Re: Can you help?

breuche (breuche@dimacs.rutgers.edu)
Wed, 30 Apr 1997 21:00:21 -0400

Duncan Chiu wrote:
>
> 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

Duncan,

If you play poker with me everybody wins!!! I wish I could have shown
you.

Ethel