Steve Landsburg posted an ambiguous question at his blog, and people disagree on the answer. That has led to betting on the right answer to the question. Here is the question:
There’s a certain country where everybody wants to have a son. Therefore each couple keeps having children until they have a boy; then they stop. What fraction of the population is female?
This is supposed to be expressed as an expectation. Phil Birnbaum does a good job of explaining where the difference in interpretation lies, although Phil puts it in terms of countries, where it really should be in terms of families.
I wrote a simulation of the problem here, a Python script. Feel free to download and play with it (Python is free). Here are a few sample runs of the program:
Girls=998007. Boys=1000000.
Average Fraction: 0.306338134795
Girls=1000045. Boys=1000000.
Average Fraction: 0.306774090892
Girls=1001524. Boys=1000000.
Average Fraction: 0.307199395218
Girls=998957. Boys=1000000.
Average Fraction: 0.306736128952
The first line of each runs shows the actual number of boys and girls produced by one million families. As you can see, the families produce an even number of boys and girls, so the expected fraction of the population that is female is 1/2.
Landsburg, however, says what he wants is the second answer, which is the average fraction of girls in a family. According to this simulation, the expected fraction of girls in a family is 3/10.
So do you bet against Landsburg? He stated his problem poorly. His statement of the problem clearly asks for the fraction of the population of the country, not a family in the country.
Steve has done this before, presented a problem with incomplete information. (At the link, he does not say if you play the game once or as many times as you like.) He should have stated the question, “What is the expectation for the fraction of girls in a family in such a country.” He should lose the bet.