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The simple resolution above assumed that the person who invented the argument for switching was trying to calculate the expectation value of the amount in Envelope A, thinking of the two amounts in the envelopes as fixed (''x'' and 2''x''). The only uncertainty is which envelope has the smaller amount ''x''. However, many mathematicians and statisticians interpret the argument as an attempt to calculate the expected amount in Envelope B, given a real or hypothetical amount "A" in Envelope A. One does not need to look in the envelope to see how much is in there, in order to do the calculation. If the result of the calculation is an advice to switch envelopes, whatever amount might be in there, then it would appear that one should switch anyway, without looking. In this case, at Steps 6, 7 and 8 of the reasoning, "A" is any fixed possible value of the amount of money in the first envelope.
This interpretation of the two envelopes problem appears in the first publications in which the paradox was introduced in its present-day form, Gardner (1989) and Nalebuff (1988).<ref>{{Cite journal|last1=Nalebuff|first1=Barry|date=Spring 1988|title=Puzzles: Cider in Your Ear, Continuing Dilemma, The Last Shall Be First, and More|journal = Journal of Economic Perspectives|volume = 2|issue=2|pages=149–156|doi = 10.1257/jep.2.2.149 |doi-access=free}} and Gardner, Martin (1989) <i> Penrose Tiles to Trapdoor Ciphers: And the Return of Dr Matrix. </i> </ref>)
It is common in the more mathematical literature on the problem. It also applies to the modification of the problem (which seems to have started with Nalebuff) in which the owner of envelope A does actually look in his envelope before deciding whether or not to switch; though Nalebuff does also emphasize that there is no need to have the owner of envelope A look in his envelope. If he imagines looking in it, and if for any amount which he can imagine being in there, he has an argument to switch, then he will decide to switch anyway. Finally, this interpretation was also the core of earlier versions of the two envelopes problem (Littlewood's, Schrödinger's, and Kraitchik's switching paradoxes); see [[Two envelopes problem#History of the paradox|the concluding section, on history of TEP]].
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Alternatively, we do go on ad infinitum but now we are working with a quite ludicrous assumption, implying for instance, that it is infinitely more likely for the amount in envelope A to be smaller than 1, ''and'' infinitely more likely to be larger than 1024, than between those two values. This is a so-called [[Prior probability|improper prior distribution]]: probability calculus breaks down; expectation values are not even defined.<ref name=":1" />
Many authors have also pointed out that if a maximum sum that can be put in the envelope with the smaller amount exists, then it is very easy to see that Step 6 breaks down, since if the player holds more than the maximum sum that can be put into the "smaller" envelope they must hold the envelope containing the larger sum, and are thus certain to lose by switching. This may not occur often, but when it does, the heavy loss the player incurs means that, on average, there is no advantage in switching. Some writers consider that this resolves all practical cases of the problem.<ref name=":2">{{Citation | first = Barry | last = Nalebuff |title = Puzzles: The Other Person's Envelope is Always Greener| journal = Journal of Economic Perspectives | volume = 3 | issue = 1 | pages = 171–181 | doi=10.1257/jep.3.1.171| year = 1989 | doi-access = free }}.</ref>
But the problem can also be resolved mathematically without assuming a maximum amount. Nalebuff,<ref name=":2" /> Christensen and Utts,<ref name=":3" /> Falk and Konold,<ref name=":1" /> Blachman, Christensen and Utts,<ref>{{cite journal |first1=NM |last1=Blachman |first2=R |last2=Christensen |first3=J |last3= Utts |year= 1996 | journal =The American Statistician |volume=50 |issue=1 |pages= 98–99 | doi = 10.1080/00031305.1996.10473551 |title=Letters to the Editor}}</ref> Nickerson and Falk,<ref name=":0" /> pointed out that if the amounts of money in the two envelopes have any proper probability distribution representing the player's prior beliefs about the amounts of money in the two envelopes, then it is impossible that whatever the amount ''A=a'' in the first envelope might be, it would be equally likely, according to these prior beliefs, that the second contains ''a''/2 or 2''a''. Thus step 6 of the argument, which leads to ''always switching'', is a non-sequitur, also when there is no maximum to the amounts in the envelopes.
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