How about this:

Lets assume we have an infinitely long array of squares. And a fair 6-sided dice.

We roll the dice an infinite number of times and write each roll's number into a square.

When we finish, how many squares have a "1" written in them? An infinite number, right?

How many squares have an even number written in them? Also an infinite number.

How many squares have a number OTHER than "1" written in them? Again, an infinite number.

Therefore, the squares with "1" can be put into a one-to-one correspondence with the "not-1" squares...correct?

Now, while we have this one-to-one correspondence between "1" and "not-1" squares set up, let's put a sticker with an "A" on it in the "1" squares. And a sticker with a "B" on it in the "not-1" squares. We'll need the same number of "A" and "B" stickers, obviously. Aleph-null.

So, if we throw a dart at a random location on the array of squares, what is the probability of hitting a square with a "1" in it?

What is the probability of hitting a square with a "A" sticker?

The two questions don't have a compatible answers, right? So, in this scenario, probability is useless. It just doesn't apply. You should have no expectations about either outcome.

BUT. NOW. Let's erase the numbers and remove the stickers and start over.

This time, let's just fill in the squares with a repeating sequence of 1,2,3,4,5,6,1,2,3,4,5,6,1,2,3,...

And then, let's do our same trick about putting the "1" squares into a one-to-one mapping with the "not-1" squares, and putting an "A" sticker on the "1" squares, and a "B" sticker on the "not-1" squares.

Now, let's throw a dart at a random location on the array of squares. What is the probability of hitting a square with a "1" in it?

What is the probability of hitting a square with a "A" sticker on it?

THIS time we have some extra information! There is a repeating pattern to the numbers and the stickers. No matter where the dart hits, we know the layout of the area. This is our "measure" that allows us to ignore the infinite aspect of the problem and apply probability.

For any area the dart hits, there will always be an equal probability of hitting a 1, 2, 3, 4, 5, *or* 6. As you'd expect. So the probability of hitting a square with a "1" in it is ~16.67%.

Any area where the dart hits will have a repeating pattern of one "A" sticker followed by five "B" stickers. So the probability of hitting an "A" sticker is ~16.67%.

The answers are now compatible, thanks to the extra "structural" information that gave us a way to ignore the infinity.

In other words, you can't apply probability to infinite sets, but you can apply it to the *structure* of an infinite set.

If the infinite set has no structure, then you're out of luck. At best you can talk about the method used to generate the infinite set...but if this method involves randomness, it's not quite the same thing.

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