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