SPOILER ALERT: solution presented here. Rot-13 would be immensely painful, so I'll just present some facts from which an LW reader can piece together a solution. The probability of drawing 11 blue balls followed by 9 green balls from an urn that's 45% blue is 7.0567033E-7. If the urn is 65% blue it's 6.8969856E-7. The log-likelihood-ratio is 0.099425348 decibans. So for practical purposes the answer is "my best guess is still 50/50" but the posterior probability is really 0.50572313. If you draw 100 rather than 20 it's 0.52858571.

I think even if it were a real-life problem I would have correctly guessed, without doing arithmetic, which class had more evidence; but I was sort of spoilered by knowing that the answer has to be the "counterintuitive" one.

ETA: another way to do the sums is that each boy provides 1.5970084 decibans of evidence for program A, and each girl 1.9629465 for program B.