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Solution: Three Envelopes Problem – in a nutshell.

April 3, 2019

Solution to Exercise

It seems at first sight that by switching envelopes you either gain £100 (£200 instead of £100) or lose £50 (£100 – £50) with equal likelihood. The flaw in the logic is in the way that the switching argument is framed, and it is contained in the possible amounts that could be found in the two envelopes. As framed in the original argument for switching, the amount could be £100, £200 or £50. More generally, there could be £X, £2X or £1/2 X in the envelopes. But we know that there are only two envelopes, so there can only be two amounts in these envelopes, not three. In order to resolve the paradox, you must label the envelopes before you make your choice, not after. So envelope 1 is labelled, say, A, and envelope 2 is labelled, say, B. A corresponds in advance to, say, £100 and B corresponds in advance to, say, £200, or to £50, but not both.

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