Paul C. Anagnostopoulos said:
It's not the problem that's pedantic, it's this discussion. By all means, though, clarify the standard wording of the problem. You're right, it won't hurt. But to claim that the problem as worded was ambiguous and thus could not serve its purpose to surprise people about their intuitions is ... well ... pedantic. In fact, I daresay that such a clarification would do absolutely no good for the very people the problem is meant to educate.
Absolutely clear. All I was saying is that seems like precisely the default assumption, given no mention of any conditions whatsoever.
Just yanking your chain. Actually, though, "randomly picking" has the connotation of picking sometimes and not picking other times, whereas "picking randomly" has the connotation you want. But maybe that's just me.
But continue on. I have no idea why the probability of picking an even integer isn't .5.
~~ Paul
If it's absolutely clear that the 2/3 vs. 1/3 solution does involve making an assumption, then I don't really have a problem with your opinion on the problem. For the record, I agree, it looks to me like whoever wrote the problem probably intended it to be interpreted the way many people do--that Monty always opens a door and offers a switch. There is that subtle ambiguity there, however, and I think it's much preferable to add that statement to the problem for clarity. What can it possibly hurt? The problem is still as tricky as ever, and now it is unambiguous.
In general, poorly worded problems frustrate me, and some textbooks have more than their share. Personal anecdote: When I was a freshman in college taking a physics class, I remember struggling with one particular homework problem. The idea of the problem was: A man throws a ball at a certain velocity (under earth's gravity). How far does the ball travel? The answer was in the back of the book, but I couldn't figure out how to get it. It seemed to me that the height of the man would matter--Why wasn't I given that information? After playing around with it for a while, I finally realized--You had to assume that at the instant the ball was thrown, its initial position was
ground level. Who the hell throws and releases a ball at ground level? A perfect example of a poorly worded problem.
Anyway, about the positive integers problem. I have no idea what sort of responses I may have gotten, but here's a quick rundown of the point I was planning on making in a Socratic dialogue:
Poster: Wouldn't it be 1/2? Only half of the integers are even, the other half are odd?
Me: What do you mean, both sets are infinite! How is one infinity twice as large as another? In fact, there is a bijection between them: f
=2n gives a 1-1 correspondence between the even integers and
all of the positive integers. The two sets are the same size, or
cardinality. There
are sets with larger cardinalities, but these both have the same cardinality--cardinality aleph-naught.
Poster: Well, what I meant was, look at how they're ordered: 1, 2, 3, 4, ....
Every other one is even.
That's what I meant when I said
half of them are even--
every other one.
Me: Ah, I see where you're going with this now! However, what makes you think this
order you're accustomed to is relevant? That's an
assumption! What if some omnipotent deity took all of the positive integers, tossed them into an infinitely large urn, and then selected one at random? Now you have infinitely many integers, and infinitely many of them are even! Furthermore, they're all scrambled up now! What now?
Poster: Well, I would imagine that each integer has an equally likely chance of coming up, but since it's infinitely many out of infinitely many, I'm not sure how to evaluate that.
Me: I only brought up the omnipotent deity as a thought experiment; if we want to follow along those lines, and assume that each integer has an equally likely chance of coming up, that is yet another
assumption! Is that what you wish to do?
Poster: Yes.
Me: Fair enough, we'll work under that assumption and see what happens. So every number is as likely as any other number to be selected? OK, let's call that probability p, where p is some number strictly between 0 and 1.
Poster: OK
Me: Now refer back to the Kolmogorov probability axioms, our probability function must be countably additive. So the probability of picking
any number (which should obviously be a full probability of 1) will be the infinite sum of all the probabilities of the individual integers. That is, the countable sum from 1 to infinity of the positive constant p. What is that sum?
Poster: It diverges. It's infinite.
Me: Hmmm...That's a problem isn't it? it was supposed to be just 1. It doesn't make any sense to have infinite probability.
Poster: Well maybe the probability of picking a particular integer isn't positive. Maybe each integer has zero probability of being picked!
Me: An interesting idea, but we've got a problem there, too. Like before, what's the sum from 1 to infinity of the constant zero?
Poster: Zero.
Me: Exactly. Again, it needs to be 1 in order for us to have a probability model.
Poster: What now?
Me: Well, we've reached a dead end with this assumption. The probability of picking an individual integer
has to be zero or one, and either possibility leads to an impossibility. Our initial assumption that each integer is just as likely as any other is faulty--that situation can't happen....
[end dialogue]
Anyway, I'm only (once again) trying to illustrate that we should always be clear when we make assumptions and what those assumptions are. In some cases, like this one, it's entirely possible to find out our assumptions don't make any sense. (Obviously that wasn't going to be a problem with the Monty Hall problem, but, in general terms,
always be aware of your assumptions).
