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

John Clark jonkc at bellsouth.net
Sun Jul 4 21:47:09 UTC 2010

```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.

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.

John K Clark

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