You're in a hospital where your son was just born. As a nurse wheels your newborn into the nursery, she remarks that yours is the only boy in the room, and the rest of the babies are girls. Once in the nursery, boys are swaddled in blue blankets and girls are wrapped in pink.
A few minutes later, another baby is brought into the nursery and the baby's father, Tom, introduces himself to you. You couldn't see if his child was a boy or a girl, and before you get a chance to ask him, Tom has gone down the hall.
A few minutes later a baby, swaddled in blue, is brought out of the nursery.
What is the probability that Tom's newborn child is a boy?
I mentioned in my first post that "If the nurse was asked to fetch a random BOY, then Tom's probability of having a boy is unchanged from the a priori 1/2.". This is, of course, because the fact that a boy came out gives us no extra information if the nurse was sent to fetch a random boy.
I was a little troubled when thinking further about this, because we still have three possibilities, and two of them correspond to the case where Tom has a son:
Tom has nurse chose You'd see
daughter your baby blue
son your baby blue
son Tom's baby blue
So doesn't this still mean that Tom's probability of having a son is 2/3?
Well, no it doesn't, I finally realized. If the nurse has been instructed to fetch a baby boy, then the three cases are no longer equally likely. The first case is twice as likely as the last two. The probability that the baby that comes out is yours is now 3/4, but the probability that Tom has a baby boy is still 1/2.
So, no contradiction.