courses)

[RocketManJames]

Yes, it's not difficult to prove that there isn't a one-to-one correspondence via Cantor's diagnolization trick. But, I know one thing for sure... there's no way I'd figure it out on my own unless I read about his diagonalization and what it meant about cardinalities of infinite sets.

In any case, the reason I brought the whole thing up was to perhaps stir up interest in those who had never heard of the idea of varying degrees of infinity.

RMJ

[RocketManJames]

BruceZ may have a much better answer to this, but this is how I understand it...

In terms of finite sets, then yes, probability of 1 implies 100% certainty, and 100% certainty implies probability 1.

However, if you're not talking about finite sets, then the implications hold only in one direction. That is, 100% certainty implies that the probability is 1. But, probability of 1 does not imply 100% certainty.

Similarly, you can see how probability 0 is not the same as impossibility.

RMJ

i hate infinity
if you believe the universe is infinite in all directions, then you can say we are at the center of the universe
also, the percentage of all rational integers that contain the number 3 is 100% (because of that damned infinity again)

Well, in my vocabulary, probability 1 and 100% certainty amounts to the same thing. As pointed out above, the problem arises when considering infinite spaces where probability zero of an event, strangely does not correspond to the event being impossible in a theoretical sense.

Anyway, the subject is mostly an interesting mind job, since picking a truly random element from an infinite set cannot be done in the real world. We can only make (almost) random selections of sets of elements from a finite set.

[Candyman]

I posted this [img]/forums/images/icons/smile.gif[/img]

[Mason Malmuth]

This guy knows his stuff.

MM