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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up vote
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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
1
This does not seem to have much to do with set theory ...
â Henning Makholm
Aug 20 at 9:34
The question presumes binary genders. Please formulate it in an inclusive manner.
â joriki
Aug 20 at 9:45
add a comment |Â
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
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
edited Aug 20 at 9:43
Asaf Karagilaâ¦
293k31408736
293k31408736
asked Aug 20 at 8:38
Jonathan Aguilera
11
11
1
This does not seem to have much to do with set theory ...
â Henning Makholm
Aug 20 at 9:34
The question presumes binary genders. Please formulate it in an inclusive manner.
â joriki
Aug 20 at 9:45
add a comment |Â
1
This does not seem to have much to do with set theory ...
â Henning Makholm
Aug 20 at 9:34
The question presumes binary genders. Please formulate it in an inclusive manner.
â joriki
Aug 20 at 9:45
1
1
This does not seem to have much to do with set theory ...
â Henning Makholm
Aug 20 at 9:34
This does not seem to have much to do with set theory ...
â Henning Makholm
Aug 20 at 9:34
The question presumes binary genders. Please formulate it in an inclusive manner.
â joriki
Aug 20 at 9:45
The question presumes binary genders. Please formulate it in an inclusive manner.
â joriki
Aug 20 at 9:45
add a comment |Â
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1
This does not seem to have much to do with set theory ...
â Henning Makholm
Aug 20 at 9:34
The question presumes binary genders. Please formulate it in an inclusive manner.
â joriki
Aug 20 at 9:45