The credit department of Lion’s Department Store in Anaheim, California, reporte

Question

The credit department of Lion’s Department Store in Anaheim, California, reported that 26% of their sales are cash, 26% are paid with a credit card, and 48% with a debit card. Twenty percent of the cash purchases, 80% of the credit card purchases, and 60% of the debit card purchases are for more than $50.

Ms. Tina Stevens just purchased a new dress that cost $120. What is the probability that she paid cash? (Round your answer to 3 decimal places.)

The credit department of Lion’s Department Store in Anaheim, California, reported that 26% of their sales are cash, 26% are paid with a credit card, and 48% with a debit card. Twenty percent of the cash purchases, 80% of the credit card purchases, and 60% of the debit card purchases are for more than $50.

Explanation / Answer

Solution: As a first step, let’s summarize some of the information given in the problem statement.

• There are three mutually exclusive and collectively exhaustive events, that is, three ways of payment.

A1 = pay by cash

A2 = pay by credit card

A3 = pay by debit card

• The prior probabilities are:

P(A1) = 0.30

P(A2) = 0.26

P(A3) = 0.44

The additional information can be either:

B1 = price >$50

B2 = price <$50

• The following conditional probabilities are given.

P(B1 | A1) = 0.20 = probability that twenty percent of the cash or check purchases are for more than $50.

P(B1 | A2) = 0.85 = probability that 85 percent of the credit card purchases are for more than $50.

P(B1 | A3) = 0.60 = probability that 60 percent of the debit card purchases are for more than $50.

Ms. Tina Stevens just purchased a new dress that cost $120. So, we want to know the probability that she paid cash or check

So we find P (A1 | B1) formally by using Bayes’ Theorem.

P( A1 | B1) = [P(A1) * P(B1|A1)] / [ P(A1) * P(B1|A1) + P(A2) * P(B1|A2) + P(A3) * P(B1|A3)]

= [0.30 * 0.20] / [ 0.30 * 0.20 + 0.26 * 0.85 + 0.44 * 0.60]

= 0.06 / ( 0.06 + 0.221 + 0.264)

= 0.06 / 0.545

= 0.110

= 0.11

So, the probability that she paid cash or check is 0.11

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