From: dseal@armltd.co.uk (David Seal)
Newsgroups: sci.math
Subject: Re: Marilyn Vos Savant's error?
Date: 16 Dec 92 17:22:12 GMT
Distribution: sci
Organization: A.R.M. Ltd, Swaffham Bulbeck, Cambs, UK

In article <1gj5grINNk05@crcnis1.unl.edu> burchell@cse.unl.edu (David
Burchell) writes:

>I ask the net Gods: Can this be correct??
>
>From ``Ask Marilyn'' by Marilyn Vos Savant, Parade magazine, December
>13, 1992.
>
>You have a hat in which there are three pancakes.  One is golden on
>both sides, one is brown on both sides, and one is golden on one side
>and brown on the other.  You withdraw one pancake and see that one side
>is brown.  What is the probability that the other side is brown?
>
>---Robert H. Batts, Acton, Mass.
>
>It's two out of three.  The pancake you withdrew had to be one of only
>two of them: the brown/golden one or the brown/brown one.  And of the
>three brown sides you could be seeing, two of them also have brown on
>the other side.

The usual problem with probability questions comes up yet again... People
almost never state exactly what is being randomised or what the distribution
is, leaving it up to the reader to make assumptions. In this particular
case, you need an assumption about the distribution of the pancake chosen:

  Assumption 1: the three pancakes are chosen with probability 1/3 each.

and one about which side of the pancake you see:

  Assumption 2 - version A: you saw a random side of your pancake, with the
  two sides being chosen with probability 1/2 each.

  Assumption 2 - version B: the words "one side is brown" are actually
  intended to mean that you look at both sides and see that at least one
  side is brown (then conveniently forget whether both sides are!).

  Assumption 2 - version C: the brown/golden pancake is always placed under
  the hat brown side up. When you pulled your pancake out, you did so by
  sliding it out, in such a way that you were certain to see the top side
  first.

Using assumption 1 and assumption 2(A), the stated solution is correct.
There are 6 *equiprobable* possibilities:

  Pancake chosen   Side seen   Colour seen
  ----------------------------------------
       B/B          Top            B
       B/B          Bottom         B
       B/G          Top            B
       B/G          Bottom         G
       G/G          Top            G
       G/G          Bottom         G

Knowing that we saw a brown side tells us that one of the first three cases
is correct, and gives us no further information. These three cases are
therefore still equiprobable, so the chance that the other side is brown is
2/3.

OTOH, using assumption 1 and assumption 2(B) gives us an answer of 1/2. The
equiprobable cases are now:

  Pancake chosen   Side seen   Brown seen?
  ----------------------------------------
       B/B          Both        Yes
       B/G          Both        Yes
       G/G          Both        No

Knowing that a brown side has been seen now tells us that we've got one of
the first two cases, which gives a probability of 1/2.

Assumption 1 and assumption 2(C) also give an answer of 1/2, since there are
now three equiprobable possibilities:

  Pancake chosen   Side seen   Colour seen
  ----------------------------------------
       B/B          Top            B
       B/G          Top            B
       G/G          Top            G

The observed facts tell us that only the first two are still possible.

Or if you want a silly case, try:

Assumption 2, version D: this is just like assumption 2(C), except that the
brown/golden pancake is always placed under the hat golden side up. Now the
possibilities are:

  Pancake chosen   Side seen   Colour seen
  ----------------------------------------
       B/B          Top            B
       G/B          Top            G
       G/G          Top            G

and the probability wanted is 1, since the observed facts exclude all but
the first possibility. Of course, you don't need to use probability for
this: under these assumptions, you can deduce logically that the other side
of the pancake is brown.

So to summarise: the error is really in the problem, in that it doesn't
state important facts about how the "experiment" was set up. There's no real
error with the stated solution, though it would have been nice if the
assumptions being made about the problem were stated explicitly rather than
having to be read between the lines.

David Seal
dseal@armltd.co.uk

All opinions are mine only...