Comments on: Creating modern probability

By: Peter Cameron

Peter Cameron — Sat, 20 Oct 2012 09:27:22 +0000

What we are looking at is the proportion of occurrences of A in infinitely many independent trials. Maybe I didn’t say it very well.

The paradox is that, if \Omega={H,T} and A={H}, then p could be anything, say 1/3. Then the event that the proportion of occurrences of A either fails to converge or converges to anything other than 1/3 is a null set. So the null sets are not just determined by the structure of the set of sequences, but depend on numbers fed in at the start.

This contrasts with Baire category where the meagre sets are determined independently of any structure on \Omega (if it is discrete).

By: Stubborn Mule

Stubborn Mule — Sat, 20 Oct 2012 08:52:54 +0000

I don’t understand your paradox example. You make reference to the independent trials A_n, but then the event you refer to makes no reference to A_n. Also, if I understand the point correctly, if \Omega is {H, T} and A = {H} then presumably p is 1/2, but isn’t is perfectly possible to define a probability measure on \Omega such that P({A}) takes on any value between 0 and 1?

By: Ralph Dratman

Ralph Dratman — Sat, 20 Oct 2012 04:08:31 +0000

So interesting.