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Subject: Re: The Computer-Chess Player And The Mathematician (was: Waltzing Matilda

Author: Dan Newman

Date: 14:33:54 08/29/98

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On August 29, 1998 at 07:14:10, Serge Desmarais wrote:

>On August 29, 1998 at 05:39:43, Dan Newman wrote:
>
>>On August 28, 1998 at 23:30:44, Serge Desmarais wrote:
>>
>>>On August 18, 1998 at 07:26:14, fca wrote:
>>>>A man has exactly two biological children.
>>>>Question (1): "One of them is a B. What is the probability that both are Bs?"
>>>>Question (2): "The older of them is a B. What is the probability that both are
>>>>Bs?"
>>>>
>>>>Now the answer to (1) is 1/3, and to (2) is 1/2.
>>>
>>>
>>>
>>>   I don't understand how you can say that the answer to 1 is 1/3? If the first
>>>one is a boy, the other child has 50% chance of being a boy and 50% of being a
>>>girl, so that should be 1/2, instead!
>>>
>>>
>>>Serge Desmarais
>>>
>>
>>The reasoning for 1/3 must go something like this:  There are four
>>possibilities for two children which we might specify as BB, BG, GB,
>>and GG, each of which is equally probable.  Now, when we are told
>>that one is a boy, we must have one of BB, BG, GB (again each of
>>which has the same probability)--hence the 1/3 for BB.
>>
>>Another way of looking at it is to think about the set of all sibling
>>pairs in the world.  Aproximately 1/4 will be BB, 1/2 BG or GB, and
>>1/4 GG.  If we toss the GG's out of the set, then the BB's will be
>>1/3 of the remaining pairs.
>>
>>-Dan.
>
>   Isn't it a little twisted? ONE of them is a boy, so BG or GB is the same in
>reality. Because the ONE that is a boy for sure is B, so the other is either B
>or G. Since there ar only 2 possible letters here and one is B, the question
>should be formulated as what are the chances for the other letter (kid) to also
>be B. Since no placement of the first mentionned B was specified, I would say
>that BG or GB is the same?
>
>
>Serge Desmarais
>

It's definitely twisted.  (I argued once with a friend for 15 minutes or so
about this--while driving a car, so I was somewhat distracted.  He gradually
convinced me that it *might* be 1/3 and not 1/2--but I still had my doubts.)

Let's change the meaning of the symbols so that BG means either BG or GB.
Now, in the collection of all sibling pairs approximately 1/4 will be
BB, 1/2 BG, and 1/4 GG.  So we might have as representative sample of 8
such pairs (BB BB  BG BG BG BG  GG GG)--which happens to obey just those
fractions.  Now, if we consider only those in this sample which have a B
in them we have (BB BB  BG BG BG BG), which as you can see has exactly
1/3 BB.

Now, if we say the older of the two (assuming one of them is always older
than the other) is the B as in question (2, we need to go back to the old
notation so we can represent birth order: let the first letter in each pair
be for the younger and the second for the older.  Then we would have
aproximately 1/4 each of BB, BG, GB, and GG.  So our representative sample
could be (BB BB  BG BG  GB GB  GG GG).  Now, If we restrict this sample
to contain only those sibling pairs with an older boy we would have
(BB BB GB GB) which gives us 1/2 for BB.

I guess what it boils down to is that probabilities have to do with a
lack of knowledge.  A change in that lack may result in a change in the
probabilities.  Question 2) implies an increase in our knowledge about
the situation (over the first question) and so the probablility goes
from 1/3 to 1/2.

-Dan.



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