markov chain accounting firm math problem

Project Description:

“general motors has three auto divisions (1, 2 and 3). it also has an accounting division and a managerial consulting division. the question is: what fraction of the cost of the accounting and management consulting divisions should be allocated to each auto division? we assume that the entire cost of the accounting and management consulting departments must be allocated to the three auto divisions. during a given year, the work of the accounting division and management consulting division is allocated as shown in table 4.
for example, accounting spends 10% of its time on problems generated by the accounting department, 20% of its time on work generated by division 3 and so forth. each year it costs $63 million to run the accounting department and $210 million to run the management consulting department. what fraction of these costs should be allocated to each auto division? think of $1 in costs incurred in accounting work. there is a .20 chance that this dollar should be allocated to each auto division, a .30 chance it should be allocated to consulting, and a .10 chance to accounting. if the dollar is allocated to an auto division, we know which division should be charged for that dollar. if the dollar is charged to consulting (for example), we repeat the process until the dollar is eventually charged to an auto division. use knowledge of absorbing chains to figure out how to allocate the costs of running the accounting management consulting departments among the three auto divisions.

[note: you must supply a fully worked solution and not just an answer].
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