Has anyone in the pop math biz tackled the mathematical side of the news story about China's (now abandoned) one-child-per-family policy? Specifically, many families adopted the family planning algorithm "Have kids till you have a son, then stop", which (under idealized assumptions) gives rise to families of average size exactly 2. A non-mathematician friend of mine asked me at dinner last night why the expected size of a family that stops when the first son is born is 2. I began to give him an intuitive argument that doesn't involve calculation, but he ended up preferring the argument that shows that 1/2 + 2/4 + 3/8 + ... = 2 by way of summing the formulas 1/2 + 1/4 + 1/8 + ... = 1 1/4 + 1/8 + ... = 1/2 1/8 + ... = 1/4 ... to obtain 1/2 + 2/4 + 3/8 + ... = 1 + 1/2 + 1/4 + ... = 2 Is there a place where this is explained, and explained well? Jim Propp
Of course, you're assuming that boys and girls are equally likely, and that there's no limit to the number of children that a couple can have, and you're ignoring multiple births. The idealized problem is discussed at https://en.wikipedia.org/wiki/Geometric_distribution, but the explanation isn't terribly enlightening. --ms On 06-Nov-15 14:42, James Propp wrote:
Has anyone in the pop math biz tackled the mathematical side of the news story about China's (now abandoned) one-child-per-family policy?
Specifically, many families adopted the family planning algorithm "Have kids till you have a son, then stop", which (under idealized assumptions) gives rise to families of average size exactly 2.
A non-mathematician friend of mine asked me at dinner last night why the expected size of a family that stops when the first son is born is 2. I began to give him an intuitive argument that doesn't involve calculation, but he ended up preferring the argument that shows that 1/2 + 2/4 + 3/8 + ... = 2 by way of summing the formulas 1/2 + 1/4 + 1/8 + ... = 1 1/4 + 1/8 + ... = 1/2 1/8 + ... = 1/4 ... to obtain 1/2 + 2/4 + 3/8 + ... = 1 + 1/2 + 1/4 + ... = 2
Is there a place where this is explained, and explained well?
Jim Propp _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
This problem was discussed extensively on MathOverflow (2010), though I don't think a unanimous conclusion was reached: http://mathoverflow.net/questions/17960/google-question-in-a-country-in-whic... <http://mathoverflow.net/questions/17960/google-question-in-a-country-in-which-people-only-want-boys>. —Dan
On Nov 6, 2015, at 11:42 AM, James Propp <jamespropp@gmail.com> wrote:
Has anyone in the pop math biz tackled the mathematical side of the news story about China's (now abandoned) one-child-per-family policy?
Specifically, many families adopted the family planning algorithm "Have kids till you have a son, then stop", which (under idealized assumptions) gives rise to families of average size exactly 2.
A non-mathematician friend of mine asked me at dinner last night why the expected size of a family that stops when the first son is born is 2. I began to give him an intuitive argument that doesn't involve calculation, but he ended up preferring the argument that shows that 1/2 + 2/4 + 3/8 + ... = 2 by way of summing the formulas 1/2 + 1/4 + 1/8 + ... = 1 1/4 + 1/8 + ... = 1/2 1/8 + ... = 1/4 ... to obtain 1/2 + 2/4 + 3/8 + ... = 1 + 1/2 + 1/4 + ... = 2
Is there a place where this is explained, and explained well?
I would think of it as: E = (1/2)*1 + (1/2)*(1+E) E = 1 + E/2 so: 2*E = 2 + E E = 2 In other words, half the time your first born is a son, so you stop at 1 child, and the other half of the time your first born is a daughter, so your expected number of children is then 1 + E. Alternatively, you always start with 1 child, and then half the time you repeat. No need for an excplicit series with this approach. Tom James Propp writes:
Has anyone in the pop math biz tackled the mathematical side of the news story about China's (now abandoned) one-child-per-family policy?
Specifically, many families adopted the family planning algorithm "Have kids till you have a son, then stop", which (under idealized assumptions) gives rise to families of average size exactly 2.
A non-mathematician friend of mine asked me at dinner last night why the expected size of a family that stops when the first son is born is 2. I began to give him an intuitive argument that doesn't involve calculation, but he ended up preferring the argument that shows that 1/2 + 2/4 + 3/8 + ... = 2 by way of summing the formulas 1/2 + 1/4 + 1/8 + ... = 1 1/4 + 1/8 + ... = 1/2 1/8 + ... = 1/4 ... to obtain 1/2 + 2/4 + 3/8 + ... = 1 + 1/2 + 1/4 + ... = 2
Is there a place where this is explained, and explained well?
Jim Propp _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
At the risk of reopening an inflammatory thread ... Quoting James Propp <jamespropp@gmail.com>:
A non-mathematician friend of mine asked me at dinner last night why the expected size of a family that stops when the first son is born is 2. I began to give him an intuitive argument that doesn't involve calculation, but he ended up preferring the argument that shows that 1/2 + 2/4 + 3/8 + ... = 2 by way of summing the formulas 1/2 + 1/4 + 1/8 + ... = 1 1/4 + 1/8 + ... = 1/2 1/8 + ... = 1/4 ... to obtain 1/2 + 2/4 + 3/8 + ... = 1 + 1/2 + 1/4 + ... = 2
Given the large collection of probability "paradoxes": Any sensible person would prefer a relatively straightforward calculation with easily understandable components, over any result based on meta-level reasoning, with its possible hidden pitfalls. For the mechanics, multiplying the series by 1 - 1/2 converts it to a familiar sum. This does require accepting that the series manipulation is safe, based on absolute convergence. (1 - 1/2) * (1/2 + 2/4 + 3/8 + ...) = (1/2 - 1/4) + (2/4 - 2/8) + (3/8 - 3/16) + ... = 1/2 + (2/4 - 1/4) + (3/8 - 2/8) + (4/16 - 3/16) + ... = 1/2 + 1/4 + 1/8 + 1/16 + ... = 1, so the original series must sum to 2. Rich
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