Game A + Game B = Game C. Parrondo's games follow this basic structure where A and B are losing games and C is a winning game-a phenomenon called Parrondo's paradox. These games can take on a wider class of definitions and exhibit these paradoxical results. In this paper, we show three paradoxical cases. (i) The successive "tossing" of a single fair quantum coin gives a biased result, a previously known result. (ii) The random tossing of two quantum coins, each with successive biased expectations, gives an average random walk position of approximately zero. (iii) The sequential periodic tossing of two quantum coins, each with successive negative biased expectations, gives an average random walk with positive expectation. Using these results, we then propose a protocol for identifying and classifying quantum operations that span the same Hilbert space for a two-level quantum system.