Exercise 2 (a) Suppose you are playing the coin game described previously, except this time you and your friend start the game with a TOTAL of twenty coins spread between you (So, your system has more states). Write the MATLAB code to construct this matrix using a for loop.

(b) If you and your friend both start with ten coins, what is the probability for you to have exactly 4 coins on turn twelve? (There is a slight technical difficulty here. You may ask, what if someone has won before the twelfth turn? To get over that conceptual problem, we can assume that once somebody wins or loses, the state remains unchanged and you can view the turns to continue. Thats what the 1s in the transition matrix mean.) 5

(c) What is the probability of someone winning the twenty coin game after eighteen turns?.

(d) Suppose you start the game with 8 coins, and your friend has twelve. What is your probability of winning the game eventually (after so many turns that the game is almost surely over)? Describe the procedure how you can get this result using MATLAB. (Explain in words. You can use MATLAB codes as illustration.) (e) Suppose that you get a coin whenever the dice comes up even. What happens if you are playing with a loaded dice? For example one that comes up even 53% of the time? Write the MATLAB code to construct the transition matrix for your probability distribution.

(f) Given that there are a total of twenty coins, and you are playing with the loaded dice above, how many coins should you start with in order to make the game as fair as possible? Being fair means each of you two eventually has approximately the same probability of winning (close to 50%).