Over in this thread, we were discussing the counterintuitive nature of probability when the sample space is infinite. I came up with what I thought was a very interesting question, deserving of its own thread. Here goes.

Let’s say you’ve got a uniform distribution on [0, 1]. IOW, if B is a subset of [0, 1], the probability of picking an element of B is just the Lebesgue measure of B (call that m(B)).

As we all know, m(**Q**) = 0. So the probability of picking a rational number is zero. Doesn’t mean it’ll never happen (see the earlier referenced thread for an explanation), but it does have probability zero.

Here’s the question: what’s the probability of picking a rational number less than 1/2, given that you’ve picked a rational number? Intuitively, it’s 1/2, but we all know what intuition’s worth in higher math.

There are two ways to look at it that I can think of.[ol]

[li]Let L denote the event that we pick a number less than 1/2, and Q denote the event that we pick a rational number. In this case, we want P(LQ|Q). By the standard formula for conditional probabilities and the idempotence of set intersection, P(LQ|Q) = P(LQ)/P(Q). If picking a rational number is independent from picking a number less than 1/2, then P(LQ|Q) = P(L)P(Q)/P(Q), and we might be able to cancel, to get P(LQ|Q) = P(L). But that cancellation is a little fishy, as P(Q) = 0.[/li][li]Let L denote the event that we pick a rational number less than 1/2, and let Q denote the event that we pick a rational number. Now we want P(L|Q) (note that LQ = L here, as L is a subset of Q). Again, by the formula for conditional probability, P(L|Q) = P(LQ)/P(Q). Since LQ = L, P(L|Q) = P(L)/P(Q). No hope here, I think.[/ol]So there’s the quandary. Is there any hope of resolving this? Is there any theoretical reason to prefer the first analysis over the second?[/li]

On preview, I’m pretty sure that in the first case, picking a rational number is independent from picking a number less than 1/2. LQ is a subset of Q, and a subset of a set of measure zero must have measure zero. Since we’ve got a uniform distribution, P(LQ) = m(LQ) = 0. P(L)P(Q) = 0, as P(Q) = 0. So the question remains, is P(L)P(Q)/P(Q) = L?