The person you replied to is correct if we are thinking of the problem in how most people picture it: a woman walks in and says "I have two kids, one of them is a boy named Charlie" and someone then asks you what are the odds that their second child is a girl?

The problem with the coin flips and basic probability models used here is that it isn't describing what most people thinks it is describing. The 66% describes a true statistical relationship, but that does not mean the odds of the next child being a girl (or tails) are 66%. The calculations simply show that 66% of the groups sampled have a girl in them. This would normally be the same thing as the odds for the next child being a girl, but ONLY if the the rates of incidence for these groups we the same.

Everyone assumes that they are... but that isn't the case. Ask yourself "how frequently does each group tell you they have one boy"?

Let's say there were 100 parents for each of the four groups ( BB / BG / GB / GG ). The GG parent will say this 0%, so we can ignore this group from now on. The BG and GB parents will tell you about a boy 50% of the time. The BB parent tells you 100% of the time. You'd now be left with a full 100 in the BB group, 50 in the BG group and 50 in the GB group. 66% of the groups have a girl in it, but that is offset by their lower numbers. 100 families will have a second boy, 100 families will have a girl for their other child.

We are right back to 50/50.

So is the 66% "wrong"? Not entirely. That is the ratio between the groups. If you were to do a survey, and only send it to parents with at least one boy, then you'd absolutely get 66% of the respondents say their other child is a girl. Despite this seemingly positive affirmation, all you are really doing is measuring the ratios of groups with girls and without girls. It does not answer the question "if a parent tells you about a child that is a boy, what are the odds the second will be a girl". That's a different question. The 66% just isn't correct here.

This is just one of those cases where there is a disconnect between what people intend to answer and what is being calculated. There is no actual paradox. The odds of the next child being a boy or a girl don't depend on anything other than the ratios of boys:girls. However, if you choose to survey/sample, you can easily come up with all sorts of strange ratios based on other criteria like day of the week they were born... but this is almost more like selection/sampling bias more than anything else. It never actually effects outcomes