I was so confused about this question and all the answers for so long, until it finally clicked. To me, this “puzzle” very obviously reads as your new acquaint

I was so confused about this question and all the answers for so long, until it finally clicked. To me, this “puzzle” very obviously reads as your new acquaintance Dave volunteering some information about his kids. This means the probabilities are completely independent, and the answer is obviously 1/2. But the intention is to make it sneaky and complicated, in the style of bad, poorly worded puzzles. While it is phrased as a conversation, it is actually about drawing a sample from a random cohort, filtering based on gender and birth day, then determining the probability of the result. Which is completely different from how the puzzle is phrased.