Thursday, December 14, 2017

What are the odds?

It's the time of year, you give out special problems in your classes. So this is mine for the blog. It is motivated by this picture of the home secretaries of the German federal states after their annual meeting as well as some recent discussions on Facebook:
I would like to call it Summers' problem:

Let's have two real random variables $M$ and $F$ that are drawn according to two probability distributions $\rho_{M/F}(x)$ (for starters you may both assume to be Gaussians but possibly with different mean and variance). Take $N$ draws from each and order the $2N$ results. What is the probability that the $k$ largest ones are all from $M$ rather than $F$? Express your results in terms of the $\rho_{M/F}(x)$. We are also interested in asymptotic results for $N$ large and $k$ fixed as well as $N$ and $k$ large but $k/N$ fixed.

Last bonus question: How many of the people that say that they hire only based on merit and end up with an all male board realise that by this they say that women are not as good by quite a margin?

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