4-2 Addition Rule and Multiplication Rule 159 One way to apply the addition rule is to add the probability of event A and the probability of event B and, if there is any overlap that causes double-counting, compensate for it by subtracting the probability of outcomes that are included twice. This approach is reflected in the above formal addition rule. INTUITIVE ADDITION RULE To find P1A or B2, add the number of ways event A can occur and the number of ways event B can occur, but add in such a way that every outcome is counted only once. P1A or B2 is equal to that sum, divided by the total number of outcomes in the sample space. FORMAL ADDITION RULE P1A or B2 = P1A2 + P1B2 - P1A and B2 where P1A and B2 denotes the probability that A and B both occur at the same time as an outcome in a trial of a procedure. TABLE 4-1 Results from Drug Tests of Job Applicants Positive Test Result (Test shows drug use.) Negative Test Result (Test shows no drug use.) Subject Uses Drugs 45 (True Positive) 5 (False Negative) Subject Does Not Use Drugs 25 (False Positive) 480 (True Negative) *Numbers in red correspond to positive test results or subjects who use drugs, and the total of those numbers is 75. Drug Testing of Job Applicants EXAMPLE 1 Refer to Table 4-1. If 1 subject is randomly selected from the 555 subjects given a drug test, find the probability of selecting a subject who had a positive test result or uses drugs. YOUR TURN. Do Exercise 11 “Texting or Drinking.” SOLUTION Refer to Table 4-1 and carefully count the number of subjects who tested positive (first column) or use drugs (first row), but be careful to count subjects exactly once, not twice. When adding the frequencies from the first column and the first row, include the frequency of 45 only once. In Table 4-1, there are 45 + 25 + 5 = 75 subjects who had positive test results or use drugs. We get this result: P1positive test result or subject uses drugs2 = 75>555 = 0.135 Disjoint Events and the Addition Rule The addition rule is simplified when the events are disjoint. DEFINITION Events A and B are disjoint (or mutually exclusive) if they cannot occur at the same time. (That is, disjoint events do not overlap.)
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