Let me explain differently. Let two events X and Y be independent. Then (by definition) we know P(X=x|Y=y)=P(X=x). The following two statements are assumed to be correct for this problem:

• P(Child 2=Boy)=0.5,
  
• Sex of any child born is biologically independent of the sex of siblings.

These points underpin statements made arguing both ways, e.g. without the opening of the door, P(BB)=P(BG)=P(GG)=P(GB)=0.25. So pretty much every argument on this thread agrees on these two points.

By the definition of independence, P(Child 2=Boy|Child 1=Boy)=P(Child 2=Boy)=0.5.

The question is asking for P(Child 2=Boy|Child 1=Boy). Because of the 2nd assumption, this value is independent of sex of Child 1. To reiterate, P(Child 2=Boy|Child 1=Boy)=P(Child 2=Boy|Child 1=Girl)=P(Child 2=Boy).

What the question is NOT asking is the probability that a family has two boys, assuming AT LEAST ONE child is a boy, i.e. P(Child 1=Boy AND Child 2=Boy l (Child 1=Boy) OR (Child 2=Boy)). Clearly in this case the dependent event is conditional on the sex of Child 1, because we are asking about the sex of Child 1 as well as Child 2.

To further clarify this difference, * P(Child 2=Boy AND Child 1=Boy | Child 1=Girl)=0 (if child 2 is a girl the probability both children are boys is 0). * P(Child 2=Boy | Child 1=Girl) is non-zero (it is in fact 0.5 as explained above).

I agree 100% that P(Child 2=Boy AND Child 1=Boy|Child 1=Boy)>P(Child 2=Boy AND Child 1=Boy), i.e. the probability both children are boys increases given the observation that Child 1=Boy, as it rules out cases where Child 1=Girl. This is like saying the probability that we will have two coin flips in a row land on heads is higher after the first flip lands on heads - before we flip at all it is a 25% chance but after the second flip it is 50%.

However for all the reasons above, P(Child 2=Boy|Child 1=Boy) is NOT greater than P(Child 2=Boy), and they are in fact equal at 0.5.

Tldr; of course a family with two children is more likely to have two boys if one child is known to be a boy, as it eliminates the case of two girls. But that isn't what the question asked. The question asked about the probability that the 2nd child is a boy given the first child is a boy. Those are two independent events, and knowing the sex of the 1st child doesn't alter the probability that the 2nd child is a boy.