164 CHAPTER 4 Probability In Example 5, if we treat the events as dependent without using the 5% guideline, we get the following cumbersome calculation that begins with 130,639,273 employees, with 4.2% of them (or 5,486,849) testing positive: a 5,486,849 130,639,273ba 5,486,848 130,639,272ba 5,486,847 130,639,271b = 0.0000740879 = 0.00007411rounded2 Just imagine randomly selecting 1000 employees instead of just 3, as is commonly done in typical polls. Extending the above calculation to include 1000 factors instead of 3 factors: I mean, come on! the 5% guideline for cumbersome calculations. The sample size of 3 is clearly no more than 5% of the population size of 130,639,273. We get P1all 3 employees test positive2 = P1first tests positive and second tests positive and third tests positive2 = P1first tests positive2 # P1second tests positive2 # P1third tests positive2 = 10.042210.042210.0422 = 0.0000741 There is a 0.0000741 probability that all three selected employees test positive. YOUR TURN. Do Exercise 29 “Medical Helicopters.” CAUTION In any probability calculation, it is extremely important to carefully identify the event being considered. See Example 6, where parts (a) and (b) might seem quite similar but their solutions are very different. Birthdays EXAMPLE 6 When two different people are randomly selected from those in your class or friend group, find the indicated probability by assuming that birthdays occur on the days of the week with equal frequencies. a. Find the probability that the two people are born on the same day of the week. b. Find the probability that the two people are both born on Monday. YOUR TURN. Do Exercise 13 “Drinking and Driving.” SOLUTION a. Because no particular day of the week is specified, the first person can be born on any one of the seven weekdays. The probability that the second person is born on the same day as the first person is 1>7. The probability that two people are born on the same day of the week is therefore 1>7. b. The probability that the first person is born on Monday is 1>7 and the probability that the second person is also born on Monday is 1>7. Because the two events are independent, the probability that both people are born on Monday is 1 7 # 1 7 = 1 49 Redundancy Reliability of systems can be greatly improved with redundancy of critical components. Race cars in the NASCAR Winston Cup series have two ignition systems so that if one fails, the other will keep the car running. Airplanes have two independent electrical systems, and aircraft used for instrument flight typically have two separate radios. The following is from a Popular Science article about stealth aircraft: “One plane built largely of carbon fiber was the Lear Fan 2100 which had to carry two radar transponders. That’s because if a single transponder failed, the plane was nearly invisible to radar.” Such redundancy is an application of the multiplication rule in probability theory. If one component has a 0.001 probability of failure, the probability of two independent components both failing is only 0.000001. R s b im r c p
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