Please help with the following problem.
An auto insurance company classifies its customers in three categories: poor, satisfactory, and preferred. Each year, 30% of those in the poor category are moved to satisfactory and 5% of those in the satisfactory category are moved to preferred. Also, 5% of those in the satisfactory category are moved to the poor category. Customers are never moved from poor to preferred, or conversely, in a single year. Assuming these percentages remain valid over a long period of time, how many customers can the company expect to have in each category in the long run?
In the long run, we should expect that the total number in each category remains constant. Let's say that in the long run, there are a poor, b satisfactory and c preferred. We immediately establish the following points,
-30% of poor goes to satisfactory (poor category becomes 0.7a, satisfactory becomes b+0.3a)
-5% satisfactory moves to preferred ...
This solution helps with a problem that explains how many customers a company can expect to have in various satisfaction categories.