# Re: Can you help?

Duncan Chiu (dchiu@idt.net)
Tue, 29 Apr 1997 22:46:41 -0400

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

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

So, if I change the [brain] to [heart] in the problem:

"> > > 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 [heart]. What is the probability that the original jar contained a
> > > [heart]? "

the probability then is 1? [Case 4) above.]

I shall test this in our next poker game --- a sure win, right? Thanks.

duncan