# Problem 121: Disc game prize fund
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 random. 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 wins £9. Find the maximum prize fund that should be
allocated to a single game in which fifteen turns are played.