[Puzzler] Counter-intuitive babies

[Mike]

I did not solve this problem. I was so confident in my answer and so confounded when I was told it was wrong that I didn't even attempt the math to try to sort it out. I wish I had. I don't know if I would have gotten it.

A hospital nursery contains 2 baby boys and some unknown number of baby girls. A new baby is born and placed in the nursery. Shortly after, a random baby is picked from the nursery (so this includes all the original babies and the newborn). The randomly selected baby happens to be a boy.

Given all of that, what is the probability that the newborn baby was a boy?

The counter-intuitive part: The answer is NOT 50%. There is a single, definite answer, and the problem gives you enough information to determine what that is.

[FlameBlade]

Very nearly fell into trap here. First, assume that boys and girls has equal chance of being born.

Two cases.

2 boys, k+1 girls. Pr = 2/k+1+2

And

3 boys, k girls. Pr = 3/ k+3

So, probability is

2.5/k+3.

Fine.

5/2k+6

[Mike]

You've calculated the odds of selecting a boy at random.

Now what are the odd that the newborn is a boy given that a randomly selected baby was a boy?

[I may not have explained the problem clearly enough.]

[FlameBlade]

Gah. Missed newborn part.

Something something Bayesian. Gonna need paper and pencil for this one.

[FlameBlade]

3/5. That's why people hate Bayes.

My intuition would have been wholly wrong.

[FlameBlade]

I figured out how explain what should have been the correct intuition from step one. Invis spell!



End invis spell.

[FlameBlade]

As a check:



Just a thought.

[Mike]

Your final formula is dead-on for the generalized case. As the number of starting boy babies approaches infinity, the probability for this situation approaches 50%.