+A bag contains one red disc and one blue disc. In a game of chance a player
+takes a disc at random and its colour is noted. After each turn the disc is
+returned to the bag, an extra red disc is added, and another disc is taken at
+The player pays £1 to play and wins if they have taken more blue discs than red
+discs at the end of the game.
+If the game is played for four turns, the probability of a player winning is
+exactly 11/120, and so the maximum prize fund the banker should allocate for
+winning in this game would be £10 before they would expect to incur a loss.
+Note that any payout will be a whole number of pounds and also includes the
+original £1 paid to play the game, so in the example given the player actually
+Find the maximum prize fund that should be allocated to a single game in which
+fifteen turns are played.
+There are two possible states: