A married couple owns a consulting firm. The firm’s weekly income is a random variable, which may equal \($10\),000, \($15\),000, or \($20\),000. The income earned next week depends only on the income earned this week. In any week, the married couple may sell the firm to either their adult son or their adult daughter. If the income this week is \($10\),000, the couple may sell the firm next week to their son with probability 0.05 or to their daughter with probability 0.10. If the income this week is \($10\),000 and they do not sell, the income next week will be either \($10\),000 with probability 0.40, \($15\),000 with probability 0.25, or \($20\),000 with probability 0.20. If the income this week is \($15\),000, then next week there is a 0.15 probability that the couple will sell the firm to their son, and a 0.20 probability that they will sell the firm to their daughter. If the income this week is \($15\),000 and they do not sell, the income next week will be either \($10\),000 with probability 0.20, \($15\),000 with probability 0.35, or \($20\),000 with probability 0.10. Finally, if the income this week is \($20\),000, the couple may sell the fi rm next week to their son with probability 0.15 or to their daughter with probability 0.10. If the income this week is \($20\),000 and they do not sell, the income next week will be either \($10\),000 with probability 0.25, \($15\),000 with probability 0.20, or \($20\),000 with probability 0.30.

(a) Formulate this problem as an absorbing multichain MCR. Represent the transition probability matrix in canonical form. Construct the reward vector.

(b) Given that the firm’s income in the first week is \($10\),000, what is the probability that the firm will eventually be sold to the son?

(c) Given that the firm’s income in the first week is \($20\),000, what is the probability that the firm will eventually be sold to the daughter?

(d) Given that the firm’s income in the first week is \($15\),000, what is the expected number of weeks before the firm is sold to the daughter?

(e) Given that the firm’s income in the first week is \($15\),000, what is the expected total income that will be earned before the firm is sold?