[ExI] Probability mind benders [was: Psi and EP]

[x at extropica.org]

2010/7/4 John Clark <jonkc at bellsouth.net>:
> On Jul 4, 2010, at 4:28 PM, BillK wrote:
>
> Be careful how you bet on this though.
>
> 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.

You must mean "not 1/4 but 1/3."  And this is true only under the
slightly incorrect prior that the ratio of male to female births is
1:1.  Like the Monty Hall "paradox", probability, not to be confused
with likelihood, changes with changing evidence.


> 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.

Your confusion here is not about probabilities, but semantics.  Did
the boy also have red hair? If the problem is clearly specified, the
probabilities are clear and unambiguous.

In both cases, context (implicit or explicit) is significant.

- Jef