Yes. This explanation makes sense.
Or, a simple reverse explanation:
Given that the jar either contains two brians or a brain and a heart,
what are the odds you will pull a brain out of each of them? It is twice
as likely you'll pull it out of the two brain jar. So, back to the original,
given that you pulled a brain out, you are twice as likely to have pulled
it from the two brain jar than the one brain jar, so 2/3 of the time there
is another brain.
But, this still assume we are not looking into the jar and are pulling out
an organ at random (that's what she said). If we can look, and pull out
a brain, it's back to 1/2 since we ALREADY KNEW each jar contained a brain.
Perry
>
>
> ---------------------
> Forwarded message:
> From: scot@flume.cs.dartmouth.edu
> To: dchiu@idt.net
> CC: scot@cs.dartmouth.edu, IraFrdmn@aol.com, drei96@dimacs.rutgers.edu
> Date: 97-04-28 23:48:55 EDT
>
> Duncan disagreed with my posting:
>
> (Lots of background deleted)
>
> > > 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.
>
> Duncan then said:
>
> > 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.
>
> It COULD have happened, but DIDN'T. Once we know that it didn't, then the
> odd on what DID happen are no longer the original ones. The probablility
> changes based on new information. While that outcome was part of the
> original sample space, it is NOT possible any more (for THIS trial only).
> Therefore we need to compute the conditional probability based on the
> new information that eliminates some of the original sample space from
> consideration.
>
> An alternate problem may make this clearer. The stockroom has jars
> containing pairs of organs. Each jar contains either two hearts or two
> brains. There is an equal number of jars containing 2 hearts and jars
> containing 2 brains.
>
> Now the alien picks a jar at random. There are two possible contents:
>
> 1) Two brains
> 2) Two hearts
>
> a priori, these two outcomes are equally likely. He reaches into the jar
> and pulls out a brain. What is the probability that the other organ in
> that jar is a brain? My claim is that the probability went up from
> 1/2 to 1 that the other organ is a brain the moment the alien pulled out
> a brain, because the original two heart possiblitity is no longer possible.
>
> Scot
>
> Scot
>