# Re: Can you help?

scot@flume.cs.dartmouth.edu
Mon, 21 Apr 1997 23:07:54 -0400 (EDT)

Ira posted the following question:

> A friend of mine gave me the following question. Ihope that you can give
>
> A jar contains one organ either a brain or a heart. An Alien puts a brain in
> the jar. He then pulls out a brain. What`s the probability that the jar
> originally contained a brain. I think that this is the "LETS MAKE A DEAL"
> problem and I feel that the answer is 2/3 but then sometimes I think it is
> 3/4?

I deleted the answer given by Bro. Pat (I hope that I remembered it correctly)
that gave an answer of 2/3, and claimed that this problem was isomorphic to
the problem of having two children, one of whom is a boy. What is the chance
that the other one is a boy?

This answer made it clear to me what was bugging me when I tried to figure
out this problem - as stated we don't know what the chance is that the jar
originally contained a brain. This is a problem in conditional probability.
The fact that the alien pulled out a brain MAY change the probabilities from
the a priori probabilities. Let me re-state the problem in a way that I
think the proposer intended it, solve it, and then explain why I believe
that the problem has to be clarified.

An alien picks a jar at random from a stockroom. The stockroom contains
an equal number of jars containing a single brain and jars containing a
single heart. The alien then adds a brain to the jar, shakes it, and
removes at random one of the two organs in the jar. The organ removed turns
out to be a brain. What is the probability that the original jar contained a
brain?

Stated this way, we can solve the problem. There are two equally likely
possibilities for the original jar - it contained a brain or a heart. The
four equally likely possibilites for the outcome of the procedure are:

1) Original brain. Picked added brain.
2) Original brain. Picked original brain.
3) Original heart. Picked added brain.
4) Original heart. Picked original heart.

The first three cases lead to the outcome observed. The fourth does not,
so cannot be the case we are in. We have three equally likely remaining
cases, and in two of them the original organ was a brain. Therefore the
probability that the original organ was a brain is 2/3.

Note that I made two important assumptions. First, the alien picks one of
the two organs at random. If he always picks the object that he just added
then we have no additional information about the original contents of the
jar. The second is that the jar containing the organ was chosen from a
distribution with equal probability that it contained a heart or a brain.
Note that if the storeroom contained 2 hearts for every brain, then the
four outcomes would not be equally likely. The first two cases would occur
half as often as the last two. Therefore when the fourth case is ruled out
the first two cases would exactly balance the third and the probability that
the original organ was a brain would be 1/2.

Scot