At Joe's Restaurant, 80 percent of the diners are new customers (N), while 20 percent are returning customers (R). Fifty percent of the new customers pay by credit card, compared with 70 percent of the regular customers. If a customer pays by credit card, what is the probability that the customer is a new customer?
A).7407
B).8000
C).5400
D).5000

Question

Quizplus · Accepted Answer

To find the probability that a customer is a new customer given they pay by credit card, we use Bayes' theorem or a basic probability approach. Let's calculate:- Probability of a new customer (P(N)) = 80% or 0.8- Probability of a returning customer (P(R)) = 20% or 0.2- Probability of paying by credit card given a new customer (P(C|N)) = 50% or 0.5- Probability of paying by credit card given a returning customer (P(C|R)) = 70% or 0.7First, find the overall probability of paying by credit card, P(C), which is the sum of the probabilities of being a new or returning customer and paying by credit card:P(C) = P(N) * P(C|N) + P(R) * P(C|R)     = 0.8 * 0.5 + 0.2 * 0.7     = 0.4 + 0.14     = 0.54Then, the probability that a customer is a new customer given they pay by credit card, P(N|C), is:P(N|C) = [P(N) * P(C|N)] / P(C)       = (0.8 * 0.5) / 0.54       = 0.4 / 0.54       ≈ 0.7407 or 74.07%So, the correct answer is A) 0.7407.

At Joe's Restaurant, 80 Percent of the Diners Are New