Alice has freely chosen to put either a gold coin or a silver coin in each of an infinite sequence of envelopes numbered 1,2,3,... Bob can open any number of envelopes and check the coins within, provided he leaves at least one envelope untouched. For all the untouched envelopes, Bob then must guess whether it's gold or silver coins within. If he gets all of them correct, Bob wins, otherwise Alice wins. Both seeks to maximize their own winning probabilities.
Clearly, Bob can just leave envelope #1 untouched and randomly guess silver or gold to ensure a 50% winning probability. Can he do better than that?
Bob can use the axiom of choice to his advantage.