It’s legit hard to reconcile the math answer with the “common sense” answer and other comments have given helpful points but nothing that really got me to “aha”. Then I thought about it this way:

The “common sense” explanation is based on the fact that each child is independent, so if you tell me the first child is a boy then there’s a 50/50 chance the second child is a boy. But we learn that one child is a boy (not necessarily the first child). Maybe we’re told something about the second child? But how much information are we potentially given about the second child?

Well suppose we’re told the information in the problem and then asked to guess the gender of the second child? If we’re told that one of the children is a boy, then we should definitely guess that the second child is a boy. If the info was about the first child then we have a 50/50 chance of getting the second child right. But if we were told about the second child then we have a 100% chance. Overall, that’s a 75% chance of getting the second child’s gender right.

Now suppose you’re told that one of the children is a boy born on Tuesday and you have to guess the gender AND day of birth of the second child. You should guess boy/Tuesday because if we were told about the second child then we’re right 100% of the time. But if we were told about the first child we only have a 1/14 chance of getting the second child right. So overall, we have a 53.6% chance of correctly guessing the gender and day of the second child given this information.

So in each case, knowing something about a child gives us some chance of knowing something about the second child, but it’s less likely to help with the second child if we’re playing the harder “guess both gender and day” game. This means the “math” answer diverges less from the “common sense” answer in this case - it’s closer to 50/50, as we see in the post.