[math-fun] Probability puzzle
(This problem is said to be one that Google asked prospective employees in job interviews. My phrasing is almost identical to how it was presented to me.) PUZZLE: Suppose that in a very large population, each family has children until the first boy is born, and then has no more. What is the percentage of girls in the population? (Of course, make all the usual simplifying assumptions in this kind of problem.) --Dan I sleep as fast as possible so I can get more rest in the same amount of time.
Making the usual simplifying assumptions, and handwaving furiously: Since boys and girls are equally likely, it takes on average two tries to get a boy, and it's given that for each family, exactly one child is a boy. So half the kids are girls. --ms On Wednesday 07 July 2010 18:41:04 Dan Asimov wrote:
(This problem is said to be one that Google asked prospective employees in job interviews. My phrasing is almost identical to how it was presented to me.)
PUZZLE: Suppose that in a very large population, each family has children until the first boy is born, and then has no more. What is the percentage of girls in the population?
(Of course, make all the usual simplifying assumptions in this kind of problem.)
--Dan
I sleep as fast as possible so I can get more rest in the same amount of time.
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
On Thursday 08 July 2010 00:30:52 Mike Speciner wrote:
Since boys and girls are equally likely, it takes on average two tries to get a boy, and it's given that for each family, exactly one child is a boy. So half the kids are girls.
Yup. Or, more simply: whenever the parents decide to stop having children, each child is independently a boy with probability 1/2 and the parents' decisions aren't going to change that. -- g
I don't think this could be right. Since each family has precisely one boy, then a family of N kids has N-1 girls and 1 boy. The probability of a family having N kids is 1/2^N, so the fraction of boys is sum( 1/N * 1/(2^N). After the 1st 3 terms, there are at least 1/2 + 1/4*1/2 + 1/8*1/3 fraction of boys, which is 16/24 or 2/3 .66666. So it is greater than 2/3 boys. ----- Message from ms@alum.mit.edu --------- Date: Wed, 7 Jul 2010 19:30:52 -0400 From: Mike Speciner <ms@alum.mit.edu> Reply-To: math-fun <math-fun@mailman.xmission.com> Subject: Re: [math-fun] Probability puzzle To: Dan Asimov <dasimov@earthlink.net>, math-fun <math-fun@mailman.xmission.com>
Making the usual simplifying assumptions, and handwaving furiously:
Since boys and girls are equally likely, it takes on average two tries to get a boy, and it's given that for each family, exactly one child is a boy. So half the kids are girls.
--ms
On Wednesday 07 July 2010 18:41:04 Dan Asimov wrote:
(This problem is said to be one that Google asked prospective employees in job interviews. My phrasing is almost identical to how it was presented to me.)
PUZZLE: Suppose that in a very large population, each family has children until the first boy is born, and then has no more. What is the percentage of girls in the population?
(Of course, make all the usual simplifying assumptions in this kind of problem.)
--Dan
I sleep as fast as possible so I can get more rest in the same amount of time.
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
----- End message from ms@alum.mit.edu -----
Never mind. I'm ignoring the fact that there are N times as many kids in a family of N kids as there are in a family with an only child. Doh! ----- Message from mbgreen@cis.upenn.edu --------- Date: Wed, 07 Jul 2010 19:57:05 -0400 From: mbgreen@cis.upenn.edu Reply-To: math-fun <math-fun@mailman.xmission.com> Subject: Re: [math-fun] Probability puzzle To: math-fun <math-fun@mailman.xmission.com> Cc: Dan Asimov <dasimov@earthlink.net>, "greenwald@cis.upenn.edu" <greenwald@cis.upenn.edu>
I don't think this could be right. Since each family has precisely one boy, then a family of N kids has N-1 girls and 1 boy. The probability of a family having N kids is 1/2^N, so the fraction of boys is sum( 1/N * 1/(2^N). After the 1st 3 terms, there are at least 1/2 + 1/4*1/2 + 1/8*1/3 fraction of boys, which is 16/24 or 2/3 .66666. So it is greater than 2/3 boys.
----- Message from ms@alum.mit.edu --------- Date: Wed, 7 Jul 2010 19:30:52 -0400 From: Mike Speciner <ms@alum.mit.edu> Reply-To: math-fun <math-fun@mailman.xmission.com> Subject: Re: [math-fun] Probability puzzle To: Dan Asimov <dasimov@earthlink.net>, math-fun <math-fun@mailman.xmission.com>
Making the usual simplifying assumptions, and handwaving furiously:
Since boys and girls are equally likely, it takes on average two tries to get a boy, and it's given that for each family, exactly one child is a boy. So half the kids are girls.
--ms
On Wednesday 07 July 2010 18:41:04 Dan Asimov wrote:
(This problem is said to be one that Google asked prospective employees in job interviews. My phrasing is almost identical to how it was presented to me.)
PUZZLE: Suppose that in a very large population, each family has children until the first boy is born, and then has no more. What is the percentage of girls in the population?
(Of course, make all the usual simplifying assumptions in this kind of problem.)
--Dan
I sleep as fast as possible so I can get more rest in the same amount of time.
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
----- End message from ms@alum.mit.edu -----
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
----- End message from mbgreen@cis.upenn.edu -----
From: Dan Asimov <dasimov@earthlink.net> To: math-fun <math-fun@mailman.xmission.com> Sent: Wed, July 7, 2010 3:41:04 PM Subject: [math-fun] Probability puzzle (This problem is said to be one that Google asked prospective employees in job interviews. My phrasing is almost identical to how it was presented to me.) PUZZLE: Suppose that in a very large population, each family has children until the first boy is born, and then has no more. What is the percentage of girls in the population? (Of course, make all the usual simplifying assumptions in this kind of problem.) --Dan ________________________________ Phooey, this isn't at all a mathematical puzzle. A social convention cannot override biology, so the proportion of boys and girls is the biologically determined one, nominally 1/2, 1/2. -- Gene
I'll try channeling von Neumann on this one: B = 1/2 + 1/4 + 1/8 + 1/16 + ... = 1 G = 0/2 + 1/4 + 2/8 + 3/16 + ... = 1 => B = G On Jul 7, 2010, at 6:41 PM, Dan Asimov wrote:
(This problem is said to be one that Google asked prospective employees in job interviews. My phrasing is almost identical to how it was presented to me.)
PUZZLE: Suppose that in a very large population, each family has children until the first boy is born, and then has no more. What is the percentage of girls in the population?
(Of course, make all the usual simplifying assumptions in this kind of problem.)
--Dan
I sleep as fast as possible so I can get more rest in the same amount of time.
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
When I coded up Monty Hall, I was convinced I had a bug when I saw the results. But for this problem, I was convinced before I finished the code. This code also ends up showing the average number of expected children per family. #include <stdio.h> #include <stdlib.h> main() { int boys=0, girls=0; int families; for (families = 0; families < 1000000; families++) { while (1) { int coin = random() % 2; if (coin) { boys++; break; } else { girls++; } } } printf("Boys: %d Girls: %d\n", boys, girls); }
Some discussion here http://mathoverflow.net/questions/17960/google-question-in-a-country-in-whic... On Wed, Jul 7, 2010 at 3:41 PM, Dan Asimov <dasimov@earthlink.net> wrote:
(This problem is said to be one that Google asked prospective employees in job interviews. My phrasing is almost identical to how it was presented to me.)
PUZZLE: Suppose that in a very large population, each family has children until the first boy is born, and then has no more. What is the percentage of girls in the population?
(Of course, make all the usual simplifying assumptions in this kind of problem.)
--Dan
I sleep as fast as possible so I can get more rest in the same amount of time.
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Thane Plambeck tplambeck@gmail.com http://thaneplambeck.typepad.com/
participants (8)
-
Dan Asimov -
Eugene Salamin -
Gareth McCaughan -
Jason -
mbgreen＠cis.upenn.edu -
Mike Speciner -
Thane Plambeck -
Veit Elser