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.

But why does it necessarily have nothing to do with biology? It seems you've added a condition to the question that does not exist. Additional data points are quite useful, and using them to find the correct "real world," outcome seems most logical.

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%.