quasi wrote: >Paul wrote: >> >>The following puzzle is copied and pasted from the internet. >> >> Alice secretly picks two different real numbers by an unknown >> process and puts them in two (abstract) envelopes. Bob >> chooses one of the two envelopes randomly (with a fair coin >> toss), and shows you the number in that envelope. You must >> now guess whether the number in the other, closed envelope >> is larger or smaller than the one you?ve seen. Is there a >> strategy which gives you a better than 50% chance of guessing >> correctly, no matter what procedure Alice used to pick her >> numbers? > >Yes. > >Let R denote the set of real numbers and let (0,1) denote >the open interval from 0 to 1. > >Let f : R -> (0,1) be a strictly decreasing function. > >Use the following strategy: > >If the initially exposed value is t, "switch" with probability >f(t) and "stay" with probability 1 - f(t). > >Suppose Alice chooses the pair x,y with x < y (by whatever >process, it doesn't matter). After Alice choose that pair, >then, by following the strategy I specified above, the >probability of guessing the highest card is exactly > > (1/2)*f(x) + (1/2)*(1 - f(y)) > >which simplifies to > > 1/2 + f(x) - f(y)
which simplifies to
1/2 + (1/2)*(f(x) - f(y))
>and that exceeds 1/2 since f is strictly decreasing. > >Of course, it's not the case that probability of guessing >correctly is more than c for any fixed c > 1/2, but the >problem didn't require that.