A sporting good company operates retail stores in the NY and

A sporting good company operates retail stores in the NY and Atlanta area. Customers were asked to rate their shopping experiences, the results of which are summarized 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 Philadelphia area

*30% f the customers who rated their store Good came from Philadelphia area

Use Bayes\' Theorem to determine the probability that a customer who shopped in the Philadelphia are rated his or her experience as good.

Solution

Bayes\' theorem. Let A₁, A₂, ... , A_n be a set of mutually exclusive events that together form the sample space S. Let B be any event from the same sample space, such that P(B) > 0. Then,

Note: Invoking the fact that P( A_k B ) = P( A_k )P( B | A_k ), Baye\'s theorem can also be expressed as

P( A_k | B ) =

SO 67% = 10

36%=16

30%=12

PROB =12/38

P( A_k ) P( B | A_k )/[P( A₁ ) P( B | A₁ ) + P( A₂ ) P( B | A₂ ) + . . . + P( A_n ) P( B | A_n )]