Suppose someone shows you two closed envelopes containing money and proposes you the following game.

• One envelope contains exactly twice the amount of the other envelope
• You choose one envelope at random and open it
• Now you may decide: Keep the money (and not touch the other envelope) or take the other envelope (and not keep the first)

Question: Should you change or not?

Naive answer: The setting is symmetric, and knowing the amount in the first envelope does not give you any hint. So it does not matter if you change or not.

Refutation: After having observed the first amount x, the probability that the second envelope contains twice as much is one half. Therefore, if you change your choice, with probability 0.5 you gain x, while with probability 0.5 you loose x/2. Therefore you should change.

What do you think?

Want to read my extended solution?