Roseann
Duncan Chiu wrote:
>
> 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
> > > the answer.
> > >
> > > 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