What if the universe were infinitely large? More specifically, what if there were infinitely many people (or agents relevantly similar to us) in the universe? This possibility raises a number of difficult philosophical problems. In this edition, I outline a paradox that arises in infinite universes. The philosopher Caspar Hare, who has written about the paradox, calls it “The Paradox of Infinite Distrust,” and it is as follows: Imagine that you are one of the inhabitants of an infinite universe. I sit you down at a table in a small enclosed room. There is an open box on the table in front of you. I toss a fair die into the box, and quickly put the lid on the box before you can see the result of the die roll.

What is the probability that the die in your box displays a 6?

Now I tell you that I have set up this exact scenario for an infinite number of other people in the universe. I have put each one of these infinite number of people into a room with an open box in front of them, tossed a fair die into the box, and covered up the lid.

What is the probability that the die in your box displays a 6?

Suppose I now pick an arbitrary point in the universe, and designate this point “the centre of the universe” – call it C. I then divide the people in the universe into two groups: those whose die displays a 6 (call them “sixers”), and those whose die displays a 1, 2, 3, 4, or 5 (call them “non-sixers”). In any given region of space, the number of non-sixers is about five times the number of sixers (remember that the dice are all fair!).

Now, I take the sixer who is closest to C, and the non-sixer who is closest to C, and call them “buddies”.

Then I take the sixer who is second-closest to C, and the non-sixer who is second-closest to C, and call them buddies too. And so on, so that every person in the universe has a buddy.

What is the probability that the die in your box displays a 6?

I now open up a Skype call between you and your buddy, who is in the exact same situation as you – they also don’t know what the die in their box displays. I now ask each of you:

What is the probability that the die in your box displays a 6?

Consider two arguments:

Argument 1: the probability is 1/6. After all, 1/6 was clearly the answer when I first asked you this question. Since then, I’ve added a bunch of new information. But none of this new information is relevant to whether or not your die displays a 6. After all, it’s just information about stuff I’ve been doing elsewhere in the universe. So nothing has changed since the first time I asked you this question, which means that the answer must still be the same – 1/6.

Argument 2: the probability that your die is a 6 must be the same as the probability that your buddy’s die is a 6. After all, you are in perfectly symmetrical situations. You have both been given identical information. Now, by construction, either your die or your buddy’s displays a 6. One of you is a sixer and the other is not. This means that the probability that your die displays a 6 is 1/2.

So which is it? We have two seemingly flawless arguments. But their conclusions contradict each other. Is the probability 1/2, or 1/6?