Question

An online clothing company decides to investigate whether offering their customers a coupon upon completion of their first purchase will encourage them to make a second purchase. To do so, the company programs the website to randomly select 100 first time customers. Sixty of these customers are randomly selected to receive a coupon for $5 off their next purchase, to be made in the next 30 days. The other 40 customers are not offered a coupon. The table below shows the number of customers in each group that made a second purchase within 30 days of their first purchase.

Based upon the table, is “yes, made a second purchase” independent of “yes, being sent a coupon”?

A) Yes, exactly half of the customers made a second purchase and half did not.
(B) Yes, the largest count in the table comes from those who were sent a coupon and made a second purchase within 30 days.
(C) No, the probability of making a second purchase is not equal to the probability of making a second purchase given that a coupon was sent.
(D) No, the probability of making a second purchase is the same whether or not a coupon was sent.
(E) It is impossible to draw a conclusion about independence because a coupon was not sent to exactly half of the customers.

Answer 1

(C) No, the probability of making a second purchase is not equal to the probability of making a second purchase given that a coupon was sent.

Explanation:

Let A = the customer makes a second purchase within 30 days and let B = customer is sent a coupon. Events A and B are independent if P(A) = P(A | B).

P(A) = P(the customer makes a second purchase within 30 days) = \frac{50}{100} = 0.5

100

50

​ =0.5

P(A | B) = P(the customer makes a second purchase within 30 days | customer is sent a coupon) = \frac{34}{60} = 0.567

60

34

​ =0.567

Because P(A) ≠ P(A | B) making a second purchase is not independent of being sent a coupon.