A worker in a food plant has a rash on his left arm and the company doctor is concerned that he may be suffering from Flaubert’s disease. This can only occur as a result of biological contamination of the raw materials that the worker handles in his job. However, the disease is rare and has not occurred in the plant during the ten years of its operation so the doctor estimates that the probability that contamination exists in the plant to be only 2%, while there is a 98% chance that the contamination is not present. The doctor has now to advise the company’s board on whether they should close the plant for two days to enable fumigation to be carried out. This would certainly eradicate any contamination but it would cost a total of $30 000. If the plant is not closed and the contamination is present then it is virtually certain that other workers would fall ill. This would leave the company open to legal action by employees. The total cost of this, and the closure of the plant for fumigation that would be imposed by the authorities, would amount to an estimated $2 million. The doctor has another option. A test for Flaubert’s disease has just been developed by a Swiss pharmaceutical company but, as yet, the test is not perfectly reliable and has only a 0.6 probability of giving a correct indication. The price of the test would have to be negotiated with the Swiss company, but it is likely to be expensive.
(a) Determine the expected value of the information from the test, assuming that the food company’s objective is to minimize expected costs.
(b) Discuss the effect on the expected value of the information from the test of the following changes (you need not perform any calculations here).
(i) The doctor’s estimate of the probability of contamination moving closer to 50%;
(ii) The doctor changing her mind so that she is certain that there is no contamination in the plant.
(c) If the test was perfectly reliable what would be the value of the information it yielded (assume that the doctor’s original prior probability applies)?