Re: [Phys-l] probability problem

[John Mallinckrodt <ajm@csupomona.edu>]

WARNING:  Spoiler below for those still thinking about Carl's card  
problem.

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This is a great problem for demonstrating the importance of "sample  
space" considerations.  I don't think the answer to case 1 is  
controversial, but, as Carl suggests, it is easy to convince yourself  
(incorrectly) that the answers to both cases 2 and 3 should be 3/51 =  
1/17.  Case 2 is the hard one and the trouble seems to be primarily  
the result of not appreciating that there are many more ways of  
having a "one ace hand" than there are of having "a one specific ace  
hand."

I easily programmed a spreadsheet to deal me a million two card hands  
in 20 sets of 50,000.  After throwing out the 19407 of 'em that  
happened to consist of two identical cards, I found that

Case 2)  0.0304 +/- 0.0006 of the hands with at least one ace had two  
aces.  (Expected value = 1/33 = 0.0303)
Case 3)  0.0592 +/- 0.0014 of the hands with the ace of spades had  
two aces. (Expected value = 3/51 = 1/17 = 0.588)

(Quoted uncertainties are standard deviations of the mean for the  
twenty trials.)

John Mallinckrodt
Cal Poly Pomona

On Jun 28, 2010, at 6:35 AM, Carl Mungan wrote:

> The following problem comes out of Boas. I have a solution on my
> website. From time to time, someone will email me and say my solution
> is nonsense. In any case, it's a cute problem:
>
> ---
>
> You're sitting across from a dealer. He shuffles a single deck of
> cards and deals you two cards face down. He then looks at them
> without showing them to you. Consider the following three distinct
> scenarios:
>
> 1. He tells you nothing.
> 2. He tells you, "You've got at least one ace."
> 3. He says, "Wow, you've got the ace of spades."
>
> For each of these three scenarios, what is the probability that if
> you now turn over the two cards you'll find that you've got two aces?
> IOW, what odds would you take to bet on it?
>
> ---
>
> The most common disagreement is about scenario 2 vs 3. You folks are
> pretty smart, so I won't spoil anyone's fun by giving away the
> possible solutions at the moment.
>
> I think the problem relates to the idea that entropy is connected
> with what we know about a system and not just about the physical
> state of the system. -Carl
> --
> Carl E Mungan, Assoc Prof of Physics  410-293-6680 (O) -3729 (F)
> Naval Academy Stop 9c, 572C Holloway Rd, Annapolis MD 21402-1363
> mailto:mungan@usna.edu     http://usna.edu/Users/physics/mungan/
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