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?

(In reply to re: Easier gedanken att: ALL by Ady TZIDON)

Just to confirm, all of the probabilities can be computed from Bayes' theorem.

Let n be the number of children in the nursery after Tom's child arrives.  Let Bt be the event "Tom's child is a boy" and Gt be the event "Tom's child is a girl."  Let Bs be the event "A boy was selected from the nursery" and Gs be the event "A girl was selected from the nursery."

P(Bt) = P(Gt) = 1/2

P(Bs) = (1/2)(1/n) + (1/2)(2/n) = 3/2n

P(Gs) = 1 - 3/2n = (2n-3)/2n

P(Bs|Bt) = 2/n

P(Bs|Gt) = 1/n

P(Gs|Bt) = (n-2)/n

P(Gs|Gt) = (n-1)/n

P(Bt|Bs) = P(Bs|Bt)P(Bt)/P(Bs) = (2/n)(1/2) / (3/2n) = (2/2n)(2n/3) = 2/3

P(Bt|Gs) = P(Gs|Bt)P(Bt)/P(Gs) = ((n-2)/n)(1/2) / ((2n-3)/2n) = ((n-2)/2n)(2n/(2n-3)) = (n-2)/(2n-3)

So this confirms that the probability that Tom's child is a boy, given that a boy is selected, is independent of the number of girls in the nursery, while the probability that Tom's child is a boy, given that a girl is selected, depends on the number of girls that existed in the nursery prior to the arrival of Tom's child.

This makes some intuitive sense, because we already knew exactly how many boys were there but we didn't know exactly how many girls there were.  If a boy is selected, we know it's either Tom's child or the only other boy that was present - there's no ambiguity, it's always definitely one of these two children.  We can enumerate the possibilities and find that in 2/3 of these cases, Tom's child is a boy, regardless of how many girls there are.

But if a girl is selected, the impact of this information varies based on how many girls were previously present.  If there were NO girls in the nursery prior to the arrival of Tom's child, and then a girl is selected, we know with 100% certainty that Tom's child is a girl.  On the other hand if there were 1000 girls in the nursery, and a girl is selected, then this doesn't tell us very much about Tom's child (it slightly increases the probability that Tom's child is a girl, but the effect is small).  As you've noted, as n increases then selecting a girl gives us less and less information about Tom's child (since it's increasingly likely that the girl selected is NOT Tom's child) and the probability that he had a boy approaches 1/2. 

Posted by tomarken on 2014-05-14 10:21:46