4-2 Addition Rule and Multiplication Rule 163 The key point of part (b) in Example 4 is this: We must adjust the probability of the second event to reflect the outcome of the first event. Because selection of the second subject is made without replacement of the first subject, the second probability must take into account the fact that the first selection removed a subject who tested positive, so only 49 subjects are available for the second selection, and 5 of them had a negative test result. Part (a) of Example 4 involved sampling with replacement, so the events are independent; part (b) of Example 4 involved sampling without replacement, so the events are dependent. See the following. Sampling In the wonderful world of statistics, sampling methods are critically important, and the following relationships hold: ■ Sampling with replacement: Selections are independent events. ■ Sampling without replacement: Selections are dependent events. Exception: Treating Dependent Events as Independent Some cumbersome calculations can be greatly simplified by using the common practice of treating events as independent when small samples are drawn without replacement from large populations. (In such cases, it is rare to select the same item twice.) Here is a common guideline routinely used with applications such as analyses of survey results: TREATING DEPENDENT EVENTS AS INDEPENDENT: 5% GUIDELINE FOR CUMBERSOME CALCULATIONS When sampling without replacement and the sample size is no more than 5% of the size of the population, treat the selections as being independent (even though they are actually dependent). Example 5 illustrates use of the above “5% guideline for cumbersome calculations” and it also illustrates that the basic multiplication rule extends easily to three or more events. Drug Screening and the 5% Guideline for Cumbersome Calculations EXAMPLE 5 In a recent year, there were 130,639,273 full-time employees in the United States. If one of those employees is randomly selected and tested for illegal drug use, there is a 0.042 probability that the test will yield a positive result, indicating that the employee is using illegal drugs (based on data from Quest Diagnostics). Assume that three employees are randomly selected without replacement from the 130,639,273 employees in the United States. Find the probability that the three selected employees all test positive for drug use. SOLUTION Because the three employees are randomly selected without replacement, the three events are dependent, but here we can treat them as being independent by applying continued he d st e, ve Independent Jet Engines Soon after departing from Miami, Eastern Airlines Flight 855 had one engine shut down because of a low oil pressure warning light. As the L-1011 jet turned to Miami for landing, the low pressure warning lights for the other two engines also flashed. Then an engine failed, followed by the failure of the last working engine. The jet descended without power from 13,000 ft to 4000 ft, when the crew was able to restart one engine, and the 172 people on board landed safely. With independent jet engines, the probability of all three failing is only 0.00013, or about one chance in a trillion. The FAA found that the same mechanic who replaced the oil in all three engines failed to replace the oil plug sealing rings. The use of a single mechanic caused the operation of the engines to become dependent, a situation corrected by requiring that the engines be serviced by different mechanics.
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