In the billing department of a local city, Arthur, Beatrice, and Carla prepare a

Question

In the billing department of a local city, Arthur, Beatrice, and Carla prepare and mail out 50%, 30%, and 20% of the real-estate tax bills, respectively, to property owners. The bills are verified by the treasurer before mailing. If 0.5% of the tax bills prepared by Arthur, 0.3% of the tax bills prepared by Beatrice, and 1.0% of them bills prepared by Carla are inaccurate, what are the probabilities that an inaccurate statement detected by the treasurer at final verification was prepared by Arthur; Beatrice; Carla?

Explanation / Answer

Here we have to find conditional probabilities.Let A and B are two different events, then conditional probability of A given B is given by,

P(A|B) = P(A and B) / P(B)                                 ------- (1)

From given details,

P(Bill prepared by Arthur) = 0.5; P(Bill prepared by Beatrice) = 0.3 and P(Bill prepared by Carla) = 0.2.

Futher,

P(Inaccurate bill by Arthur) = 0.005;

P(Inaccurate bill by Beatrice) = 0.003;

P(Inaccurate bill by Carla) = 0.01;

Hence from (1), we have to find

P(Bill prepared by Arthur | Inaccurate bill) = P(Inaccurate bill by Arthur) / P(Inaccurate bill)

and similarly for others.

Here to evaluate probabilities, we need P(Inaccurate bill) which is not given.

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