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Re: [Phys-l] probability problem



On 06/29/2010 05:33 AM, Philip Keller wrote:
As to
JD's objections, I agree that this is not a real life problem. But I
think it could be worded well enough so that the exercise is valid.
Take the dealer's comments out of the scenario and instead let it be
a computer game (guaranteed to be programmed fairly). You are dealt
the two cards face down. There is a button you can click to ask for a
hint. The computer can respond:

1. no hints today -- I am tired
2. you have at least one ace -- I don't know what suit it is
3. you have at least one ace and it is the ace of spades.

Then I think the exercise is intact and it shows the non-intuitive
ways that additional information can affect probability.

That addresses some of the objections in a technical way,
but I'm still not happy with it.

I have observed that whenever I start with an answer and
try to work backwards, constructing a question that will
produce that answer, I usually regret it. It usually
leads to convoluted and confusing questions, not resembling
real-world questions.

The three cases enumerated above don't cover all the
important possibilities. At the very least we have:

1) The one-scenario game, where you don't get any
information.

2) The two-scenario game, where if there is an ace
it will be reported to you. If there is no ace
the non-report is very informative.

3) The three-scenario game, where the ace of spades
will be reported and otherwise any generic ace will
be reported. Both kinds of non-reports are very
informative.

Remember the curious incident of the dog in the night-time.
http://en.wikipedia.org/wiki/Silver_Blaze
This is a very real and very important concept. Rewording
the problem to evade or conceal this concept seems like a
Bad Idea. I see no practical or pedagogical advantage to
doing the combinatorial calculation in a situation where
even if you do the combinatorics correctly you have
calculated something that is conceptually unsound or at
best drastically disconnected from the real world.

*) I think it is a good idea to teach the importance of
"curious" non-reports.
*) Even if you don't want to get into that, I would hope
there are less-contrived ways of motivating combinatorial
calculations of this kind.