Probability of that n children are daughters given at least one daughter, using set theory

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Here is a question that I would like to dolve using set theory. I know the answer but was not able to solve it with Set Theory. The question goes as follows: a father has n children, given that the father has at least 1 daughter, what is the probability of n daughters?
Here's my attempty using Set Theory:
Let G be the event of all girls defined to be g1^g2^...^gn and let B be the event of all boys defined to be b1^b2^...^bn. Let !B be defined as the complement of all boys which is also the event of at least one girl. So here !B = !(b1^b2^...^b3) which is then !B = !b1/!b2/.../!bn. Then I tried applying Bayes Rule such that P(G|!B) = P(G^!B)/P(!B). So here is where I run into a problem, P(G^!B) = P(G^(!b1/!b2/.../!bn)) = P((G^!b1)/(G^!b2)/.../(G^!bn)) = P(!b1/!b2/.../!bn). This makes sense to me as the set of all girls intersected by the set of at least one girl should be equivalent to the set of at least 1 girl. However this would mean that P(G|!B) = P(!B)/P(!B) = 1. Obviously this is wrong but I don't understand where. Your advice is very much appreciated




probability




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  • 1




    This does not seem to have much to do with set theory ...
    – Henning Makholm
    Aug 20 at 9:34










  • The question presumes binary genders. Please formulate it in an inclusive manner.
    – joriki
    Aug 20 at 9:45














up vote
-3
down vote

favorite












Here is a question that I would like to dolve using set theory. I know the answer but was not able to solve it with Set Theory. The question goes as follows: a father has n children, given that the father has at least 1 daughter, what is the probability of n daughters?
Here's my attempty using Set Theory:
Let G be the event of all girls defined to be g1^g2^...^gn and let B be the event of all boys defined to be b1^b2^...^bn. Let !B be defined as the complement of all boys which is also the event of at least one girl. So here !B = !(b1^b2^...^b3) which is then !B = !b1/!b2/.../!bn. Then I tried applying Bayes Rule such that P(G|!B) = P(G^!B)/P(!B). So here is where I run into a problem, P(G^!B) = P(G^(!b1/!b2/.../!bn)) = P((G^!b1)/(G^!b2)/.../(G^!bn)) = P(!b1/!b2/.../!bn). This makes sense to me as the set of all girls intersected by the set of at least one girl should be equivalent to the set of at least 1 girl. However this would mean that P(G|!B) = P(!B)/P(!B) = 1. Obviously this is wrong but I don't understand where. Your advice is very much appreciated




probability




share|cite|improve this question


















  • 1




    This does not seem to have much to do with set theory ...
    – Henning Makholm
    Aug 20 at 9:34










  • The question presumes binary genders. Please formulate it in an inclusive manner.
    – joriki
    Aug 20 at 9:45












up vote
-3
down vote

favorite









up vote
-3
down vote

favorite











Here is a question that I would like to dolve using set theory. I know the answer but was not able to solve it with Set Theory. The question goes as follows: a father has n children, given that the father has at least 1 daughter, what is the probability of n daughters?
Here's my attempty using Set Theory:
Let G be the event of all girls defined to be g1^g2^...^gn and let B be the event of all boys defined to be b1^b2^...^bn. Let !B be defined as the complement of all boys which is also the event of at least one girl. So here !B = !(b1^b2^...^b3) which is then !B = !b1/!b2/.../!bn. Then I tried applying Bayes Rule such that P(G|!B) = P(G^!B)/P(!B). So here is where I run into a problem, P(G^!B) = P(G^(!b1/!b2/.../!bn)) = P((G^!b1)/(G^!b2)/.../(G^!bn)) = P(!b1/!b2/.../!bn). This makes sense to me as the set of all girls intersected by the set of at least one girl should be equivalent to the set of at least 1 girl. However this would mean that P(G|!B) = P(!B)/P(!B) = 1. Obviously this is wrong but I don't understand where. Your advice is very much appreciated




probability




share|cite|improve this question














Here is a question that I would like to dolve using set theory. I know the answer but was not able to solve it with Set Theory. The question goes as follows: a father has n children, given that the father has at least 1 daughter, what is the probability of n daughters?
Here's my attempty using Set Theory:
Let G be the event of all girls defined to be g1^g2^...^gn and let B be the event of all boys defined to be b1^b2^...^bn. Let !B be defined as the complement of all boys which is also the event of at least one girl. So here !B = !(b1^b2^...^b3) which is then !B = !b1/!b2/.../!bn. Then I tried applying Bayes Rule such that P(G|!B) = P(G^!B)/P(!B). So here is where I run into a problem, P(G^!B) = P(G^(!b1/!b2/.../!bn)) = P((G^!b1)/(G^!b2)/.../(G^!bn)) = P(!b1/!b2/.../!bn). This makes sense to me as the set of all girls intersected by the set of at least one girl should be equivalent to the set of at least 1 girl. However this would mean that P(G|!B) = P(!B)/P(!B) = 1. Obviously this is wrong but I don't understand where. Your advice is very much appreciated





probability





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edited Aug 20 at 9:43









Asaf Karagila♦

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asked Aug 20 at 8:38









Jonathan Aguilera

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  • 1




    This does not seem to have much to do with set theory ...
    – Henning Makholm
    Aug 20 at 9:34










  • The question presumes binary genders. Please formulate it in an inclusive manner.
    – joriki
    Aug 20 at 9:45












  • 1




    This does not seem to have much to do with set theory ...
    – Henning Makholm
    Aug 20 at 9:34










  • The question presumes binary genders. Please formulate it in an inclusive manner.
    – joriki
    Aug 20 at 9:45







1




1




This does not seem to have much to do with set theory ...
– Henning Makholm
Aug 20 at 9:34




This does not seem to have much to do with set theory ...
– Henning Makholm
Aug 20 at 9:34












The question presumes binary genders. Please formulate it in an inclusive manner.
– joriki
Aug 20 at 9:45




The question presumes binary genders. Please formulate it in an inclusive manner.
– joriki
Aug 20 at 9:45















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