So the clear Bayesian version is: Mathematician says "I have two children", and you say, "Please tell me the sex of one of them", and she says "male". What's the chance both are boys?

One step back, though. The prior probability of being asked: "One's a girl. What's the chance both are boys?" is probably close to 0.

So the correct question to avoid that prior is: "What's the distribution of probabilities over 2 girls, one of each, and 2 boys?", not "What's the chance both are boys?"

So the clear Bayesian version is: Mathematician says "I have two children", and you say, "Please tell me the sex of one of them", and she says "male". What's the chance both are boys?

One step back, though. The prior probability of being asked: "One's a girl. What's the chance both are boys?" is probably close to 0.

So the correct question to avoid that prior is: "What's the distribution of probabilities over 2 girls, one of each, and 2 boys?", not "What's the chance both are boys?"