Randomly pick one of my children. He is a boy. What is the probability of the other being a boy? 
  Yes, that would be 1/2.

To be clear, that probability would be 1/3. But I believe i understand where you are getting at. You have to phrase the question in the form "pick one of my 2 children, he is a boy with property x=x0, what s the prob that the other is a boy?" .

The probability is prob = ((N/2)-1)/(N-1) where N=2 * 2 * (number of different property values for x). For N>>1, prob tends to 1/2