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BGonline.org Forums
Bad news from Belgium - OT questions
Posted By: Timothy Chow In Response To: Bad news from Belgium - OT questions (Casper van der Tak)
Date: Monday, 11 October 2010, at 2:12 p.m.
If you want to say that the probability is "infinitesimal" then first you need to define carefully what an infinitesimal is and how to do arithmetic with infinitesimals. There are various ways that this could be done, but none of them gives rise to a good theory in which the probability of picking a real number is the reciprocal of "the number of real numbers."
Here's the kind of paradox that can arise. Let c denote the "number of real numbers." The "Cantor set" is by definition the set of real numbers between 0 and 1 whose base-three expansion has no 1's in it (i.e., the expansion is an infinite string of 0's and 2's). There are c real numbers in the Cantor set; indeed, by pretending that the 2's are 1's and reading the string as a binary string, we get a one-to-one correspondence between the numbers in the Cantor set and all the real numbers between 0 and 1. So what is the probability of picking a number in the Cantor set? Logically it would seem to be c * (1/c) = 1, right? There are c numbers each of which can get picked with probability 1/c.
On the other hand, one can show that the probability of picking something not in the Cantor set is given by an infinite series that converges to 1. So logically, the probability of picking something in the Cantor set should be infinitesimal. But certainly 1 is not infinitesimal.
There seems to be no way around this paradox and others like it, even by appealing to infinitesimals, except by arbitrarily declaring that the probability of picking any individual real number is zero. This declaration does lead to a good theory in the sense that it gives you sensible results for any calculation you really want to do. The one thing you lose is the pre-theoretical intuition that "probability zero" should mean "impossible." For under the standard theory, as you note, probability zero events happen all the time. However, people have decided that this counterintuitive result is something one can live with and get used to, since it doesn't lead to any contradictory calculations. Besides, the "probability zero events that happen nevertheless" are only theoretical entities. A point of infinitesimal size never gets picked in reality; in reality you always pick some number with a finite number of decimal places, which corresponds to an interval of very small but nonzero width, and the probability of picking something in that interval is certainly nonzero. So as long as you take care not to over-idealize your real-world problem, these strange probability-zero events won't ever arise. (You could even take the point of view that it's good that the theory sets these probabilities to zero because it thereby warns you that you may be over-idealizing, while at the same time doing the right thing to prevent you from getting into trouble if you insist on considering such things.)
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