A law firm employs three types of lawyers: junior lawyers, senior lawyers, and partners. At the end of each year, there is a 0.1 probability that a junior lawyer will be promoted to senior lawyer and a 0.05 probability that he or she will leave the firm. Also, there is a 0.2 probability that a senior lawyer will be promoted to partner and a 0.15 probability that he or she will leave the firm. There is a 0.03 probability that a partner will leave the firm. The firm never demotes a lawyer. Let X be the status of an employee at the end of th year after he or she has been hired, it could be one of the three types of lawyers (junior, senior, or partner) or two types of leaving (either leaving as a partner or leaving as a non-partner). (a) Construct the (one-step) transition matrix. (b) Find P(6). Then use it to answer the following two questions. What is the probability that a newly hired junior lawyer will leave the firm during the next 6 years before becoming a partner? If John is a senior lawyer currently, what is the probability that he will be a partner at the end of the 6th year? (c) What is the probability that a newly hired junior lawyer will eventually leave the firm before becoming a partner? (Hint: consider P(n) for very large n, such as n=800, although it is not realistic.)

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