174 CHAPTER 4 Probability becomes P1positive test result subject uses drugs2 = P1subject uses drugs and had a positive test result2 P1subject uses drugs2 = 45>555 50>555 = 0.900 By comparing the intuitive approach to the formal approach, it should be clear that the intuitive approach is much easier to use, and it is also less likely to result in errors. The intuitive approach is based on an understanding of conditional probability, instead of manipulation of a formula, and understanding is so much better. b. Here we want P(subject uses drugs positive test result). This is the probability that the selected subject uses drugs, given that the subject had a positive test result. If we assume that the subject had a positive test result, we are dealing with the 70 subjects in the first column of Table 4-1. Among those 70 subjects, 45 use drugs, so P1subject uses drugs positive test result2 = 45 70 = 0.643 Again, the same result can be found by applying the formula for conditional probability, but we will leave that for those with a special fondness for manipulations with formulas. INTERPRETATION The first result of P(positive test result subject uses drugs) = 0.900 indicates that a subject who uses drugs has a 0.900 probability of getting a positive test result. The second result of P(subject uses drugs positive test result) = 0.643 indicates that for a subject who gets a positive test result, there is a 0.643 probability that this subject actually uses drugs. Note that P(positive test result subject uses drugs) ≠ P(subject uses drugs positive test result). This shows that in general, P1B A2 ≠ P1A B2. See “Confusion of the Inverse” that follows. YOUR TURN. Do Exercise 13 “Denomination Effect.” Confusion of the Inverse Note that in Example 2, P(positive test result subject uses drugs) ≠ P(subject uses drugs positive test result). This example proves that in general, P1B A2 ≠ P1A B2. There could be individual cases where P1A B2 and P1B A2 are equal, but they are generally not equal. DEFINITION Confusion of the inverse is to incorrectly think that P1B A2 and P1A B2 are equal or to incorrectly use one value in place of the other. Confusion of the Inverse EXAMPLE 3 Consider these events: D: It is dark outdoors. M: It is midnight. The Prosecutor’s Fallacy The incorrect use of probabilities occurs in court trials when experts confuse the probability of being guilty with the probability of evidence found against the defendant. For example, the probability of a defendant matching a description is not the same as the probability that someone who matches a description is guilty of the charge. Although it is commonly referred to as “prosecutor’s fallacy,” the same errors are often made by defense attorneys. In such cases, attorneys must be very careful with conditional probability. For real examples, see the Sally Clark case, the O.J. Simpson case, and the People vs. Collins case. T u a in w c p
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