Consider the following brand-loyalty marketing model. There are three brands of a similar product brands A, B, and E. Market research has established that for this product customers may switch back and forth among the three brands several times before they settle on a permanent brand. The marketing department has concluded that once a customer has chosen the brand A for two successive purchases then that customer is considered to be a permanent brand-A customer. Likewise, once a customer has chosen the brand E for three successive purchases then that customer is considered to be a permanent brand-E customer. From past data marketing has estimated that 80% of the customers who are not yet permanent brand-A customers and who have just purchased a brand-A product, will next be equally likely to purchase a brand B, or E product. The remaining 20% again purchase a brand-A product. Likewise a not-yet-permanent customer who has just purchased a brand-E product will next purchase a brand E product 40% of the time, otherwise those customers will be equally likely to next purchase a brand A, or B product. Customers who just purchased a brand-B product will next purchase any of the three brands will equal probability. The market model assumes that new customers who have yet to make their first purchase are equally likely to initially purchase any of the brands. Likewise, customers who just purchased a brand-B product will next purchase any of the three brands with equal probability. Be careful in developing your model.

(i) Draw the state transition diagram and write the single-step transition matrix, P in Standard Form, and label the states.

(ii) ) Describe two ways to numerically compute the probability of a customer becoming a permanent brand-A customer. Do not make use of P ^() in your solution.

(iii) Describe how to numerically compute the mean number of purchases a customer makes before they become a permanent customer of some sort