702 CHAPTER 11 Probability Example 1 illustrates that when determining the probability of A or B, we add the probabilities of events A and B and then subtract the probability of both events occurring simultaneously. MATHEMATICS TODAY Slot Machines You pull the handle, or push the button, to activate the slot machine. You are hoping that each of the three reels stops on Jackpot. The first two reels show Jackpot. When the third reel comes to rest, though, it does not show Jackpot. You say to yourself, “I came so close” and try again. In reality, however, you probably did not come close to winning the Jackpot at all. The instant you pull the slot machine’s arm, or push the button, the outcome is decided by a computer inside the slot machine. A random number generator within the computer is used to determine the outcomes. Although each reel generally has 22 stops, each stop is not equally likely. The computer assigns only a few random numbers to the stop that places the jackpots on all three reels at the same time and many more random numbers to stop the reels on stops that are not the Jackpot. If you do happen to hit the Jackpot, it is simply because the random number generator happened to generate the right sequence of numbers the instant you activated the machine. For more detailed information, check HowStuffWorks.com. Why This Is Important A slot machine is one of many machines that makes use of probability. Solution We are asked to determine the probability that the number selected is even or is greater than 6. Let’s use set A to represent the statement “the number is even” and set B to represent the statement “the number is greater than 6.” Fig. 11.12 is a Venn diagram, as introduced in Chapter 2, with sets A (even) and B (greater than 6). There are a total of 10 numbers, of which five are even (2, 4, 6, 8, and 10). Thus, the probability of selecting an even number is . 5 10 Four numbers are greater than 6: the 7, 8, 9, and 10. Thus, the probability of selecting a number greater than 6 is . 4 10 Two numbers are both even and greater than 6: the 8 and 10. Thus, the probability of selecting a number that is both even and greater than 6 is . 2 10 If we substitute the appropriate statements for A and B in the formula, we obtain PAB PA PB PA B P P P P ( or ) ( ) ( ) ( and ) even or greater than 6 (even) greater than 6 even and greater than 6 5 10 4 10 2 10 7 10 = + − ⎛ ⎝⎜ ⎞ ⎠⎟ = + ⎛ ⎝⎜ ⎞ ⎠⎟ − ⎛ ⎝⎜ ⎞ ⎠⎟ = + − = Thus, the probability of selecting an even number or a number greater than 6 is . 7 10 The seven numbers that are even or greater than 6 are 2, 4, 6, 7, 8, 9, and 10. 7 Now try Exercise 9 Even and greater than 6 Greater than 6 Even B A 1, 3, 5 2 6 4 8 10 9 7 U Figure 11.12 Example 2 Using the Addition Formula Consider the same sample space, the numbers 1 through 10, as in Example 1. If one piece of paper is selected, determine the probability that it contains a number less than 5 or a number greater than 8. Solution Let A represent the statement “the number is less than 5” and B represent the statement “the number is greater than 8.” A Venn diagram illustrating these statements is shown in Fig. 11.13. P P (number is less than 5) 4 10 (number is greater than 8) 2 10 = = Rocco Macri/123RF
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