Chapter 9.1, Problem 87E

Advertising. A television advertising campaign is conducted during the football season to promote a well-known brand $X$ shaving cream. For each of several weeks, a survey is made, and it is found that each week, $80\%$ of those using brand $X$ continue to use it and $20\%$ switch to another brand. It is also found that of those not using brand $X,20\%$ switch to brand $X$ while the other $80\%$ continue using another brand.

(A) Draw a transition diagram.

(B) Write the transition matrix.

(C) If $20\%$ of the people are using brand $X$ at the start of the advertising campaign, what percentage will be using it $1$ week later? $2$ weeks later?