Quote:
Originally Posted by omk3
I don't really see why the exact same information would produce different probability results depending on how you got the information. The hard facts we get from the Scientific American puzzle and pdurrant's puzzle are exactly the same. Does it really matter how they are presented?
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This is a very good summary of the meta-problem. In both cases, we end up knowing that a particular person has two children, and one of the is a boy born on a Tuesday. It does seem peculiar for me to say that in one case the probability that the other child is a boy is 1/2 and in the other it's 13/27ths.
Let me give some concrete numbers to show what's going on. I'm going to use low numbers, and expect the probabilities to work exactly. Of course, in an actual trial it wouldn't work with such low numbers. Just imaging that when I say "196" I mean "196 million", etc.
We take 196 fathers of exactly two children.
I think we can agree that we'd expect 49 to have BB, 49 to have BG, 49 to have GB and 49 to have GG.
We ask them all to make a true statement about their children using the template "I have at least one [boy|girl]"
On average, we'd expect 98 of them to state "I have at least one boy" (because for 49 (BB) this is the only true statement they can make, 49 (GG) can't state this, and of the remaining 98 (BG, GB) we'd expect half to choose the boy statement and half to choose the girl statement)
Now we ask those 98 (49 BB and 49 (BG or GB) ) to make a true statement about their children of the form "I have at least one boy born on a [day of week]"
On average, we'd expect 14 to say "I have at least one boy born on a Tuesday". ( and 14 to say "..Wednesday, 14 say "Thursday", etc) And we'd of that 14, we'd expect to have 7 BB and 7 (GB or BG).
So now we have 14 fathers who've all made a true statement about their children "I have at least one boy born on a Tuesday". How many of them have BB? Exactly 1/2.
Now, let's take that same 196 fathers we started with. This time we /ask/ them "who has at least one boy?". This time we expect to get 147 saying yes - all 49 BB, 49 BG and 49 GB.
And now we ask those 147 "who has at least one boy born on a tuesday?"
We expect to get 7 BG and 7 GB saying yes, but we expect to get 13 BB saying yes, because either of their children could have been born on a Tuesday and their still have a boy born on a Tuesday.
So we now have 27 fathers who can all make a true statement about their children "I have at least one boy born on a Tuesday". But only 13/27 have BB - not 1/2.
So - we have the same information about each individual in each group - we just arrived at that information a different way. And the probability for individuals from each group having two boys is different, even through we know exactly the same information about them.
Well, actually, we don't have exactly the same information - we also know the information about how we obtained that information, and it's that that makes the difference.