930 CHAPTER 13 Counting and Probability If events E and E are disjoint so that ∩ = ∅ F F , we say they are mutually exclusive . In this case, ( ) ∩ = P E F 0, and formula (5) takes the following form: THEOREM Mutually Exclusive Events If E and F are mutually exclusive events, that is, if ∩ = ∅ E F , then ( ) ( ) ( ) ∪ = + P E F P E P F (6) DEFINITION Complement of an Event Let S denote the sample space of an experiment, and let E denote an event.The complement of E, denoted E, is the set of all outcomes in the sample space S that are not outcomes in the event E. Computing Probabilities of the Union of Two Mutually Exclusive Events If ( ) = P E 0.4 and P F 0.25, ( ) = and E and F are mutually exclusive, find ( ) ∪ P E F . EXAMPLE 8 Solution Since E and F are mutually exclusive, use formula (6). ( ) ( ) ( ) ∪ = + = + = P E F P E P F 0.4 0.25 0.65 Now Work PROBLEM 47 4 Use the Complement Rule to Find Probabilities Recall that if A is a set, the complement of A, denoted A, is the set of all elements in the universal set U that are not in A. We similarly define the complement of an event. The complement of an event E, denoted E, in a sample space S has the following two properties: ∩ = ∅ ∪ = E E E E S Since E and E are mutually exclusive, it follows from formula (6) that ( ) ( ) ( ) ( ) ( ) ( ) ∪ = = + = = − P E E P S P E P E P E P E 1 1 1 We have the following result: THEOREM Computing Probabilities of Complementary Events If E represents any event and E represents the complement of E, then ( ) ( ) = − P E P E 1 (7) TIP Sometimes it is easier to calculate P E( ) than to calculate P E . ( ) In such cases, we use formula (7). j Computing Probabilities Using Complements On the local news the weather reporter stated that the probability of rain tomorrow is 40%. What is the probability that it will not rain? EXAMPLE 9 Solution The complement of the event “rain” is “no rain.” ( ) ( ) = − = − = P P no rain 1 rain 1 0.4 0.6 There is a 60% chance of no rain tomorrow. Now Work PROBLEM 51
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