Racing Pundits

[El Salsero Gringo]

Yes it is. Or, formally, let P be the probability black comes up, and P(n) the probability black comes up in the first n tries. Then we have P(n) <= P (because if it comes up in n tries, it certainly comes up). So we have P(n) <= P <= 1. That is, 1-2^-n <= P <= 1. This is true for all n, so P = 1.

[Looking at it really formally, there's a set of disjoint events E_n = "black comes up for the first time on trial n". So that P(E_n) = 2^-n. P(black comes up) = P(union E_n); I can't remember which convergence theorem you use, but basically the fact that everything is disjoint and bounded and there's only a countable number of sets means all the limits have to come out as you'd expect].

I'll take that under advisement, and get back to you. But since it's conceivable that white comes up for ever, the Probability that black ever turns up cannot be exactly 1.

Here's a different example. Suppose X is uniformly distributed on [0,1]; i.e. P(X <=y ) = y for any y in [0,1]. Then the probablilty X equals any particular number is 0. E.g. P(X = .5) = 0. But that doesn't mean X can't equal 0.5; it's got to equal something, and .5 is as likely (or unlikely) as anything else.

Actually, the fact that P(X=0.5) = 0 means that X cannot be 0.5. In fact, it can't be any single number that you nominate. Bizarre, but true. But that's a confusion that you've introduced by introducing a continuous random variable whereas before we were talking about discrete RV's.

Does that help? It's a difficult concept, not least because it's one of the areas where idealised theory and "the real world" don't match. The whole concept of "random number" in the real world is very dubious. For example, if I say pick a number between 0 and 1, the number you give me is very likely to be computable, despite the fact computable numbers are only a countable subset of [0,1] and in an ideal world the probability of getting one would be zero.

Bringing psychology and the inability for me to pick an uncomputable number truly 'at random' doesn't help either!

[David Bailey]

I'll take that under advisement, and get back to you. But since it's conceivable that white comes up for ever, the Probability that black ever turns up cannot be exactly 1.

Mmm... if "for ever" is "infinity", then I believe the infinite series as described does actually converge do 1, from what I recall - bit counter-intuitive, but there you go.

And I could well be wrong, I'm already suffering "Maths degree" flashbacks, so I'll happily accept anyone's word on the matter.

[David Franklin]

I'll take that under advisement, and get back to you. But since it's conceivable that white comes up for ever, the Probability that black ever turns up cannot be exactly 1.

Quoting from wikipedia:

Suppose that a coin is flipped again and again. A sequence heads, heads, heads, ..., ad infinitum, without ever coming up tails, is possible in some sense -- it does not violate any laws of physics to suppose that tails never appears -- but it is very, very improbable. In fact, such a sequence has probability zero.

To argue the other points would be more than a little tedious; we're at the stage where part of the problem is definitions; I'm (unless mistaken!) using standard probability measure theory, but we have all mercilessly abused notation. But the key point is a set can have (probability) measure zero and be non-empty.