11.5 Or and And Problems 701 The words or and and are part of our everyday vocabulary. These words also play an important role in the study of probability. For example, consider the following two questions related to drawing a single card from a standard 52-card deck. What is the probability of drawing an ace or a heart? What is the probability of drawing an ace and a heart? In this section, we will discuss these and similar probability problems that involve the words or and and. SECTION 11.5 Or and And Problems LEARNING GOALS Upon completion of this section, you will be able to: 7 Understand and solve probability problems that involve the word or. 7 Understand and solve probability problems that involve the word and. Why This Is Important Probability involving the words or and and is used in a wide variety of science, business, computer science, and engineering applications. A basic understanding of the probability we present in this section is an important skill that can be applied to a wide variety of occupations. In Section 11.4, we showed how to work probability problems by constructing sample spaces. Often it is inconvenient or too time consuming to solve a problem by first constructing a sample space. For example, if an experiment consists of drawing two cards with replacement from a standard 52-card deck, there would be 52 52, ⋅ or 2704, points in the sample space. Trying to list all these sample points could take hours. In this section, we learn how to solve compound probability problems that contain the words or or and without constructing a sample space. Or Problems The or probability problem requires obtaining a “successful” outcome for at least one of the given events. For example, suppose that we roll one die and we are interested in determining the probability of rolling an even number or a number greater than 4. For this situation, rolling either a 2, 4, or 6 (an even number) or a 5 or 6 (a number greater than 4) would be considered successful. Note that the number 6 satisfies both criteria. Since 4 of the 6 numbers meet the criteria (the 2, 4, 5, and 6), the probability of rolling an even number or a number greater than 4 is 4 6 or . 2 3 A formula for determining the probability of event A or event B, symbolized P(A or B), follows. Probability of A or B To determine the probability of A or B, use the following formula. PAB PA PB PA B ( or ) ( ) ( ) ( and ) = + − Since we add (and subtract) probabilities to determine P A B ( or ), this formula is sometimes referred to as the addition formula. We explain the use of the or formula in Example 1. Example 1 Using the Addition Formula Each of the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10 is written on a separate piece of paper. The 10 pieces of paper are then placed in a hat, and one piece is randomly selected. Determine the probability that the piece of paper selected contains an even number or a number greater than 6. Charnsitr/Shuttersock
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