SECTION 13.3 Probability 929 (b) In probability, the word or means the union of the two events.The event E or F is { } { } { } ( ) ∪ = ∪ = ∪ = E F n E F 1, 3, 5 1, 2 1, 2, 3, 5 4 (c) Use formula (4). Then ( ) ( ) ( ) ( ) ( ) ( ) = = = = = = P E n E n S P F n F n S 3 6 1 2 2 6 1 3 (d) ( ) ( ) ( ) ∩ = ∩ = P E F n E F n S 1 6 (e) ( ) ( ) ( ) ∪ = ∪ = = P E F n E F n S 4 6 2 3 3 Find Probabilities of the Union of Two Events The next formula can be used to find the probability of the union of two events. THEOREM For any two events E and F, ( ) ( ) ( ) ( ) ∪ = + − ∩ PEF PE PF PEF (5) This result is a consequence of the Counting Formula discussed in Section 13.1. For example, formula (5) can be used to find ( ) ∪ P E F in Example 6(e). Then PEF PE PF PEF 1 2 1 3 1 6 3 6 2 6 1 6 4 6 2 3 ( ) ( ) ( ) ( ) ∪ = + − ∩ =+−=+−== as before. Computing Probabilities of the Union of Two Events If ( ) ( ) = = P E P F 0.2, 0.3, and ( ) ∩ = P E F 0.1, find the probability of E or F. That is, find ( ) ∪ P E F . EXAMPLE 7 Solution Use formula (5). ( ) ( ) ( ) ( ) = ∪ = + − ∩ = + − = E F PEF PE PF PEF Probability of or 0.2 0.3 0.1 0.4 A Venn diagram can sometimes be used to obtain probabilities. To construct a Venn diagram representing the information in Example 7, draw two sets E and F. Begin with the fact that ( ) ∩ = P E F 0.1. See Figure 7(a). Then, since ( ) = P E 0.2 and ( ) = P F 0.3, fill in E with − = 0.2 0.1 0.1 and fill in F with − = 0.3 0.1 0.2. See Figure 7(b). Since ( ) = P S 1, complete the diagram by inserting ( ) − + + = 1 0.1 0.1 0.2 0.6 outside the circles. See Figure 7(c). Now it is easy to see, for example, that the probability of F but not E is 0.2. Also, the probability of neither E nor F is 0.6. Figure 7 (a) E F S 0.1 (b) E F S 0.1 0.2 0.1 (c) E F S 0.1 0.2 0.6 0.1 Now Work PROBLEM 45
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