The Addition Rule 3.3 SECTION 3.3 The Addition Rule 157 Mutually Exclusive Events The Addition Rule A Summary of Probability What You Should Learn How to determine whether two events are mutually exclusive How to use the Addition Rule to find the probability of two events Mutually Exclusive Events In Section 3.2, you learned how to find the probability of two events, A and B, occurring in sequence. Such probabilities are denoted by P1A andB2. In this section, you will learn how to find the probability that at least one of two events will occur. Probabilities such as these are denoted by P1A or B2 and depend on whether the events are mutually exclusive. Two events A and B are mutually exclusive when A and B cannot occur at the same time. That is, A and B have no outcomes in common. DEFINITION The Venn diagrams show the relationship between events that are mutually exclusive and events that are not mutually exclusive. Note that when events A and B are mutually exclusive, they have no outcomes in common, so P1A andB2 = 0. A Sample Space B B A and B A Sample Space A and B are mutually exclusive. A and B are not mutually exclusive. Recognizing Mutually Exclusive Events Determine whether the events are mutually exclusive. Explain your reasoning. 1. Event A: Roll a 3 on a die. Event B: Roll a 4 on a die. 2. Event A: Randomly select a male student. Event B: Randomly select a nursing major. 3. Event A: Randomly select a blood donor with type O blood. Event B: Randomly select a female blood donor. SOLUTION 1. Event A has one outcome, a 3. Event B also has one outcome, a 4. These outcomes cannot occur at the same time, so the events are mutually exclusive. 2. Because the student can be a male nursing major, the events are not mutually exclusive. 3. Because the donor can be a female with type O blood, the events are not mutually exclusive. EXAMPLE 1 Study Tip In probability and statistics, the word or is mostly used as an “inclusive or” rather than an “exclusive or.” For instance, there are three ways for “event A or B” to occur. (1) A occurs and B does not occur. (2) B occurs and A does not occur. (3) A and B both occur.
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