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?

(In reply to It really does seem too easy. by Sara)

In some regards, people are incredible probability measurers...for example, we all know that 'e' is a more commonly used letter than 'x'...how many of you had to count to check?
But unfortunately, Bayes' Theorem just isn't something we're good at. It's not intuitive. Anyone who answered 1/2 for this problem should look up a good website about it and read. Seriously, it's fun stuff. Or read this (a bash you over the head it's so simple problem):

Suppose there is an empty nursery. A baby is added to it. Later on we select a baby at random from the room (there's only one in there to choose) - and voila, we picked a boy. Now, what was the probability that the baby we added was a boy?...

I hope nobody answers 50%. Before we peaked, the probability may have been even, but now we have new information. Bayes' Theorem is a way of understanding the implications of this new information for less trivial examples.

A really famous example of how people get themselves into trouble with this is the Monty Hall problem: On a game show, there are 3 doors, A, B, and C. Behind one is a new car. The contestant picks a door, say he picks A. Then the host opens one of the other doors that is empty, say B is empty.
We all know the contestant had a 1 in 3 chance of being right to begin with. Could this new information have changed things? Could Bayes be involved? (sort of, but there are several ways to think about good ol' Monty).

If anyone is still reading, another interesting problem people have is extrapolating probabilities. I mean, we're good at subconsciously counting that 'e' is more common than 'x', but do a lousy job forming theoretical probabilities when you can't really count. I'm floundering while trying to describe this example I once heard, so maybe I better just give the example (simplified):

Janet was a very bright girl, but she never cared much for school. She thought there was too much wrong with the world so she dropped out and joined a group that organizes rallies to save the forests. What probability would you assign to Janet having since done the following:
A) become a librarian.
B) learned to play acoustic guitar.
C) become a bank teller.
D) climbed to head of the forest-rally organization.
E) become a kindergarten teacher.
F) gotten married and raised a large family.
G) become a bank teller, even though she continues to fight for the forests in her spare time.
H) fought off a drug addiction.

I'll explain after a few responses =DDD Oh, and hope I'm not repeating too much stuff already out there...first time here. But fun fun fun!

And, Charlie - you're my hero.

Posted by Ben on 2003-11-28 01:14:00