150 CHAPTER 3 Probability Using the Multiplication Rule to Find Probabilities For anterior cruciate ligament (ACL) reconstructive surgery, the probability that the surgery is successful is 0.95. (Source: The Orthopedic Center of St. Louis) 1. Find the probability that three ACL surgeries are successful. 2. Find the probability that none of three ACL surgeries are successful. 3. Find the probability that at least one of three ACL surgeries is successful. SOLUTION 1. The probability that each ACL surgery is successful is 0.95. The chance of success for one surgery is independent of the chances for the other surgeries. P1three surgeries are successful2 = 10.95210.95210.952 ≈ 0.857 So, the probability that all three surgeries are successful is about 0.857. 2. Because the probability of success for one surgery is 0.95, the probability of failure for one surgery is 1 - 0.95 = 0.05. P1none of three are successful2 = 10.05210.05210.052 ≈ 0.0001 So, the probability that none of three surgeries are successful is about 0.0001. Note that because 0.0001 is less than 0.05, this can be considered an unusual event. 3. The phrase “at least one” means one or more. The complement to the event “at least one is successful” is the event “none are successful.” Use the complement found in part 2 to find the probability. (To avoid rounding in the second step below, use (0.05)(0.05)(0.05), not the rounded result.) P1at least one is successful2 = 1 - P1none are successful2 = 1 - (0.05)(0.05)(0.05) ≈ 0.9999. So, the probability that at least one of three surgeries is successful is about 0.9999. Note that this probability is not rounded to three decimal places because the result would be 1.000, which implies the event is certain. Even though it is highly likely that at least one of the three surgeries is successful, it is not a certain event. TRY IT YOURSELF 4 The probability that a particular rotator cuff surgery is successful is 0.9. (Source: The Orthopedic Center of St. Louis) 1. Find the probability that three rotator cuff surgeries are successful. 2. Find the probability that none of three rotator cuff surgeries are successful. 3. Find the probability that at least one of three rotator cuff surgeries is successful. Answer: Page A38 In Example 4, you were asked to find a probability using the phrase “at least one.” Notice that it was easier to find the probability of its complement, “none,” and then subtract the probability of its complement from 1. In general, this probability can be written as P(at least one occurrence of event A) = 1 - P(no occurrence of event A). EXAMPLE 4 For help with translating verbal phrases, see Integrated Review at MyLab Statistics
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