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j9rb-1108.qxd 11/13/03 1:50 PM Page 75 LESSON Name Date 11.8 Study Guide For use with pages 626–632 GOAL Find the probability that event A or event B occurs. VOCABULARY Disjoint events, or mutually exclusive events, are events that have no outcomes in common. Overlapping events are events that have one or more outcomes in common. Two events are complementary events if they are disjoint events and one event or the other must occur. EXAMPLE 1 Identifying Disjoint and Overlapping Events Tell whether the events are disjoint or overlapping. a. Randomly select a number from b. Randomly select a card from a 1 to 20. 52-card deck. Event A: Select an even number. Event A: Select an Ace. Event B: Select a number less than 2. Event B: Select a Club. EXAMPLE Solution a. The outcomes for event A are 2, 4, 6, 8, 10, 12, 14, 16, 18, and 20. The outcome for event B is 1. There are no outcomes in common. b. Because one card is an Ace of Clubs, the events have an outcome in common. Answer: The events are disjoint. Answer: The events are overlapping. 2 Finding the Probability of Disjoint Events Lesson 11.8 A bag of marbles contains 25 yellow marbles, 40 red marbles, 5 purple marbles, and 30 blue marbles. You randomly draw a marble from the bag. What is the probability that you draw a yellow or a purple marble? Solution The events are disjoint because the marble cannot be both yellow and purple. Event A: You choose a yellow marble. Event B: You choose a purple marble. P(A or B) ⫽ P(A) ⫹ P(B) 25 5 100 100 30 3 ⫽ ᎏᎏ ⫽ ᎏᎏ 100 10 ⫽ ᎏᎏ ⫹ ᎏᎏ Probability of disjoint events Substitute probabilities. Add. Then simplify. 3 10 Answer: The probability that you draw either a yellow or a purple marble is ᎏᎏ. Copyright © McDougal Littell/Houghton Mifflin Company All rights reserved. Chapter 11 Pre-Algebra Resource Book 75 j9rb-1108.qxd 11/13/03 1:50 PM Page 76 LESSON Name Date 11.8 Study Guide Continued EXAMPLE For use with pages 626–632 3 Finding the Probability of Overlapping Events You randomly choose a number from 1 to 30. What is the probability that you choose a prime number or a number greater than 20? Solution There are 10 prime numbers from 1 to 30: 2, 3, 5, 7, 11, 13, 17, 19, 23, and 29. Two of these prime numbers are greater than 20. There are 10 numbers greater than 20 but less than or equal to 30. Event A: The number is prime. Event B: The number is greater than 20. P(A or B) ⫽ P(A) ⫹ P(B) ⫺ P(A and B) 10 30 10 30 2 30 3 5 ⫽ ᎏᎏ ⫹ ᎏᎏ ⫺ ᎏᎏ ⫽ ᎏᎏ Probability of overlapping events Substitute probabilities. Then simplify. Answer: The probability that you choose a prime number or a number greater 3 5 than 20 is ᎏᎏ. Exercises for Examples 1–3 You randomly choose a number from 1 to 20. For the specified events A and B, tell whether the events are disjoint or overlapping. Then find P(A or B). 1. Event A: Choose a number divisible by 5. Event B: Choose a number divisible by 3. Lesson 11.8 2. Event A: Choose a number divisible by 12. Event B: Choose a number divisible by 5. EXAMPLE 4 Finding the Probability of Complementary Events 1 You spin a spinner. The probability that you spin red is ᎏᎏ. What is the probability 3 that you do not spin red? Solution The events red and not red are complementary events because they are disjoint and one or the other must occur. P(not red) ⫽ 1 ⫺ P(red) 1 3 2 3 ⫽ 1 ⫺ ᎏᎏ ⫽ ᎏᎏ Probability of complementary events 1 3 Substitute ᎏᎏ for P(red). Then subtract. 2 3 Answer: The probability that you do not spin red is ᎏᎏ, or about 66.67%. Exercises for Example 4 Given P(A), find P(not A). 3. P(A) ⫽ 44% 76 Pre-Algebra Chapter 11 Resource Book 4. P(A) ⫽ 0.01 6 7 5. P(A) ⫽ ᎏᎏ Copyright © McDougal Littell/Houghton Mifflin Company All rights reserved.