[Chinjut]

It's not that the probability is close to 1/2 that makes it paradoxical for most people. It's that the probability differs from 1/2 at all. As in the OP of this very thread saying "Somehow knowing the day of the week the boy was born changes the result. It's completely bizarre."

[justinpombrio]

It's the fact that it differs from 2/3. If the day of the week was not mentioned the (conventional) answer is exactly 2/3. Not 1/2.

[Chinjut]

Yes, that's also "paradoxical", though probably not the paradox that would trip people up first unless they'd seen the other problem first. But, you're right that I may have misread which departure from expected answer was bugging the OP. Nonetheless, everything else I stated still holds.