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Topic: probability question
Replies: 9   Last Post: Mar 13, 2008 4:08 PM

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quasi

Posts: 12,057
Registered: 7/15/05
Re: probability question
Posted: Mar 12, 2008 5:42 AM
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On Wed, 12 Mar 2008 00:33:19 -0700, The World Wide Wade
<aderamey.addw@comcast.net> wrote:

>In article <b1set3l134nn2kt3buh893q8jjfg3kau6s@4ax.com>,
> quasi <quasi@null.set> wrote:
>

>> On Tue, 11 Mar 2008 21:56:27 EDT, Steven <sgottlieb60@hotmail.com>
>> wrote:
>>

>> >Suppose you meet me on a street corner and I introduce you to my son who is
>> >with me. I inform you that I have another child at home. What is the
>> >probability that my other child is a girl.

>>
>> The problem is not adequately specified.
>>
>> It depends on how the child accompanying the father is selected.
>>
>> If the child that accompanies the father is selected at random by a
>> flip of a fair coin, then the probability that the other child is a
>> girl is 1/3.

>
>The sample space for the children is (b b), (b g), (g b), (g g) where
>the first slot is the youngest child, the second slot is the oldest.
>These oredered pairs all have probability 1/4. Now we select a child
>at random for a walk. We get a new sample space: (b b b), (b g g), (b
>g b), (g b g), (g b b), (g g g), with the probabilities being 1/4 for
>the first and last triples, and 1/8 for the others. The probability
>the other child is a girl given the randomly selected child out with
>daddy is a boy is thus
>
>p((b g b) (g b b))/p((b b b) (b g b) (g b b))
>
> = (1/8 + 1/8)/(1/4 + 1/8 + 1/8) = 1/2.


Yes, you're right -- my mistake.

However, as far as I can see, for the rest of the cases I discussed,
my conditional probabilities are correct.

>> If the father always chooses the younger child to go with him, then
>> there is no information, so the probability that the other child is a
>> girl is 1/2.
>>
>> Similarly, if the father always chooses the older child to go with
>> him, then again there is no information, so the probability that the
>> other child is a girl is 1/2.
>>
>> If a boy has _priority_ over a girl to go with the father, then the
>> probability that the other child is a girl is 2/3.
>>
>> If a girl has _priority_ over a boy to go with the father, then the
>> probability that the other child is a girl is 0.
>>
>> Thus, the probability critically depends on the selection mechanism.
>>
>> In the absence of any information, the problem is not well posed.
>>
>> Of course, in a given situation. you can always make a subjective
>> judgement as to the type of selection, in which case, you can then
>> derive an answer based on that extra assumption.
>>
>> For example, thinking about it culturally, if I _had_ to guess, I
>> would guess that the father would choose to take the _younger_ child,
>> in which case, the probability is 1/2.


In any case, my main point still holds -- the required probability
can't be determined unless you know the method for the selecting the
child who goes with the father.

quasi



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