Eldon has been snooping around in Marilyn’s book, “The Power of Logical Thinking,” and has found an example where Marilyn proves herself wrong. I have the first edition: February 1996. The following problem appears on Page 23.
Dear Marilyn:
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, look at one side, and see that it is brown. What is the probability that the other side is brown?
Robert H. Batts, Acton Mass.
Dear Robert:
It’s two out of three. The pancake you withdrew had to be one of only two of them: And of the three brown sides you could be seeing, two of them also have brown on the other side.
MvS
Then followed a couple of dissenters; both with outstanding credentials. On page 24 Marilyn offered this explanation.
Dear Readers:
Don’t worry. Whenever I’m wrong, we announce it in this column loud and clear. Some readers are still certain that earlier probability answers are wrong and that I simply won’t admit an error. But that’s not the case, and this one wasn’t an exception. The original answer is correct. Note that it’s easier to discover a brown side on a brown/brown pancake than on a brown/gold pancake.
Here’s another way to look at it. Before you pull out any pancake at all, what are the chances that you’ll pull out a pancake with sides that match? They’re two out of three. So if you pull out a pancake and see a gold side, the chances that the other side is also gold is 2/3. Likewise, if you pull out a pancake and see a brown side, the chances that the other side is also brown is 2/3.
MvS
Eldon agrees with this answer and thinks the last paragraph is somewhat genius. The jury is still out as to whether she admits when she is wrong. The last paragraph disagrees with her answer to Our Question.
Eldon's friends in Australia disagree with Eldon's analysis of the prior paragraph. They think that when the pancake was pulled from the hat, and a gold side was discovered, had the statement been "at least one side is gold," then the probability for two golds would be 1/2.
I still agree with Marilyn, count how many were there before the selection. For the distinguishability advocates, distinguish the side which we discovered vs. the side still unknown. Want to change the odds? Change the numbers in the hat. Don’t change hats:
Use the same hat, add one brown/gold pancake and use the same argument.
In the hat you now start with Brown/brown, gold/brown, brown/gold, and gold/gold.
Now, as per Marilyn’s own argument: Before you pull out any pancake at all, what are the chances that you’ll pull out a pancake with sides that match? It's a half.
It depends upon the starting ratio of pancakes in the hat.
For Our Question: A woman and a man (unrelated) each have two children. At least one of the woman’s children is a boy and the man’s older child is a boy. Do the woman and man have equal chances for two boys?
Answer: They do if they started with an equal number of pancakes in the hat, and they did.
As the pancake question contradicts the “at least one is a boy” question, Marilyn was wrong one way or the other. Eldon finds it hard to believe that the same person who wrote the “genius” paragraph could miss the “at least one is a boy” question. One way or the other Marilyn didn’t do her homework.
For my delightful friends in Australia.
We discovered "at least one side is gold."
What did we discover about the undiscovered side?