Martin Gardner popularized the Penney-Ante game in which player 1 chooses a length-three string of heads/tails. Aware of player 1's choice, then player 2 also chooses such a string. A coin is then tossed repeatedly until one of the players' strings appears. The player's string that appears first wins the game. If you are player 2, what is your best response to a player 1's choice? The answer for a fair coin is known, but demonstrates some unusual behavior. What if the coin is not fair? Using a tree-based method and a Markov chain based method, I'll determine the best-response for player 2 for a p-coin (where the probability of the coin landing on heads is p). Consequently, this will also address how player 1 should choose a string, too. Finally, I will determine the optimal strategies for a simultaneous-move version of the Penney-Ante game. This work is joint with Riley Dickinson and Stanley R. Huddy.