Wednesday, April 19, 2017

How likely are you to be in a random finite subset of an infinite set?

Suppose that out of a set of infinitely many people, including you, a finite subset is chosen at random. How likely are you to be in that subset? Intuitively, not very likely. And the larger the infinity, the less likely.

But how do you pick out a finite subset at random? Here’s a natural way. First, pick out a subset at random, by flipping a fair coin for each person in the original set, and including a person in the subset if the comes up heads. Almost surely, this will generate an infinite subset (a consequence of the law of large numbers). But suppose this experiment is repeated—perhaps uncountably infinitely often—until the set picked out is finite. (This construction requires that the set of potential repetitions be well-ordered.) Or maybe you just get lucky, and to everybody’s surprise the set picked out is finite.

So now we have a method for picking out a finite subset at random (though it may take some luck). How likely are you to be in that finite subset?

Well, think about it step-by-step. Before you learned that the set picked out by the heads was finite, your probability that you were in the set was the probability that your coin landed heads, i.e., 1/2. Then you learn that the set of people for whom heads was rolled is finite. But this fact tells you nothing about your coin toss. For the claim that the set of people with heads is finite is logically equivalent to the claim that the set of people other than you with heads is finite. And the latter claim tells you nothing about your coin toss.

So, your probability needs to stay at 1/2.

Thus, the probability that a random finite subset of the infinitely many people includes you is finite. This is a little counterintuitive when the infinity is countable. And it becomes far more counterintuitive the larger this infinity gets. It is a stupendously implausible claim when that infinity is large, say ℶω.

Causal finitism blocks the story by making it impossible for you to find out that the set of people who got heads is finite.


Dagmara Lizlovs said...

This sounds like selection for jury duty in my county. I just got hit with it again so I must be in the right subset. :-)

IanS said...


It is both intuitive and true (or at least reasonably arguable) that conditional on any particular finite number of heads, your probability of heads is zero (0r infinitesimal). But nothing need follow about your probability of heads conditional on an unspecified finite number of heads.

To be sure, non-conglomerability is in itself counterintuitive. But granted this, I don’t see setups with bigger infinities of people as any more counterintuitive.

An interesting question: Is there an argument or intuition for zero probability of inclusion that applies directly to ‘an unspecified finite number of heads’, without implicitly assuming conglomerability? I can’t think of one, but there may be.

Alexander R Pruss said...

Maybe, but suppose we replace finite by countable, and make the set of coins have continuum cardinality.

Then conditional on a *specified* number (namely, aleph-0) of heads much smaller than the continuum, your probability of heads should be zero. But it's not.

Of course, I've also smuggled conglomerability into this argument. While the *number* is specified, the set is not.

So maybe one can chalk this up to yet another way of highlighting how strange non-conglomerability is.

Here's another paradox in the vicinity. Suppose all the coins are labeled with the real numbers between 0 and 1. You learn that there are only finitely (or countably) many heads. So by the argument in the post, you still have probability 1/2 of heads.

Suppose you get a prize on tails. Let S be the set of coins that came out heads. Now, you are offered a special deal: you can switch your coin for a coin uniformly chosen from the interval [0,1]. It seems that you should switch, since the probability that the you will get a number in S is zero. But it seems silly to think you should always switch.

Alexander R Pruss said...

Maybe the second point is still dependent on conglomerability. For maybe gambling on a known set of known measure is different from gambling on an unknown set of known measure.