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

Author: Serge Desmarais

Date: 16:21:48 08/29/98

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On August 29, 1998 at 17:33:54, Dan Newman wrote:

>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
>>>>>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.
>>   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.

So, since we know one is a boy (B), in 1/3 of the cases the second kid will be a
boy, in 1/3 of the cases it will be a girl (G) and in the last third of cases,
it will be ???   :)

Serge Desmarais

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