Suppose that of 100 circuits sent to a manufacturing plant, 5 are defective. The plant manager receiving the circuits randomly selects 2 and tests them. If both circuits work, she will accept the shipment. Otherwise, the shipment is rejected. What is the probability that the plant manager discovers at least 1 defective circuit and rejects the shipment?

Approach To determine the probability that at least one of the tested circuits is defective, consider four possibilities. Neither of the circuits is defective, the first is defective while the second is not, the first is not defective while the second is defective, or both circuits are defective. Note that the outcomes are not equally likely. To find the probability the manager discovers at least 1 defective circuit, we may use one of two approaches.

Approach I: Use a tree diagram to list all possible outcomes and the General Multiplication Rule to compute the probability for each outcome. Then determine the probability of at least 1 defective by adding the probability that only the first is defective, only the second is defective, or both are defective, using the Addition Rule (because they are disjoint).

Approach II: Compute the probability that both circuits are not defective and use the Complement Rule to determine the probability of at least 1 defective. We will illustrate both approaches.