1100 CHAPTER 11 Further Topics in Algebra In Example 1(c), E5 = S. Therefore, event E5 is certain to occur every time the experiment is performed. On the other hand, in Example 1(d), E6 = ∅ and P1E62 = 0, so E6 is impossible. EXAMPLE 1 Finding Probabilities of Events A single fair die is rolled. Write each event in set notation and give the probability of the event. (a) E3: the number showing is even (b) E4: the number showing is greater than 4 (c) E5: the number showing is less than 7 (d) E6: the number showing is 7 SOLUTION (a) Because E3 = 52, 4, 66, we have n1E32 = 3. As given earlier, n1S2 = 6. P1E32 = 3 6 = 1 2 (b) Again we have n1S2 = 6. Event E4 = 55, 66, and thus n1E42 = 2. P1E42 = 2 6 = 1 3 (c) E5 = 51, 2, 3, 4, 5, 66 and P1E52 = 6 6 = 1 (d) E6 = ∅ and P1E62 = 0 6 = 0 S Now Try Exercises 7, 9, 13, and 15. Probability Values and Terminology • A certain event—that is, an event that is certain to occur—always has probability 1. • The probability of an impossible event is always 0 because none of the outcomes in the sample space satisfies the event. • For any event E, P1E2 is between 0 and 1 inclusive of both. Complements and Venn Diagrams The set of all outcomes in the sample space that do not belong to event E is the complement of E, written E′. For example, in the experiment of drawing a single card from a standard deck of 52 cards, let E be the event “the card is an ace.” Then E′ is the event “the card is not an ace.” From the definition of E′, for an event E, E∪E′ =S and E∩E′ =∅.* * The union of two sets A and B is the set A∪B of all elements from either A or B, or both. The intersection of sets A and B, written A∩B, includes all elements that belong to both sets.
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