Question

A large retail store sells MP3 players. A customer who purchases an MP3 player can pay either by cash or credit card. An extended warranty is also available for purchase. Suppose that the events \(M=\) event that the customer paid by cash \(E=\) event that the customer purchased an extended warranty are independent with \(P(M)=0.47\) and \(P(E)=0.16\). a. Construct a "hypothetical 1000 " table with columns corresponding to cash or credit card and rows corresponding to whether or not an extended warranty is purchased. (Hint: See Example 5.9) b. Use the table to find \(P(M \cup E)\). Give a long-run relative frequency interpretation of this probability.

Short answer

Using the probabilities given and after constructing the table, the probability that a customer pays in cash or purchases an extended warranty is computed to be 0.57. This means that in the long-run, out of every 1000 customers, about 570 customers are either likely to pay cash or buy the extended warranty or do both.

Step-by-step solution

Constructing the Table

Let's create a hypothetical 1000 purchases table, using the probabilities given: Cash payments (\(M\)) = 0.47 * 1000 = 470Credit card payments = 1000 - 470 = 530Extended warranty purchases (\(E\)) = 0.16 * 1000 = 160Without extended warranty = 1000 - 160 = 840The assumption of independence of events allows for calculation of joint occurrences in the following way:Cash and Warranty (\(M \cap E\)) = \(P(M) * P(E) * 1000 = 470 * 0.16 = 75.2 \approx 75\)Credit Card and Warranty = 1000 - 75 = 925Cash and No Warranty = 470 - 75 = 395 Credit Card without Warranty= 530 - 85= 445

Calculate the Probability

We've been asked to find the probability that a customer either pays by cash or buys the extended warranty (\(P(M \cup E)\)). In probability theory, this is found by: \(P(M \cup E) = P(M) + P(E) - P(M \cap E)\)Substituting gives us:= 0.47 + 0.16 - (0.47*0.16)= 0.57

Interpretation

This probability of 0.57 means that in the long-run, out of every 1000 customers, 570 of them are expected to either pay in cash or buy an extended warranty, or do both.