hemahemahema post=18.73797.849132 said:
Wow, you are actually a good logician. A slightly confused one though. Let's just start with the card example, for which your analogy had two errors.
A), you equated p(1)'showing a black side' to p(2)'at least one is male' where 'black' is equivalent to 'male'. And your arguement is that since p(1) makes the possibility of the other side being black higher than 33.33%, p(2) should also make the possibility of the other dog being male higher than 33.33%, and you assert it is 50%. Now the first sign the analogy is wrong is that the possibility of the other side is black is actually 66.66% (2/3), so how can the possibility of the other dog being male be 50%?
Well, first off, thanks! As for the Three Card Problem, I wasn't saying it was a direct analog, just that the same reasoning applies, only in this case, if we're bringing biology into this, picking a pair with a male makes it equally likely that you've gotten the one all male pair than either of the two mixed pairs, for the same reason that picking a black side makes it more than likely that you've gotten the all black card.
Just in this case, the Three Card Problem is a Four Card Problem, with one all black card, two mixed cards, and one all-white card, representing the all male/mixed/all female puppy combinations.
B) In the Three Card Problem, you have been told there are 3 cards, two of them have black sides and one of them is all black. There is no mention of such information in the puppy question.
My point was that if we're going to take the 'at least one dog is male' statement so literally that we don't think about how it was arrived at, then we should take the shopkeeper woman's statement that they are either a male pair, a female pair, or a mixed pair as describing the actual pool of possible pairs, not just the possible combinations of male and female you find in them, so she's really describing a Three Card Problem.
Can't use one kind of logic to read one half of a problem, and a different kind of logic to read the other half, you know? This isn't the Second Amendment, with some kinda distinction between a prefatory and an operative clause!
And to be honest, I don't think anyone on this Forum even begins to understand your obsession with this conspiracy theory that some person or organisation is going around screening pairs of dogs to keep down the number of straight pairs, despite the fact there is no mention whatsoever of this grand project in the question.
My obsession is explained by the fact that people are thinking this describes a paradox that holds true in the real world every time you hear "at least one is male." Look at the survey that Marilyn Vos Savant took to try and prove she was 'right': she asked for parents with two children and at least one male to write in--sounds a whole lot like someone 'screening pairs of kids to keep out any FF pairs' now, doesn't it!
My point was that seeing this, people will start to think this is the way the whole world works. Now think of another scenario where the statement "there is at least one male" is every bit as true: tell every parent with two children to pick one of their children at random, and if that child is male, write in with the sex of the pair. Half the responses will be from parents with two male children, even though they only make up 25% of the population by average, just like in a Four Card Problem.
In short, brain teasers are cute, but really understanding why the math works the way it does is more important. That a few of the people arguing with me really don't get the deep workings of the math is proven by the fact that they kept trying to argue that I read a word problem wrong by throwing more math at me, or complaining about the fact that a word problem had come down to "semantics": that's like complaining about a problem about the sex of puppy pairs coming down to biology!
Basically, it seems a lot of people felt very smart for 'knowing' it was 33%, and didn't like that I came along and pointed out that you could read the problem a different way from the one they did without falling into the 'trap' that the problem throws at you.
One could say that I harshed on the squee of some people who know one math trick, and they got all emo about it!
Not to say that's true of everyone who disagreed with me, but, it happened. In the end I won, though, because I now know a lot more about probability than I did before. Even if I still don't know what werepossum was talking about with set vs. sequential probability.
