Help with probability stats question

Question

The credit department of Lion’s Department Store in Anaheim, California, reported that 26% of their sales are cash, 20% are paid with a credit card, and 54% with a debit card. Twenty percent of the cash purchases, 85% 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

Answer 1

Let:

A1 = cash                        B1 = price > $50

A2 = credit card               B2 = price <= $50

A3 = debit card

 

Probabilities:

P(A1) = 0.26

P(A2) = 0.20

P(A3) = 0.54

 

Conditional Probabilities:

P(A1 | B1) = 0.20

P(A2 | B1) = 0.85

P(A3 | B1) = 0.60

 

Find P(A1| B1) using Bayes' Theorem:

 

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

                

                 =[0.26 * 0.20] /[ 0.26 * 0.20 + 0.20 * 0.85 + 0.54 * 0.60]

 

                 = 0.052 / 0.052 + 0.17 + 0.324

 

                 = 0.052 / 0.546

 

                 = 0.095