You are horribly butchering Bayes' Theorem. Both your examples are incorrect because your two events ('2 girls' and 'at least 1 girl') are not oberved independently of one another, which is a necessary condition to using Bayes' Theorem with a valid result. Great tool but wrong job.
What? Bayes' theorem does not require anything about the events, even more so it is useful when the events are not independent. If events are independent, Bayes' theorem has no usefulness. https://en.wikipedia.org/wiki/Bayes%27_theorem#Statement_of_...
Edit: however now that I look into it, I should rather have used "conditional probabilities" name rather than Bayes'. I always mix both of them, my bad.
