What, if any, is the relationship between Monty Hall and the Boy or Girl puzzles?

The two puzzles seem like they share a very similar structure and a very similar piece of intuition (or counter-intuition). I wonder if anyone has put more thought into clarifying what exactly the relationship is between the two puzzles?

Here are the hopefully-unambiguous statements of each puzzle:

Monty Hall puzzle: At a game-show, there are three doors. Behind one lies the grand prize. The contestant first selects one door but it isn't opened for her. The host then always opens one of the other two doors, always revealing that it is empty. The host then gives the contestant of sticking with the door she first chose or switching to the remaining unopened door. Should she switch?

Boy or Girl puzzle: Consider all two-child families in the world that contain at least one boy. Pick at random one such family. What is the probability that this family contains two boys?

Solution 1:

The problems can be restated to make the answers less counterintuitive.

Monty Hall: Of the two doors that Monty does not open there is exactly one that is hiding a goat, so what is the probability that this is the door you picked, when given that there is probability of two-thirds that it was the door you picked.

• You had an unbiased choice of three doors.   One hid the car, and two hid the goats.   Picking a goat hiding door means you win a car if you switch.

Boy-Girl: From the families of two children, the one you pick has at least one boy, so what is the probability that the family has one boy, when given that of the four equally-possible types of families, one has two boys, two have one boy pair, and the other is the type you did not pick (ie two-girls).

• There are four equally likely family structures: $\{BB, BG, GB, GG\}$. You know that you did not pick the later, so...

• By excluding one type of family structure, you made an unbiased choice from three family structures.   One with two boys, two with one boy (who is the eldest in one type, or youngest in the other).

There is no deep coincidence here.   They are both essentially the same situation.   This is just obscured by the original counterintuitive phrasing of the problems.