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othercriteria comments on binomial variance problem - Less Wrong Discussion
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Oops, you're right. The variant of the problem I mentioned above got rid of the assumption of binomially distributed boys (equivalently, girls).
The following setup should work, though:
In words, this says that to generate the i-th class, you flip a coin to tell whether it's in program A or program B, conditioned on the program, the proportion of boys is drawn from a program-specific beta distribution, and then the number of boys is drawn from the corresponding binomial distribution. Under the constraints that%20=%200.65)
and %20=%200.45)
, the average proportion of boys matches up with the problem.
However, by taking
or 
small (where 
and 
are adjusted accordingly to maintain the constraint), you can play with the variance so that the observed 55% boys class is more likely under either of the programs. If you had available repeated trials, you might be able to learn 
and 
. In a single trial, you can't be sure that your strategy will do worse than chance.