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Consider the envelope paradox problem:

There are two envelopes both of which contain some money. One envelope contains twice the amount of the other (but other than that, you do not know how much is in them). You select one of the envelopes randomly and see the amount of money inside.

You can opt to either keep the money in this envelope, or switch envelopes, which do you choose?

A (false) argument for switching envelopes:

Let the amount of money in the envelope you selected be $X$.

There is a $50\%$ chance that the envelope with more money is selected, and a $50\%$ chance that the envelope with less money is selected.

Therefore, there is a $50\%$ chance that there is $0.5X$ in the other envelope, and a $50\%$ chance that there is $2X$ in the other envelope.

So the expected payout for switching is ${2X + 0.5X\over2} = 1.25X$ which is better than the payout of $X$ for not switching.

Therefore you should switch.

This argument is wrong... where exactly does it fall?

Shuri2060
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  • Once you see the amount you have more information (since the distribution used to place the money in the envelops is not uniform), and thus it is not correct to say that there is 50% chance the chosen envelop contains the larger amount. – Ittay Weiss Aug 27 '16 at 20:23
  • @IttayWeiss "(since the distribution used to place the money in the envelops is not uniform)" — could you clarify what you mean by this? – Shuri2060 Aug 27 '16 at 20:26
  • The expected payout for not swapping is also $1.25X$ so it is the same for both options and there is no advantage. – John11 Aug 27 '16 at 20:47
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    @John11 $X$ is the amount of money in the envelope you first chose. If you don't swap, you'll definitely get $X$. How could the expected payout be $1.25X$ for not swapping? – Shuri2060 Aug 27 '16 at 20:48
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    @QuestionAsker I meant to use the same argument for the enveloppe you're holding. If, $Y$ was the amount in the other enveloppe, there's a 50/50 chance that what you are holding is twice/half what is in the other enveloppe. The variables $X$ and $Y$ are inter-dependent. – John11 Aug 27 '16 at 21:16

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