# A quote from a book I have

Here: (from the first non-blank line after this one to the end)

Carpenter: You would agree that the probability of picking 1 from the first 10 numbers is 1/10?

Walrus: Of course.

C: And that the probability of picking 1 from the first 100 numbers is 1/100.

W: Of course.

C: And that as you increase the number to be selected from, the probability decreases to zero in the limit?

W: (after a pause) Yes, and this logic applies to any number.

C: Would you also agree that the probability of picking 1 or 2 from the first 10 numbers is 2/10? That the probability of picking one or the other is the sum of the probabilities of picking either?

W: (pause) Yes, I would.

C: And, therefore, that the probability of picking 1 or 2 from the set of all integers is thus the sum of zero and zero - in other words, zero?

W: That must be so.

C: And that furthermore, the probability of picking any of a set of alternative integers from the set of all integers is zero?

W: Of course. The probability of picking any one of the integers from the set of all integers is also zero.

C: So pick an integer and tell it to me.

The Walrus is struck dumb.

(Graham Cleverly, personal communication, 2004)

Here's the problem. The probability of picking any of a finite set of integers is zero. But, say, the probability of picking any of the set of even integers is 1/2.

Not zero, but 1/the number of integers in the set.

...which would mean the set of even integers is of length 2???

No, it would be $$\frac{\infty}{2}$$ probably

I don't think he meant that

Right. (I forgot which it was.) But I think it should actually be $$\aleph_0\over2$$. (Aleph null over two)

It's actually 1/2 times the length of the set of integers that you're picking even ones from. (not quite, if the length of that set is odd, but...)

I confused $$\aleph$$ with $$\mathbb{N}$$; maybe you should make that more clear. (I didn't even know that that symbol even existed, I had to look it up)

I can't change the font size!

It's the Hebrew letter "aleph", and it's used in math for various levels of infinity. $$\aleph_0$$ is the smallest one, the number of integers. Oh and $${\aleph_0\over2}=\aleph_0$$, because the number of even integers is the same as the number of integers, i.e., they can be paired with, say, (x, 2x).

I meant that you could explain it in the post

Done:

I'm confused. I think I'm just going to delete my post.

So am I, kind of. (Infinity is confusing.)

Wait, no, what the book was talking about was the probability of choosing any of a given set, when choosing from all the integers. E.g., the probability of getting any of {1,2,3,5,8} when choosing a random integer.

But if the target set (the equivalent to {1,2,3,5,8} above) is infinite, such as the set of even numbers, then the probability can be nonzero.

You're just confusing me more.

In this conversation, the big set from which numbers are being chosen is always the set of all the integers.

The target set starts out as a single target number, "what's the probability of choosing 1?" (Answer: probability zero.)

Then they get to larger target sets, "what's the probability of getting 1 or 2?" And they argue that that's P(1)+P(2)=0+0=0.

This argument is correct for any finite target set. But just by common sense you have to know that the probability of choosing an even integer, out of all the integers, is 1/2.

Hmm... I think the math actually says you're right there, but only if you don't use limits: $${{\aleph_0\over2}\over\aleph_0}={\aleph_0\over2}\times{1\over\aleph_0}={\aleph_0\over2\aleph_0}={1\over2}$$

You're unnecessarily confusing yourself by using transfinite numbers to think about it. Half the integers are even, end of story.

(Not that transfinite numbers aren't fascinating and fun! I don't want to discourage you from learning more set theory.)

All this talk of infinity reminds me of Hilbert's Hotel.