Topics Probability

Question Suppose that you had two bags each with an infinite number of blue marbles. Suppose you also had another bag of infinity red marbles. If you mixed those three bags what are your odds of getting a red marble? Obviously this isn't a realistic experiment but is it 1 in 3 or 50%?

Accepted:
May 3, 2012

Comments

The intuitive answer seems to be "1 in 3," and I think that's the right answer if each infinite set of marbles has the same size (or "cardinality"). I take it you're wondering if the infinite size of the sets invalidates the intuitive answer. I don't think it does. Maybe this analogy will help. There are infinitely many even whole numbers and infinitely many even plus odd whole numbers, but there aren't twice as many of the latter as there are of the former: the cardinality of the two sets is the same. Yet the odds that a randomly chosen whole number is even are surely only 1 in 2 (rather than 1 in 1). If that reasoning is sound, then the fact that the various sets are infinite doesn't affect the probability.

I'd suggest that there needn't be a determinate answer without adding more detail. In particular, the notion of "mixing" the three collections would need to be spelled out. Suppose the "mixing" works this way: take 10 marbles from the red bag and one from each of the blue bags. Put in an infinite vat and stir. Repeat ad infinitum. (We could imagine the first operation is performed in 1 minute, the second in half a minute, the third in a 1/4 minute…) The intuitive thought is that a "random" draw is most likely to give you a red marble. (10 chances out of 12).

This may seem contrived, but only because we have some other loose, unspecified idea of mixing that we're comparing it to. The point is simply that the problem, as stated, doesn't determine the answer.