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 was about to post the exact same solution as Charlie, which I endorse 100%. The solution assumes two things:
1) a priori, the probability that Tom's child is a boy is 1/2.
2) the child brought out is selected randomly from all children.
The 2nd assumption is crucial. If, for instance, the nurse was asked to fetch a random boy, then Tom's probability is unchanged from the a priori 1/2.
Since Charlie beat me to it, I'll post the Bayesian solution instead:
By Bayes theorem,
P(Tom had a boy|random is blue) =
p(random is blue|Tom had a boy) * P(Tom had a Boy)/
p(random is blue)
Assume n children in total in the nursery.
Then
p(random is blue|Tom had a boy) = 2/n
P(Tom had a Boy) = 1/2 (a priori)
p(random is blue) = (2/n)*(1/2) + (1/n)*(1/2)
P(Tom had a boy|random is blue) =
((2/n)*(1/2))/((2/n)*(1/2) + (1/n)*(1/2)) =
(1/n)/(3/2n) = 2/3
Alternative approach
