Ms. Tina Stevens just purchased a new dress that cost $120. What is the probability that she paid cash?

Question

The credit department of Lion’s Department Store in Anaheim, California, reported that 28% of their sales are cash, 24% are paid with a credit card, and 48% with a debit card. Twenty percent of the cash purchases, 86% of the credit card purchases, and 58% 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.)

Answer 1

Draw a tree diagram with all labelled probabilities.

Want to find P(cash | greater > 50) = P(cash and > 50)/P(> 50)

P(cash and > 50) = 0.28 * 0.20 = 0.056

P(> 50) = P(cash and > 50) + P(credit and > 50) + P(debit and > 50) = 0.28 * 0.2 + 0.24 * 0.86 + 0.48 * 0.58 = 0.5408

so P(cash | greater 50) = 0.056/0.58 = 0.097

Answer 2

Letting C = cash, CC = credit card, and DC = Debit card, You are give P_C, P_CC, and P_DC. You are also given P(>50|C), P(>50|CC), and P(>50|DC) in that order. You are asked for P(C|>50)

Use Bayes Equation: P(C|>50) = P(>50|C)P(C)/[P(>50)P(C) + P(>50|CC)P(CC)+P(>50|DC)P(DC)]

Notice that the first term in the denominator is the same as the term in the numerator.