A & B are two close friends. They go to an ATM centre and withdraw arbitrary amounts of money, the range being from 0 to infinity. After they come out, what's the probability that A has more money than B? Assume that they had no money before goint to the ATM.

One way to approach would be to say that since A has no preference over B, that A has more money than B and that B has more than A, should be equally likely. They are mutually exclusive too. But they are not exhaustive since the case that A and B have equal amounts of money is not counted.

But whatever be the case, the two cases stated above ARE equally likely.

Now, consider this game :-

A & B take out their wallets and whoever has lesser amount of money gives all of it to the other person. That's it.

Consider this game from A's point of view. In any case, what he can lose is obviously less than what he can gain. So the game is advantageous for A. But the same argument holds for B. So the game is advantageous for B too! That's not possible!