Similarly if you had the knowledge “at least one is a boy born on May 11” then it would be very close to but slightly less than 50%. So we can see in the limit

Similarly if you had the knowledge “at least one is a boy born on May 11” then it would be very close to but slightly less than 50%. So we can see in the limit as the information becomes more and more specific it turns into the unconditional probability. That is, the case of “the first is a boy, what is the probability both are boys” (50%). I think this clarifies the situation in the OP pretty well.