Question

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?

Answer 1

0.7407 = 74.07% probability that the customer is a new customer.

Explanation:

Conditional Probability

We use the conditional probability formula to solve this question. It is

`P(B|A) = (P(A \cap B))/(P(A))`

In which

P(B|A) is the probability of event B happening, given that A happened.

`P(A \cap B)` is the probability of both A and B happening.

P(A) is the probability of A happening.

In this question:

Event A: Pays by credit card

Event B: New customer.

Probability of a customer paying by credit card:

50% of 80%(new customers).

70% of 20%(regular customers). So

`P(A) = 0.5*0.8 + 0.7*0.2 = 0.54`

Probability of a customer paying by credit card and being a new customer:

50% of 80%, so:

`P(A \cap B) = 0.5*0.8 = 0.4`

What is the probability that the customer is a new customer?

`P(B|A) = (P(A \cap B))/(P(A)) = (0.4)/(0.54) = 0.7407`

0.7407 = 74.07% probability that the customer is a new customer.