[ExI] Probability mind benders

[Ben Zaiboc]

John Clark <jonkc at bellsouth.net> riddled:

> 
> Yes, a lot of stuff about probability is not intuitive. I
> was reading about a really good one by Gary Foshee at the
> recent "Gathering for Gardener" conference in Atlanta.
> 
> Suppose I tell you that I have 2 children and one of them
> is a boy, what is the probability I have 2 boys? the correct
> answer is not 1/2 but 1/3. How can that be? Well there are 4
> possible combinations, BB,GG,BG and GB but but at least one
> is a boy so you can get rid of GG. So all that's left is
> BB,BG and GB; and in only one of those 3 possibilities do I
> have two boys.
> 
> But Foshee was just getting warmed up!
> 
> Now I tell you that I have 2 children and one of them is a
> boy born on a Tuesday. What is the probability that I have 2
> boys? You may think that Tuesday is not useful information
> in this matter so the answer would be the same as the
> previous example but you would be wrong. The correct answer
> is 13/27. How can that be?
> 
> Well there are 14 possibilities for EACH kid:
> B-Mo, B-Tu, B-We, B-Th, B-Fr, B-Sa, B-Su
> G-Mo, G-Tu, G-We, G-Th, G-Fr, G-Sa, G-Su
> 
> But I told you the one of my kids (the first or the second)
> was a boy born on a Tuesday so that narrows down the field
> of possibilities to: 
> 
> First child: B-Tu, second child: B-Mo, B-Tu, B-We, B-Th,
> B-Fr, B-Sa, B-Su, G-Mo, G-Tu, G-We, G-Th, G-Fr, G-Sa, G-Su.
> Second child: B-Tu, first child: B-Mo, B-We, B-Th, B-Fr,
> B-Sa, B-Su, G-Mo, G-Tu, G-We, G-Th, G-Fr, 
> G-Sa, G-Su.
> 
> No need to put B-Tu in the second row because it's already
> accounted for in the first row.
> So now just count them out, 14+13= 27 possibilities. How
> many result in 2 boys? Count them out again 7+6=13. So 13
> out of 27 possibilities give you 2 boys.
> 

I've read this one before, and have to say, even though I've just admitted I'm a duffer at maths, and especially statistics, this is rubbish.

Or rather, it's a trick question.  I'm sure it's not actually rubbish in the world of statistics.  I'm sure the maths is correct. But in the real world?  Given that they already have a boy, and have one more child, who in their right mind would say "I could have a boy and a boy, a boy and a girl, or a girl and a boy"?  G/G is obviously not possible, and B/G is the *same thing* as G/B.  We already know one child is a boy, so the question "what's the probability I have 2 boys?" is a trick question.  It's a different question to "what's the probability that my second child is a boy?", which is a sensible question in the circumstances.

So, what's the probability that my second child is a boy born on a Tuesday? (I've no idea, and don't really know how to work it out, but I strongly suspect the answer is in no way surprising).

Ben Zaiboc