162 CHAPTER 4 Probability CAUTION Don’t think that dependence of two events means that one is the direct cause of the other. Having a working light in your kitchen and having a working light in your bedroom are dependent events because they share the same power source. One of the lights may stop working for many reasons, but if one light is out, there is a higher probability that the other light will be out (because of the common power source). Drug Screening and the Basic Multiplication Rule EXAMPLE 4 Let’s use only the 50 test results from the subjects who use drugs (from Table 4-1), as shown below: Positive Test Results: 45 Negative Test Results: 5 Total: 50 a. If 2 of these 50 subjects who use drugs are randomly selected with replacement, find the probability that the first selected person had a positive test result and the second selected person had a negative test result. b. Repeat part (a) by assuming that the two subjects are selected without replacement. YOUR TURN. Do Exercise 13 “Drinking and Driving.” SOLUTION a. With Replacement: First selection (with 45 positive results among 50 total results): P1positive test result2 = 45 50 Second selection (with 5 negative test results among the same 50 total results): P1negative test result2 = 5 50 We now apply the multiplication rule as follows: P11st selection is positive and 2nd is negative2 = 45 50 # 5 50 = 0.0900 b. Without Replacement: Without replacement of the first subject, the calculations are the same as in part (a), except that the second probability must be adjusted to reflect the fact that the first selection was positive and is not available for the second selection. After the first positive result is selected, we have 49 test results remaining, and 5 of them are negative. The second probability is therefore 5>49, as shown below: P11st selection is positive and 2nd is negative2 = 45 50 # 5 49 = 0.0918 Is Ryanair’s Seating Allocation Random? Ryanair is an Irish airline company that supposedly assigns seats at random when passengers do not pay extra to reserve a seat. In one case, 15 women were traveling as part of a bachelorette party, and none of them paid for reserved seats. The 15 women were all separated, and they were all given middle seats instead of aisle or window seats. To test the randomness claim of Ryanair, four researchers were all sent on four different Ryanair flights, and every one of the 16 seat assignments was a middle seat. The researchers were able to identify the available seats on these flights and they found that on each of these four flights, there were fewer middle seats available than aisle or window seats. Probability can be used to show that Ryanair is not assigning seats using a method of random selection. (See Review Exercise 11.) c s s
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