Fun with Bayes' Theorem

From: Lee Corbin (lcorbin@ricochet.net)
Date: Fri May 04 2001 - 02:01:15 MDT

You don't really need to know that

P(Ai|B) = [P(Ai)P(B|Ai)]/P(B)

(it probably doesn't help). But it certainly helps to
consult your imagination for the following problems.

1. There are two bags, each with 10 coins. Half the coins
in the first bag are counterfeit, and all the coins in
the second bag are counterfeit. Someone hands you a bag,
and you happen to pull a coin out of that bag and examine
it. If you see that the coin is counterfeit, what is the
probability that you were handed the counterfeit bag?

2. A little girl's father discovers that his wife is
pregnant again (but they don't know the sex of the
unborn child). The man decides to visit a mathematician.
"I have two children, sir", he says, "and one of them
is a girl. What is the probability that the other is
a boy?" What did the mathematician tell him?

3. You're on the Monte Hall show, and there is a big prize
behind one curtain, and junk prizes behind the other two
curtains. You pick one of the three at random. Monte
then opens one of the other curtains and shows you a
junk prize, and asks if you still want to keep playing.
You say, "Yes, but let me switch my choice to the other
curtain!". Monte says, "That's weird," but allows the
switch. What is the probability of your getting the big
prize? (As is widely known, Marilyn Vos Savant humiliated
some experienced mathematicians with this old problem.)

4. You wash up on a desert island where it is known that Long
John Silver has hid a lot of gold in one of three strongly
built shacks. You begin dismantling one of the three. A
lightning bolt comes out of the overcast sky and strikes
one of the other shacks, destroying it and revealing that
the gold wasn't there. What is the probability that the
gold will turn up in the shack that you are not working on?

Hint: one of the answers is 1/2 and the others have the same