In this lab you will create some transition matrices and use
MATLAB to study how the probability distribution changes over time.

Simple Dice Game

Here's a simple game one person can play:
  Roll a single die getting the numbers 1 thru 5 in order, if you 
  roll a 6 before you get to 5, you have to start over.

Here's how to set up the transition matrix:
  There are 6 states you can be in: have nothing, have 1, 2, 3, 4, 5.
  Let's number these 0, 1, 2, 3, 4, 5.
  Then until you get to 5, you have a 1/6 chance of moving to the
  next state and a 1/6 chance of going back to 0.  Thus the
  transition matrix looks like:
        0    1    2    3    4    5
    0  5/6  1/6  1/6  1/6  1/6   0
    1  1/6  4/6   0    0    0    0
    2   0   1/6  4/6   0    0    0
    3   0    0   1/6  4/6   0    0
    4   0    0    0   1/6  4/6   0
    5   0    0    0    0   1/6   1

  Enter this in MATLAB, call it T  (don't enter the indices 0-5)
  To check it type sum(T) you should get all 1s.

  Let P = [1; 0; 0; 0; 0; 0] this is your starting distribution.
  You can also create it by P = zeros(6,1); P(1) = 1;

  Now if you type T*P you get the distribution after one turn.
  If you type T^2*P you get the distribution after two turns.
  If you type T^3*P you get the distribution after three turns.

Questions: (Either e-mail the answers to me, or hand them in)

     1. What is the probability that you will be back at 0 after 3 turns?
     2. What is the probability that you will be back at 0 after 10 turns?
     3. What is the probability that you have reached 5 after 10 turns?
     4. What is the average score obtained after 10 turns? (Expected Value)
     5. How many turns before the probability that you have reached 5
        exceeds 1/2?

Two Player Dice Battle

This is the game we played in class.
   There are 2 players, A and B.  For each round, each rolls a dice, 
   if A rolls a number greater than or equal to B then A wins.  
   Otherwise B wins.  The first to get 3 wins is the winner.

Building the transition matrix:  
   List all the possible states the game could be in.  
      For example, you start with A having 0 wins and B having 0 wins 
      (write as (0,0)), at some time A could have 2 wins and B have 1 
      win (write as (2,1)).

   Number the states from 1 to N, where N is the number of states.

   Build the matrix T (it will be NxN).  Use the probabilities we
   determined in class and your numbering of the states to fill in
   the values.  Remember that a winning state (either player has 3
   wins) is an absorbing state.

   I'd suggest you put it in a m-file "dice.m" and start it like:
      N = 
      T = zeros(N);    % starts T out as an NxN matrix of zeros
      % 1 = (0,0)      % list what all the states are
      % 2 = ..
      %
      % N = ..
      T( , ) =         % fill in all the non-zero values


      
  You could also enter T as
      T = [0 0 0 ... 0 0  whatever
                             more stuff
                            
                             ];

   Each column (except for winning states) should have 2 entries:
   one if A wins and one if B wins.

   To check your result, type dice and then
   type sum(T) to see if it is all 1s.

Questions: (Either e-mail the answers to me, or hand them in)

    1. What is the probability of being at (2,0) after 2 rounds?
       (0,2)?  How about (3,0) after 3 rounds?   
    2. What is the probability of being down (0,2) and then winning (3,2)?
    3. How many rounds do you have to play before you are guaranteed 
       to have a winner?
    4. What is the probability that A wins?
    5. What is the expected number of rounds before there is a winner?

E-mail: ccollins@math.utk.edu