# Re: Can you help?

Duncan Chiu (dchiu@idt.net)
Sun, 27 Apr 1997 13:50:30 -0400

scot@flume.cs.dartmouth.edu wrote:
>
> 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

Scot,

I think the probability still is 1/2. As you stated above, it should be
2/4 the jar contains an original brain [1) and 2) of the four.] I don't
see why the case 4) has to be discarded. Isn't it also a part of the
original sample space? Just because it does not happen (the alien does
not pick a heart) doesn't mean it's still *couldn't* happen. Therefore
it is still a possibility, making the chances 2 out of 4.

duncan