EXAMPLE 4.14 Determining Independence Using the Multiplication Rule

Consider a manufacturer that uses two suppliers for supplying an identical part that enters the production line. Sixty percent of the parts come from one supplier, while the remaining 40% come from the other supplier. Internal quality audits find that there is a 1% chance that a randomly chosen part from the production line is defective. External supplier audits reveal that two parts per 1000 are defective from Supplier 1. Are the events of a part coming from a particular supplier—say, Supplier 1—and a part being defective independent?

Define the two events as follows:

We have and . The product of these probabilities is

However, supplier audits of Supplier 1 indicate that . Given that , we conclude that the supplier and defective part events are not independent.