Maybe I didn’t express myself well, but this strategy should work regardless of the distribution I choose. For example, if I choose a distribution in which 1 has probability 0, than your strategy yield 1⁄2 chance.
If that kind of selection of distributions is possible then there is no free lunch to be found.
For any strategy of envelope switching a hostile distribution selector who knows your strategy in advance can trivially select distributions to thwart it.
Have you looked at the “solution”? There really isn’t a counter-strategy that reduces it to 1⁄2 chance, although there are strategies moving it arbitrarily close to 1⁄2.
If that kind of selection of distributions is possible then there is no free lunch to be found.
For any strategy of envelope switching a hostile distribution selector who knows your strategy in advance can trivially select distributions to thwart it.
Have you looked at the “solution”? There really isn’t a counter-strategy that reduces it to 1⁄2 chance, although there are strategies moving it arbitrarily close to 1⁄2.