The analysis you linked to does not refute anything said here. In fact, the conclusions of the study you showed are a necessary assumption for this problem (that each gender has an equal probability and independent of birth order). This problem has nothing to do with biology but rather using posterior information to come up with a probability given a statistical experiment.

However, I did not vote you down.

It was modded down because it has nothing to do with the statistical riddle, read the other comments and you'll see why it's 66%.