(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