Three pancakes

Three pancake problem was discussed in the popular "Ask Marylin" question-and-answer column of the Parade magazine. Details can also be found in the "Power of Logical Thinking" by Marylin vos Savant, St. Martin's Press, 1996.

Marylin received the following question:

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 Batts
Acton, Massachusetts

Solutions are available below. Not all give the same result. Which one(s) is (are) incorrect?

Solution #1

The chance is two out of three. The pancake you withdraw had to be one of only two of them: the brown/brown one or the brown/golden one. And of the three brown sides you could be seeing, two of them also have brown on the other side.

Solution #2

Only two pancakes have brown sides, and one of them has brown on only one side. There is a 50% chance that you are looking at the one with brown on both sides.

Solution #3

Here's a solution that makes use of the Principle of Proportionality. At the outset, there are three likely events to extract a pancake: brown/brown, brown/golden, golden/golden. A pancake is taken out and shows a brown side. Relative to the three possibilities, the probabilities of this happening are 1, 1/2, and 0. Since the probabilities ought to add up to 1, the conditional probabilities become 2/3, 1/3, 0. Only in the first case the chosen pancake will have the second side brown. This will happen with the probability of 2/3.

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