> I tell you I have two children and that (at least) one of them is a boy, and ask you what you think is the probability that I have two boys.
I want to find
P(you have two boys|you tell me that you have two children including at least one boy) = P(you have two boys|you tell me that X)
where for convenience I use the notation X=“you have two children including at least one boy”
I can write
P(you have two boys|you tell me X) P(you tell me X) = P(you tell me X|you have two boys) P(you have two boys)
Assuming that you don’t lie I know that X is true and I can restrict the analysis accordingly.
P(you have two boys|X and you tell me X) P(you tell me X| X) = P(you tell me X|X and you have two boys) P(you have two boys|X)
That conditioning information is redundant in some terms but not always.
If the probability of a boy is 1/2 and there is no correlation we can find that P(you have two boys|X)=1/3
(Using the notation girl=not-boy we can also write P(you have one boy and one girl|X)=2/3)
But we’re after something that could be different: P(you have two boys|you tell me that X)
P(you have two boys|you tell me X) = P(you tell me X|you have two boys) P(you have two boys|X) / P(you tell me X| X) = P(you tell me X|you have two boys) P(you have two boys|X) / ( P(you tell me X|you have two boys) P(you have two boys|X) + P(you tell me X|you have one boy and one girl) P(you have one boy and one girl|X) )
Without additional assumptions we can’t go beyond
P(you have two boys|you tell me X)
being equal to
1/3 P(you tell me X|you have two boys)
divided by
1/3 P(you tell me X|you have two boys) + 2/3 P(you tell me X|you have one boy and one girl)
If you consider it possible they might have told you "I have two children and (at least) one of them is a girl" instead of the statement about a boy (were they to have both a boy and a girl), the reasoning given in the article is wrong.
I want to find
P(you have two boys|you tell me that you have two children including at least one boy) = P(you have two boys|you tell me that X)
where for convenience I use the notation X=“you have two children including at least one boy”
I can write
P(you have two boys|you tell me X) P(you tell me X) = P(you tell me X|you have two boys) P(you have two boys)
Assuming that you don’t lie I know that X is true and I can restrict the analysis accordingly.
P(you have two boys|X and you tell me X) P(you tell me X| X) = P(you tell me X|X and you have two boys) P(you have two boys|X)
That conditioning information is redundant in some terms but not always.
If the probability of a boy is 1/2 and there is no correlation we can find that P(you have two boys|X)=1/3
(Using the notation girl=not-boy we can also write P(you have one boy and one girl|X)=2/3)
But we’re after something that could be different: P(you have two boys|you tell me that X)
P(you have two boys|you tell me X) = P(you tell me X|you have two boys) P(you have two boys|X) / P(you tell me X| X) = P(you tell me X|you have two boys) P(you have two boys|X) / ( P(you tell me X|you have two boys) P(you have two boys|X) + P(you tell me X|you have one boy and one girl) P(you have one boy and one girl|X) )
Without additional assumptions we can’t go beyond
P(you have two boys|you tell me X)
being equal to
1/3 P(you tell me X|you have two boys)
divided by
1/3 P(you tell me X|you have two boys) + 2/3 P(you tell me X|you have one boy and one girl)