Chapter 9, Problem 38RE

Product switching. A company's brand $\left(X\right)$ 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 $X^{\prime }$ ), 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-stale 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?