A baby is added to a hospital nursery. Before the baby was added there were two boys in the nursery and an uncounted number of girls. After the new baby is added a baby is selected at random among all the babies. The selected baby is a boy.

What is the probability that the added baby was a girl?

Let G be the number of girls originally in the nursery.

The probability that the newest born was a girl given that a random selection of the latest population chose a boy is the probability that the newest born was a girl and that a boy would be chosen, divided by the probability that a boy would be chosen regardless of which sex the latest born was. This comes out to:

(1/2)(2/(2+G+1)) / ((1/2)(2/(2+G+1)) + (1/2)(3/(3+G)))

Of course 2+G+1=3+G, the new baby population of the nursery, with the former representation emphasizing the original 2 boys would still be the only 2 boys, and the latter representation showing the new 3-boy component.

But the equality of the two allows multiplying the numerator and denominator by this value. At the same time we can divide the numerator and denominator by the (1/2) factor that they share.

The result is 2/5. That's the probability that the latest born was a girl given that a random selection of the new population resulted in a boy being selected.

This assumes a priori that the probability was 1/2 of any given baby being a girl.

Posted by Charlie on 2003-10-08 10:25:35