Thursday, October 23, 2014

Yet another probability paradox

You know for sure that infinitely many people, including yourself, each are independently tossing fair coins. You don't see your coin's result. But then you learn for sure something amazing: only finitely many of the coins came up heads. This is extremely unlikely—indeed, by the Law of Large Numbers it has zero probability—but it seems nonetheless possible. What probability should you now assign to your coin being heads?

Intuition: Very small, maybe zero, maybe infinitesimal.

Here's an argument, however, that you should stick to your guns and continue to assign 1/2. Let F be the proposition that only finitely many of the coins landed heads. Let G be the proposition that of the coins other than yours, only finitely many of the coins landed heads. Learning G does not affect your probability that your coin landed heads. The coins are all independent, so no information about the other tosses tells you about yours. But, now, necessarily (given the setup that you toss only one coin) F is true if and only if G is true. For your coin won't make the difference between infinitely and finitely many heads. So learning F does not affect your probability that your coin landed heads.

To make sticking to your guns even more amazing, note that this works for any infinity of people, even a very high uncountable infinity. Wow!


Mike Almeida said...

It definitely sounds right that you should assign .5 to the chances that your specific coin falls heads. But you can imagine not knowing which coin is yours, which I think is equivalent to not knowing which coin-flipper you are. In that case, the chances that you are a heads-tossing-coin-flipper is affected by the knowledge that there are just finitely many of them. You should conclude that the chances are 0 that you are one of those.

Alexander R Pruss said...

But you can stipulate for your specific coin a name, say "Bill". And then you will assign .5 to your credence that Bill landed heads. But you also know for sure that Bill landed heads iff you are one of the heads-tossers, so you should assign the same credence to that as to Bill landing heads.

Alexander R Pruss said...

And your formulation inspired some aspects of my post for today. Thanks!

Alexander R Pruss said...

Observe that the argument for sticking to your guns is one you can apply to each of the people tossing the coins. So for each of infinitely many persons, you should assign probability 1/2 that she got heads. While, nonetheless, thinking that it's certain that only finitely many got heads.