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Jan 4 |
comment |
Google question: In a country in which people only want boys
I'm not assuming any specific model, but pointing out that differences between your answer and 1/2 arise from artificial (i.e., gender-asymmetric) conditioning of the problem. Assuming $k$ families as in Zare's model or your present suggestion, is equivalent to assuming "at most $k$ boys in population", or exactly $k$ boys if it is also assumed the families complete their reproduction. No such asymmetric conditioning was part of the Google problem. Your calculations show that a symmetrical distribution can be approximated by asymmetric ones, not that the Google problem is asymmetric. |

Jan 4 |
comment |
Google question: In a country in which people only want boys
Steven, that is incorrect. The issue is whether the proportion of boys, denoted $f(B,G)$ above, is convex as a function of two variables so that $E[f(B,G)] > f(E[B],E[G]) = 1/2$. It isn't convex, as simple calculations demonstrate. (See the comments on convexity and Jensen's inequality under Douglas Zare's posting). It is convex if you condition on B (i.e., restrict f to lines B=constant), and concave if you condition on G. Such conditioning is foreign to the Google problem and imposed artificially in Doug's model. |

Jan 4 |
comment |
Google question: In a country in which people only want boys
Ratios versus differences doesn't address the main point, which is whether the family reproduction rule can break boy/girl symmetry in the underlying distribution of $(B,G)$. [It does gender-asymmetrize the allocation of boys and girls into sets called "families", but this extra structure does not play a role in the calculation requested by Google.] If the distribution is symmetrical then the proportion of girls will have expected value 1/2, because the random variables "proportion of girls" and "proportion of boys" will have the same probability distribution, and their sum is equal to 1. |

Jan 3 |
comment |
Google question: In a country in which people only want boys
@DZ: the statement "[allowing unfinished families] ... there is a higher proportion of girls when the population is larger" is in general false. It is true only in your a priori asymmetrical model conditioned on the number of families. The asymmetry arises not from the stopping rule, but because the stopping rule allows phrasing of boy/girl asymmetric conditions ("the number of boys is at most $k$") in equivalent terms without direct reference to $B$ or $G$ (i.e., "the number of families is $k$", as in your model allowing unfinished families). This asymmetry is foreign to the Google puzzle. |