While perusing through the past posts, I found out that the "two envelope paradox" has been already discussed here - to explain it, not to use it.

I propose the following principle to use this very mind-puzzling problem. If someone is willing to test this, I'd be very interested to hear about the result !

Thema of the patter : "The world changes once you understand a paradox"

The set-up : Many pairs of envelopes each containing a piece of paper : one normal with a number X written of it (X very different for each pair), and one with a two sided one, with 10X and X/10 written on it.

Ie : (100,1000/10); (4000, 40000/400); (0.1, 1/0.01); etc...

You start to "explain" the "two envelope paradox" with an ungaffed pair 50/500, but without mentioning the word "paradox". In fact you present it so that it is obvious that switching is always the best strategy (I used the 10 factor, because the "result" is that you have "in average" 5.05 times more in the second enveloppe, which is more impressive than the usual 1.25). You open the envelope with 50. You "explain" you've got 50% chances of having 500, and 50% of having 5, so it's better to switch. And indeed you win.

Then, proceed to make the audience always win "if you had switched". If the normal envelope is chosen, open the gaffed one on the 10X side to show that "you have won/you'd have won if you had switched". If the gaffed one is chosen first, open it on the X/10 side.

After several such trials, stop and explain the paradox the best you can Well, just point out that the fact that your reasoning does not depend on the value you read, so that you can decide to switch without opening the first envelop, which is a kind of a problem, because the same could be said for the second one. Give your audience some of the scientific papers written on this problem, filled with symbols of Bayesian statistics nobody will understand. Give the "+9X vs. -9X" odds explanation, etc...

[Note that these scientific papers will all describe the problem somewhere in the text as the "two envelope paradox", which instantly gives a lot of credibility to the fact you're using envelopes in the first place, not just pieces of paper in the hands etc... !]

The idea is that once your explanation is over, you can then change the gaffed odds so that they're back to reality, or even changed into forever "you'd have lost if you had switched" ! Do some other tests, and explain that "see, now that you've understood your first reasoning was wrong, the reality around you has changed to adapt to your new, correct reasoning. That's why the mathematicians never gamble : since they know they can't win agains the casino, they can't win. Only the ones who do not care about maths manage to win."

Then you've set up an atmosphere that helps make you convince your audience that some events that will take place soon will seem impossible to them, but that's because their views of the world are the "too simple ones", not the "correct" ones...

IMO, it's a bit like the tricks where someone doesn't manage to push a straw inside a potatoe, but succeeds once the guru has "convinced" him he had enough strength - while putting the thumb at the end of the straw is the real reason, which is never given. Then the guru continues with a show about our un-used physical strenghts. Here with the envelopes, it may be a good start to continue the show with un-used mind abilities defying all the logics...