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The profit of a firm depends on three positive input factors and as follows:

P(x₁,x₂) = 30x₁ − x₁² + 60ln x₂ − 4x₂

The firm AC/DC Associates manufactures ballet tutus and tights, and is in intense competition with Bob Dylan Co. and Queen Plc. All three companies are competing fiercely over the quality of their tutus and tights, and this has been a challenge recently. Ballet companies have been switching between the three companies in pursuit of finding the best tutus and tights for their dancers. Last season, the Gaga Ballet Company purchased items from all three companies as follows:

Table 1. Gaga Ballet Company's Purchases

The cost of these items are given in the following chart:

Table 2: Cost of items

Let Q be the 2x3 matrix corresponding to the purchases, and let C be the 2x3 matrix corresponding to the costs per item.

(This one is harder, so don't worry if you find this one too difficult)

Let p denote the probability that a firm fiddles its books. Suppose that p = 0.1 . Also suppose that a financial professional services firm keeps auditing firms until they have found 3 'bad' ones.