P 20

Problem 2, p. 20

Only three brands of beer are availablefor sale in Metropolis. From time to time, people try one or another of these brands. Suppose that at the beginning of the month people change the beer they drink according to the following rules:

30% of people who prefer brand 1 switch to 2

20% of people who prefer brand 1 switch to 3

30% of people who prefer brand 2 switch to 3

30% of people who prefer brand 3 switch to 2

10% of people who prefer brand 3 switch to 1

x_i = # people who prefer brand i at the beginning of the month
y_i = # people who prefer brand i at the end of the month

Matrix formulation: A x = y

a_ij = people who prefer brand j at the beginning and switch to brand i
Note: the columns should sum to 1. Assume everybody who does not switch is on the diagonal.


         .5  .0  .1      

   A =   .3  .7  .3    

         .2  .3  .6