# Re-Statisitics puzzle i0524

Eddy Sean sre at al.cam.ac.uk
Wed Jun 15 15:02:56 EST 1994

```In article <1994Jun15.065418.2854508058 at cbrc.mgh.harvard.edu> Joanne_Ownbey at CBRC.MGH.HARVARD.EDU writes:
>Being mathematically challenged- I'm confused.  How can the probability
>that he holds the ace of hearts go up and down?  Also, unless the ace's are
>somehow linked why should the fact that west holds the ace of d. influence
>whether he holds the ace of hearts, (other than by cutting the possibilties
>(ie, number of cards he holds) down by one?

The probabilities shift to reflect new information about the
situation. Take an extreme case; say I flip fifty-one cards face up,
leaving west with a single card; none of the face-up cards is the ace
of hearts.  The probability that west holds the ace of hearts is now
1.  The crux of the problem is how to smoothly merge observed data
with the prior probability of models, and Bayes' theorem is a
straightforward way to chew through a problem like this.

The reason it matters whether west holds the ace of diamonds is that
originally there were three a priori equiprobable models consistent
with spying a red ace in his hand: he's got both, he's got the heart,
or he's got the diamond.  When we see him lead the diamond, we
eliminate the model that he's got only the heart; this causes the
probability that he's got the heart to drop, since the only way he can
have it now is if he had both of them.

The 50-50 answers reflect only the prior probability; they aren't
taking into account the extra observed data of spying a red ace in
west's hand, or of seeing him lead the diamond.

--
- Sean Eddy
- MRC Laboratory of Molecular Biology, Cambridge, England
- sre at mrc-lmb.cam.ac.uk

```