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Bayes' Theorem - DVDs & Disease

A store sells four brands of DVD players. The least expensive brand, B1, accounts for 40% of the sales. The other brands (in order of their price) have the following percentages of sales: B2, 30%; B3, 20%; and B4, 10%. The respective probabilities of needing repair during warranty are 0.10 for B1, 0.05 for B2, 0.03 for B3, and 0.02 for B4. A randomly selected purchaser has a DVD player that needs rep air under warranty. What are the four conditional probabilities of being brand Bi where i = 1,2,3,4'?

There is a new diagnostic test for a disease that occurs in about 0.05% of the population. The test is not perfect, but will detect a person with the disease 99% of the time. It will, however, say that a person without the disease has the disease about 3% of the time. A person is selected at random from the population, and the test indicates that this
person has the disease. What are the conditional probabilities that
(a) the person has the disease?
(b) the person does not have the disease?

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Solution Preview

Let B_1, B_2, B_3 and B_4 denote the events that the sales of the brands B_1, B_2, B_3 and B_4 respectively. Let A denote the event that randomly selected purchaser has a DVD player that needs repair under warranty.

Here it is given that,

P(B_1) = 0.40 and ...

Solution Summary

The solution illustrates an application of Baye's theorem to DVDs and a disease test in an attachment.