KGS math club/solution 11 4

The maximum expected gain from your bet is 0.

Suppose there is a strategy that does better than this. There must be some point at which a user of this strategy believes it is profitable to bet rather than waiting for the last card. He will be betting that the current top card of the pack is at that point more likely to be black than the bottom card is. This is absurd, at any time the two cards are equally likely to be black.