A company’s brand (X) has 20% of the market. A market research firm finds that if a person uses brand X, the probability is .7 that he or she will buy it next time. On the other hand, if a person does not use brand X (represented by X′), the probability is .5 that he or she will switch to brand X next time. 

(A) Draw a transition diagram. 

(B) Write a transition matrix. 

(C) Write the initial-state matrix. 

(D) Find the first-state matrix and explain what it represents. 

(E) Find the stationary matrix. 

(F) What percentage of the market will brand X have in the long run if the transition matrix does not change?