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

Assignment

Pick one of the options below and create the transition matrix.
Besides the questions given, think of 4 or more additional questions 
you might want to answer and use the transition matrix and the right 
operations to answer the questions.

Turn in a description of what the states are in your model,
the transition matrix, and your questions with answers.

Options

RISK - using the data about the results from one round, create
       a transition matrix for a complete battle where each player
       starts with several armies.  Figure out for various starting
       configurations what the probabilities are for the Attacking
       Player to win.

       Extra rules:
          Defender can only use as many dice as he has armies (up to 2)
          Attacker can only use as many dice as he has armies-1 (up to 3)
          Thus the battle stops when Defender has 0 or Attacker has 1

       I've got a program riskt.m which creates
       the transition matrix.  After you save it, read the beginning
       which describes how to use it and how it orders the states.


 
 
 
 1 
 2 
 
 1 
 (7/12, 5/12) 
 (161/216,55/216) 
 
 2 
 (91/216,125/216) 
 (35/108,35/108) 
 
 3 
 (49/144,95/144) 
 (1420/4229,1420/4229) 

     
 
 
 
 2 
 
 2 
 (581/1296,295/1296) 
 
 3 
 (799/2731,792/2131) 



Best of 5 - like the last lab where you play to 3 wins.  In this
       case, instead of making the probabilities 7/12 and 5/12,
       make them p and q (q = 1-p), where p is the probability that
       A will win and q is the probability that B will win.
       Look at how the overall probability of A winning varies as
       p varies.  Look at other properties (like number of games)
       as well.

Disease - make up a disease that has at least 5 stages and model it
       with a transition matrix.  Look at two versions of the disease,
       one where there is no vaccination and one where there is.
       Compare the maximum infection/death probabilities for the two cases.
       If you don't like diseases, use rumors or any other transferable
       item.

Game - create your own non-trivial game (with at least 10 states).
       Explain the rules and then use the transition matrix to
       study how the game plays.  For example, determine how long
       the average game lasts.

Done?

When you get done, either e-mail or print out your results and give
them to me.

E-mail: ccollins@math.utk.edu