1104 CHAPTER 11 Further Topics in Algebra (b) “Drawing a 3” and “drawing a king” are mutually exclusive events because it is impossible to draw one card that is both a 3 and a king. P13 or K2 = P132 + P1K2 - P13 and K2 Probability of compound events = 4 52 + 4 52 - 0 Find and substitute known probabilities. = 8 52 Add and subtract fractions. = 2 13 Write in lowest terms. S Now Try Exercise 23(d). EXAMPLE 5 Finding Probabilities of Compound Events Suppose two fair dice are rolled. Find each probability. (a) The first die shows a 2, or the sum of the two dice is 6 or 7. (b) The sum of the dots showing is at most 4. SOLUTION (a) Think of the two dice as being distinguishable—one red and one green, for example. (Actually, the sample space is the same even if they are not apparently distinguishable.) A sample space with equally likely outcomes is shown in Figure 18, where 11, 12 represents the event “the first die (red) shows a 1 and the second die (green) shows a 1,” 11, 22 represents “the first die shows a 1 and the second die shows a 2,” and so on. Event B Event A (6, 1) (6, 2) (6, 3) (6, 4) (6, 5) (6, 6) (5, 1) (5, 2) (5, 3) (5, 4) (5, 5) (5, 6) (4, 1) (4, 2) (4, 3) (4, 4) (4, 5) (4, 6) (3, 1) (3, 2) (3, 3) (3, 4) (3, 5) (3, 6) (2, 1) (2, 2) (2, 3) (2, 4) (2, 5) (2, 6) (1, 1) (1, 2) (1, 3) (1, 4) (1, 5) (1, 6) Figure 18 Let A represent the event “the first die shows a 2,” and B represent the event “the sum of the two dice is 6 or 7.” See Figure 18. Event A has 6 elements, event B has 11 elements, and the sample space has 36 elements. P1A2 = 6 36 , P1B2 = 11 36 , and P1A¨B2 = 2 36 P1A´B2 = P1A2 + P1B2 - P1A¨B2 Probability of compound events = 6 36 + 11 36 - 2 36 Substitute known probabilities. = 15 36 Add and subtract fractions. = 5 12 Write in lowest terms.
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