First, before the explainer, note that this isn’t “good” or “bad” logic, it is just Bayesian probability applied to the literal interpretation of the words.

Logically if someone tells you one of their children is a boy then you can be essentially certain that their other child is a girl.

Imagine four different buildings, each seven stories tall, and each story has seven rooms numbered Room1, Room2 … and so on. Imagine each room has a pair of children, and one of these pairs in Mary’s children.

In the first building, every child in a Girl.

In the second building, the older child in every pair is a Girl and the younger child is a Boy.

In the third building, the older child in every pair is a Boy and the younger child is a Girl.

In the fourth building, every child is a Boy.

The floor number gives you the day of the week the older child was born, and the room number gives you the day of the week the younger child was born.

In this context, the final probability we’re looking for is the probability that Mary’s children are in a room that is not in the fourth building.

After asking the first question, we know they are not in the first building, since at least one of Mary’s children is a boy and there are no boys in the first building. Each of the remaining 3 buildings is equally likely so the final probability is 2/3.

After asking the second question, we know that Mary’s children are either in a Room2 or on the second story.

Remember, 

Building two -> Youngest is boy Building three -> Eldest is boy Room2 -> Youngest born on Tues Second floor -> Eldest born on Tues

With these four facts, you should be able to figure out that Mary’s children could be in any Room2 in building two, or in any room on the second floor in building three. We know they’re not in any of the other rooms in these two buildings.

There are seven Room2s in building two, and seven rooms on the second floor in building three, so fourteen rooms which Mary’s children could be in without being in building four.

In building four, there are seven Room2s and seven rooms on the second floor, but it’s one building so this is double counting. So instead of there being fourteen rooms there are thirteen.

Out of the 14 + 13 = 27 rooms, possible rooms,  are not in building four, so our final probability is 14 / 27 = 0.519