Calculating the expected first passage time from state 1 to 3: M13 = 1 + P M13 + P12 H23 M23 = 1 + P21 413 + P22 123 M13 = 4.58 Therefore, it takes 45.8 h (d) What are the long-term ratios? State 3 is absorbent, so the long-term ratios are 0 for s, and s2, and 1 for $3. i57 10.21 Market Research into Cars for Fleets of Vehicles FORD is investigating how the market in which firms buying cars for their fleets behaves. Preliminary research work has revealed that firms buy cars of several brand names, but that they tend to maintain a percentage assigned to each auto- mobile manufacturer. Every year when new models are announced, firms amend the percentages assigned to each brand name. For the purpose of performing a preliminary analysis, FORD considers that the set of the values of the percentages that these firms assign to the FORD car brand name is discrete, and is: 100, 70, 50 or 20 %. An in-depth study resulted in a 1-year transition probabilities matrix, as shown in Table 10.6. The current situation is that 30 % of the firms maintain an assigned percentage of 70 %, 45 % maintain an assigned percentage of 50 %, and 20 % maintain an assigned percentage of 20 %. (a) Fill in the ? in the matrix (b) What percentage of firms has, in the long term, a percentage assigned to FORD of 100 %? And the percentage for 70 %? And for 50 %? And finally for 20 %? (c) What are these percentages for the following year