probability on countable infinite sets

Question

My question relates to probabilities on countable infinite sets. For example, what is the probability of choosing an even number from the positive integers. Believe it or not I am interested in this question from a practical standpoint. I am writing a paper on Boltzmann's brains (Brains that occur spontaneously from the vacuum in De Sitter space). Specifically the problem is, is it more likely that I am a Boltzmann brain or a regular brain if space is infinite. There are of course no good calculations about what the odds of a Boltzmann brain identical to mine are vs. the odds of an identical brain evolving on a twin earth (my money would be that a copy on a twin earth is much more likely). But would the specific odds matter if space is infinite and there are an infinite number of both, regardless of which is more common? I know the answer would of course depend on your philosophy of probability (frequency, Bayesian, etc.) so I would be interested in the answer for each of the main theories in probability.

Answer 1

I can't tell you anything about Boltzmann's brains, but let me say something about probabilities. For simplicity, let's say that your countably infinite set is $S=\{1,2,3,\ldots\}$.

You need to assign a probability $p_n$ to each positive integer $n\geq 1$, such that $$p_1+p_2+p_3+\cdots = 1.$$

There are several ways of doing this, and the method you choose depends very much on your application. One way of doing it is to choose $p_n = (1/2)^n$. Then $p_1 = 1/2$, $p_2 = 1/4$, and so on, and it is true that $$1/2 + 1/4 + 1/8 + \cdots = 1.$$

For this choice of probabilities, the probability of getting an even number is

$$p_2+p_4+p_6+\cdots = 1/4 + 1/16+1/64+\cdots = 1/3.$$

But this is just one case of choosing the probabilities. One thing that is for certain is that you cannot choose all the $p_n$ to be the same number, but the way you otherwise choose them will depend on your application, and I'm sure you know more about which probabilities are appropriate for the Boltzmann's brains applications than I do.

Answer 2

OK, I think I may have an answer based on the response from HowDoIMath, which I posted below their comment, but I want to expand on that possible answer.

My original question brings up a related statistical question about Boltzmann brains. First, let me give a little more background, Boltzmann brains are brains that spontaneously generate from random particles in the vacuum of space. According to some physicists, this is a problem for cosmological models that include infinite space. They worry that in this infinite space, an infinite number of Boltzmann brains would be generated and so (using anthropic reasoning) I (or anyone else) are more likely to be a random Boltzmann brain, and thus all my memories are false so we can't make any inductive arguments in science. When we unpack this statement a little, physicists acknowledge that a twin earth is also possible where an identical brain has evolved (although they assume this is less likely than a Boltzmann brain).

My problem with this whole line of reasoning is that there is no guarantee that even one Boltzmann or Twin earth brain will be generated, let alone infinite numbers of them. To make it easier, lets talk about flipping a coin. The probability is 1 that you will get a billion heads in a row if you flip the coin an infinite number of times, but by "1" here all we can say is "this is almost certain to happen" but we can never say for sure that it will happen. For example, it is possible that we could flip tails an infinite number of times. The probability of this is 0, but we need to interpret this 0 as "almost certainly won't happen" rather than that it won't happen. So with this line of thought the whole Boltzmann brain argument never gets off the ground because they may not even by one of them in an infinite universe. However, is my reasoning correct? Also, if I am reasoning correctly, it is equally likely (almost certain), that a billion heads will be flipped an countably infinite number of times.

Answer 3

In conventional countably additive probability, in order to assign a probability to the set of even positive integers, you would need to have a probability assigned to each integer, and then you add up the ones for the even integers.

You can't assign equal probabilities to all positive integers, since then either they're all $0$ and their sum is $0$, whereas the sum should be $1$, or they're all some positive number as the sum is $\infty$, whereas it should be $1$.

However, there is also the idea of letting the probability of each set $A\subseteq\{1,2,3,\ldots\}$ be the "density" of $A$, which is $$ \lim_{n\to\infty} \frac{|A\cap\{1,\ldots,n\}|} n. $$ If you do that, then probabilities are finitely additive but not countably additive, and then not all of the theorems of conventional probability hold.