Bertrand's box paradox

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Bertrand's Box Paradox is a logic paradox which first appeared in Joseph Bertrand's Calcul des probabilités (1889):

You have three boxes, each with one drawer on each of two sides. Each drawer contains a coin. One box has a gold coin on both sides, one a silver coin on both sides, and the third gold on one side and silver on the other. You choose a box at random, open one drawer, and find a gold coin. What is the chance of the coin on the other side being silver?

The correct answer is one-third; the coin you see is equally likely to be any of the three gold coins, only one of which is opposite a silver coin. However, there is a tendency to fall into the following fallacious reasoning, which has been compared to the Monty Hall problem:

• You cannot be looking at the box SS; so you must be looking at GG or GS
• You were equally likely to pick either one.
• So there must be a 50/50 chance of GG or GS now.

In reality, though, you were not choosing boxes, but drawers. Now that you have a gold coin, all the possibilities are as follows:

• You chose G of GS, and the other drawer contains a silver coin (⅓)
• You chose G₁ of GG, and the other drawer contains a gold coin (⅓)
• You chose G₂ of GG, and the other drawer contains a gold coin (⅓)

This provides for a combined ⅔ chance of the other coin being a gold coin, and thus, the chance of the coin in the other drawer being silver is ⅓.

See also

• Three cards problem
• Monty Hall problem

References

• Michael Clark, Paradoxes from A to Z, p.16;
• Howard Margolis Wason, Monty Hall, and Adverse Defaults.

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