A gambler faces off against a small casino. She begins with $1000, and the casino with $11,000. In each round, the gambler loses 10% of her current funds to the casino, and the casino loses 2% of its current funds to the gambler.
a. Find the amount of money each has after one round.
b. Find a discrete-time dynamical system for the amount of money the gambler has and another for the amount of money the casino has.
c. Find the discrete-time dynamical system for the fraction p of money the gambler has.
d. Find the equilibrium fraction of the money held by the gambler.
e. Using the fact that the total amount of money is constant, find the equilibrium amount of money held by the gambler.