## Expectations and bets

More on controversial probability puzzles.

Professor Landsburg discusses an interesting probability puzzle from Google interviews:

http://www.thebigquestions.com/2010/12/22/a-big-answer-2/

He is even betting $15,000 that he is [quote] *right (with detailed terms described below)* [unquote]:

http://www.thebigquestions.com/2010/12/27/win-landsburgs-money/

The original problem statement as appears at http://www.businessinsider.com/15-google-interview-questions-that-will-make-you-feel-stupid-2010-11#in-a-country-in-which-people-only-want-boys-3 is

In a country in which people only want boys, every family continues to have children until they have a boy. If they have a girl, they have another child. If they have a boy, they stop.

What is theproportion of boys to girlsin the country?

This is a scary question (later I’ll explain why), and I agree that it ought to be reformulated as “**what fraction of the population is female**”.

I personally would never interpret this question as “**what is the expected value of the fraction over populations with a fixed number of families**”. I would interpret it the usual way: “**how frequently are girls born in a country consisting of a large number of families**”. And I would indeed come up with the “official” answer – 50%. Still, Landsburg gives an interesting analysis and explanations, and I must agree that his interpretation is more mathematically entertaining than the “official” one.

Now let me explain why I always become suspicious when I hear about “**average value of the ratio of two variables that are not independent**”.

We break a stick at a random (uniformly distributed) point. What is the ratio of the length of the longer piece to the length of the shorter piece?

Do you want to compute the expectation of the ratio? Good luck. But if we have 1,000 broken sticks and I bet that the average ratio is less than 17, will you take the bet? Note that “**Average fraction**” is an anagram of “**covering fat area**” and “**Be nominated**” is an anagram of “**Bet on median**”.

Happy New Year!

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