2. A market research company is conducting a controlled survey to determine people's preference in tooth- paste. Participants are asked to try two brands of toothpaste over a period of several months. Of those using Brand A in any month, 75% continue to use it the following month, while 25% switch to Brand B; of those using Brand B in any month, 15% continue to use it the following month, while 85% switch to Brand A. A Markov process is used to model changes in participant preferences. (a) By using column vectors, write the stochastic matrix. (b) Suppose 60% of the participants are using Brand A when the survey begins. What fraction (or percentage) of the particpants are likely to be using Brand B after 1 month? (c) Find the steady-state (or long term state) vector for this Markov process. (d) What is the probability that after many months a specific participant is using Brand A toothpaste? Does it matter if that person is using Brand B today?