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Paradox of Envelopes with Money


Imagine you are on a television show. The presenter asks you to choose one of two sealed envelopes. Both envelope A and envelope B contain some money, but the host does not tell you how much money is in each. The only thing he says is that an envelope contains the twice as much money as the other.

You choose envelope A, open the envelope and find that inside it is $ 100.00 (one hundred reais). The presenter then offers you the following offer: you can either get the $ 100 or choose to change envelopes by choosing envelope B.

This way, you can do the following reasoning: Because one envelope contains twice as much money as the other, envelope B can contain either $ 200 or $ 50, with the same probability for each. Since I have more to gain (+ R $ 100) than to lose (-R $ 50), I should make the switch.

But just as you decide to tell the presenter that you want to change envelopes, a thought arises: If you had chosen envelope B - whether you had earned $ 200 or $ 50 - you would have arrived exactly the same conclusion. Therefore, if the above argument is valid, you should change envelopes regardless of the initial choice. Does that make any sense?