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Three pancakesThree 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:
Below is a simulation device you may use to gain insight into the problem. To make a selection click on one of the radio buttons. Once you did the program will show your guess. It is one of the four combinations G/g, B/b, G/b, or B/g where the first (capital) letter describes the front color while the second (lower case) one describes the back color. Selection is erased after one second and computer reshuffles the "pancakes" so that you can make another selection. If you hit the B/b combination you win. You lose if your selection is B/g. Other cases do not count. The simulation works the following way. First, I randomly select a position for the B/b combination. Next, the G/g combination is randomly placed into one of the remaining two position. Lastly, I randomly select one of the two combinations B/g or G/b and display it in the remaining position. Note that this way the combination B/b is twice as likely to appear as the combination B/g. The reason for this is that the combination B/b has two brown sides and, when the pancake is selected, one does not know which side one looks at whereas, when B/g is selected, the observed side is clearly specified - this is its only brown side. Two different solutions are given after the simulator device. Which is the right one? Important note:You run a simulation. During a simulation you are allowed to make as many selections as indicated in the "To Go" control before your first selection. Remember also that after each selection the device needs approximately 1 second to clear up controls. Please wait till it does. To start a new simulation please press the "Reset" button. 
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.
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.
Copyright © 1996-2008 Alexander Bogomolny |
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