# 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*

*> > 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