Customers at a Publix grocery store in Charleston, South Carolina, can pay for purchases with
cash, a debit card, or a credit card. Fifty-five percent of all customers use cash and 38% use a
debit card. Careful research has shown of those paying with cash, 75% use coupons; of those
using a debit card, 35% use coupons; and of those using a credit card, only 10% use
coupons. Suppose a customer is randomly selected. (15 marks)
a. What is the probability that the customer pays with a credit card and does not use
coupons?
b. What is the probability that the customer does not use coupons?
c. If the customer does not use coupons, what is the probability that he paid with a debit
card?
What is the probability that the customer does not use coupons?
Math
To answer these questions, we need to use probability calculations based on the given information.
Let's define the following probabilities:
- P(C): Probability of paying with cash
- P(D): Probability of paying with a debit card
- P(Cr): Probability of paying with a credit card
According to the information provided:
- P(C) = 0.55 (55% pay with cash)
- P(D) = 0.38 (38% pay with a debit card)
- P(Cr) = 1 - P(C) - P(D) = 1 - 0.55 - 0.38 = 0.07 (7% pay with a credit card)
a. To find the probability that a customer pays with a credit card and does not use coupons, we need to multiply the probability of paying with a credit card (P(Cr)) by the probability of not using coupons given paying with a credit card (1 - 0.10 = 0.90).
P(Cr and not using coupons) = P(Cr) * (1 - 0.10)
= 0.07 * 0.90
= 0.063
So the probability that the customer pays with a credit card and does not use coupons is 0.063 (or 6.3%).
b. To find the probability that a customer does not use coupons, we need to consider the different payment methods.
P(not using coupons) = P(C and not using coupons) + P(D and not using coupons) + P(Cr and not using coupons)
We already calculated P(C and not using coupons) = P(C) * (1 - 0.75) = 0.55 * 0.25 = 0.1375 (or 13.75%).
And we can calculate P(D and not using coupons) and P(Cr and not using coupons) using similar logic.
P(D and not using coupons) = P(D) * (1 - 0.35)
= 0.38 * 0.65
= 0.247 (or 24.7%)
P(Cr and not using coupons) = P(Cr) * (1 - 0.10)
= 0.07 * 0.90
= 0.063 (or 6.3%)
P(not using coupons) = 0.1375 + 0.247 + 0.063
= 0.4475
So the probability that the customer does not use coupons is 0.4475 (or 44.75%).
c. To find the probability that a customer who does not use coupons paid with a debit card, we can apply Bayes' theorem. Bayes' theorem allows us to calculate the conditional probability of one event given another event.
We need to find P(D|not using coupons), which is the probability of paying with a debit card given that the customer does not use coupons.
Using Bayes' theorem:
P(D|not using coupons) = (P(D) * P(not using coupons|D))/P(not using coupons)
= (0.38 * (1 - 0.35))/0.4475
Calculating this expression:
P(D|not using coupons) = 0.38 * 0.65/0.4475
≈ 0.557
So the probability that a customer who does not use coupons paid with a debit card is approximately 0.557 (or 55.7%).