138 CHAPTER 3 Probability Complementary Events The sum of the probabilities of all outcomes in a sample space is 1, or 100%. An important result of this fact is that when you know the probability of an event E, you can find the probability of the complement of event E. The complement of event E is the set of all outcomes in a sample space that is not included in event E. The complement of event E is denoted by E′ and is read as “E prime.” The Venn diagram at the left illustrates the relationship between the sample space, event E, and its complement E′. DEFINITION For instance, when you roll a die and let E be the event “the number is at least 5,” the complement of E is the event “the number is less than 5.” In symbols, E = 55, 66 and E′ = 51, 2, 3 ,46. Using the definition of the complement of an event and the fact that the sum of the probabilities of all outcomes is 1, you can determine the formulas below. P1E2 + P1E′2 = 1 P1E2 = 1 - P1E′2 P1E′2 = 1 - P1E2 Finding the Probability of the Complement of an Event The frequency distribution from Example 7 is shown below. Find the probability of randomly selecting a user of a social media application who is not 25 to 34 years old. Ages Frequency, f 13 to 17 84 18 to 24 459 25 to 34 765 35 to 44 546 45 to 54 432 55 to 64 369 65 and over 345 Σf = 3000 SOLUTION From Example 7, you know that P1age 25 to 342 = 765 3000 = 0.255. So, the probability that a user is not 25 to 34 years old is P1age is not 25 to 342 = 1 - 765 3000 = 1 - 0.255 = 0.745. TRY IT YOURSELF 9 Use the frequency distribution in Example 7 to find the probability of randomly selecting a user who is not 18 to 24 years old. Answer: Page A38 EXAMPLE 9 The area of the rectangle represents the total probability of the sample space 11 = 100%2. The area of the circle represents the probability of event E, and the area outside the circle represents the probability of the complement of event E. E′ E Sample Space
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