Re: [Phys-l] probability problem

[Philip Keller <PKeller@holmdelschools.org>]

Similar method:

Case 1 --  There are 4C2 = 6 ways to get two aces (from the 4 available in the deck) and 52C2 = 1326 ways to pick two cards from the full deck.  So it's 6/1326 = .0045...

Case 2 -- There are still 6 ways to get the two aces, but now the denominator is reduced to 52C2 - 48C2 = 198.  Essentially, we are subtracting 48C2 because that's the number of pairs containing no aces, leaving behind the number that contain at least one.  6/198 = .0303...

Case 3 -- Now there are only 3 ways to get two aces: you have the ace of spades and there are only three other aces available.  And the denominator is now 52C2 - 51C2 = 51.  In other words, this time we have subtracted 51C2 because that's the number of pairs that do not contain the ace of spades, leaving behind the number of pairs that do contain the ace of spades.  (I agree that there is an easier way to land on 51, but I like this way because it extends from and compares to the method for case 2.)

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.
________________________________________
From: phys-l-bounces@carnot.physics.buffalo.edu [phys-l-bounces@carnot.physics.buffalo.edu] On Behalf Of John Mallinckrodt [ajm@csupomona.edu]
Sent: Monday, June 28, 2010 9:28 PM
To: Forum for Physics Educators
Subject: Re: [Phys-l] probability problem

On Jun 28, 2010, at 5:31 PM, Ann Reagan wrote:

> Kudos to John for his analysis. I had a much-less elegant, low-tech
> approach to the same result:

On the contrary!  Yours is the more "high tech" approach, what I
referred to as the "sample space" approach, the approach taken also
by Boas and by Carl in their solutions.  My spreadsheet (dealing out
a million hands!) is the "brain dead" approach.  I used it because I
was not quite convinced by the sample space arguments and I have
noticed in the past (for instance, in trying to understand the Monty
Hall problem the first time I saw it) that the simple process of
programming a simulation often provides sufficient insight to make
running it unnecessary.  Such was not the case here, but it did in
any event corroborate the "sample space" solution (as, now quite
obviously, it must!)

BTW, the Monty Hall problem is best understood by realizing that you
have a one in three chance of having picked the right curtain
initially and EVERY other time, the car ends up behind the one you
can switch to!

John Mallinckrodt
Cal Poly Pomona
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