A box contains two marbles. One black and the other white. In each turn the player takes a marble out at random and notes its colour. After each turn, the marble is replaced in the bag and an extra black marble is added. The player pays Rs. 1 to play and wins if he/she has taken more white marbles than black marbles at the end.

If the game is played for three turns, the probability of the player winning is exactly $\frac{7}{24}$, and so the maximum prize fund that should be allocated for winning in this game should be Rs. 3 before the organizer would expect to incur a loss. The maximum prize is a whole number and includes the Rs. 1 to be given by the player, so the player gets Rs. 2. Find the maximum prize fund to be allocated to a game with sixteen turns.