A sporting goods company operates retail stores in the New York and Atlanta area

Question

A sporting goods company operates retail stores in the New York and Atlanta areas. Customers were asked to rate their shopping experiences, the results of which are sum marized in the following table: Consider the following information: 67% of the customers who rated their store Poor came from the Philadelphia area. 36% of the customers who rated their store Average came from the Philadelphia area. 30% of the customers who rated their store Good came from the Philadelphia area. Use Bayes' Theorem to determine the probability that a customer who shopped in the Philadelphia area rated his or her experience as Good.

Explanation / Answer

Let A1 be the event that the customer rated the experience as Good, A2 be the event he/she rated as average and

A3 be the event he/she rated as poor. B be the event that he/she shopped in Philadelphia.

We have, P(A1) = 40 / 100 = 0.4, P(A2) = 45 / 100 = 0.45, P(A3) = 15 / 100 = 0.15, P(B | A1) = 30% = 0.3,

P(B | A2) = 36% = 0.36 and P(B | A3) = 67% = 0.67.

P(B) = P(B | A1) * P(A1) + P(B | A2) * P(A2) + P(B | A3) * P(A3) = (0.3 * 0.4) + (0.36 * 0.45) + (0.67 * 0.15) = 0.3825

Hence, P(A1 | B) = [P(B | A1) * P(A1)] / P(B) [Using Bayes Theorem]

= 0.12 / 0.3825 = 0.3137. (Ans).

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