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Boys and Girls: Solution not quite right

Just because the sum isn't infinite doesn't give *any* more reason to suppose slightly more boys than it gives to suppose slightly more girls.

The fact is, there's no need to sum the infinite series to solve this problem. If 50% of births are boys, no strategy for when to *stop* having children is going to change that. That's like saying that I can beat the house at Vegas, even by a little bit (given the non-infinite nature of time) by playing until I win exactly once. When I decide to stop playing won't change house odds.

Avrom Roy-Faderman
Friday, October 08, 2004

Another way to see the problem with the posted solution:

The posted solution states, "There will be N boys". But there *won't* be, for the same reason the series isn't really infinite: People won't literally keep trying forever until they get a boy. They're likely to stop before they have, say, 100 girls, even if they are yet to produce a son. If you assume that people really *do* keep trying until they have a boy (even if they have to have a billion children to do so), then yes, there will be N boys--but then, the series will be infinite, too, so there will also be N girls.

(This is probabilistic, of course--there may be more or fewer girls than N. But the expected value will still be N. Exactly N, not N minus some small error term.)

Avrom Roy-Faderman
Monday, October 11, 2004

Yes, the first poster has got it right.  Here's a clean way to explain it:  At each step, some subset of the couples will have a baby.  50% will get boys, 50% will get girls.  This adds an even number of girls and boys to the population, regardless of who was having babies.  Since we made no assumption about what subet of couples have a baby at any step, then we don't even need to talk about the stopping strategy.  So there will NOT be slightly more boys.  For the same reason, we also needn't talk about lifespans, or new couples entering into the population because they grew up, etc.

Graham Fyffe
Sunday, November 07, 2004

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