When do we sample? This is frustratingly obvious. If we sample when everyone is done reproducing, but ah, no, we have a problem with that.

So we have to pretend that we have a finished, stable society. Hmmm, but ok.

Let’s take a better simplification. We have 2 families. Probably 1 will have one boy. And the other is likeliest to have a girl and a boy, or 2 girls and a boy. There are 3 or 4 children in this society, and 2/3 or 2/4 are boys.

Let’s go bigger. Four families. 2 have one boy. 1 has a girl and a boy. The last has 2 or 3 girls and a boy. So we have 4 boys out of 7 or 8 children.

Bigger: Eight families. 4 with one boy. 2 with a girl and a boy. 1 with 2 girls and a boy. and 1 with 3 or 4 girls and a boy. 8 boys out of 15 or 16 children.

Clearly we are moving toward LSK’s series.

]]>There’s a 1/2 chance of having no girls (B), a 1/4 chance of having 1 girl (GB), a 1/8 chance of having two girls (GGB), etc. In general, there is a 1/(2^n) chance of getting n – 1 girls.

To find the expected value, you can add up the sum of the series (n – 1)/2^n. This is a sum of an infinite number of geometric series:

1/2 + 1/4 + 1/8 +…

+ 1/4 + 1/8 + 1/16 +…

+ 1/8 + 1/16 + 1/32 +…

and so on. The sum each series is 1, 1/2, 1/4, 1/8, etc. The sum of the sums is therefore [2]. ]]>

In the first method, a member of a large family has less chance of being chosen. Since large family also has many girls, girls are less likely to be chosen, accounting for the probability of less than 50%.

]]>https://mathoverflow.net/questions/17960/google-question-in-a-country-in-which-people-only-want-boys ]]>

n

——

2^(n+1)

from n=1 to infinity, we get a total of 1, for an average of 1 girl per family. And, by definition, there is 1 boy per family. Hence, the ratio is 50-50.

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