4-3 Complements, Conditional Probability, and Bayes’ Theorem 175 In the following, we conveniently ignore the Alaskan winter and other such anomalies. P1D M2 = 1 1It is certain to be dark given that it is midnight.2 P1M D2 = 0 1The probability that it is exactly midnight given that it is dark is almost zero.2 Here, P1D M2 ≠ P1M D2. Confusion of the inverse occurs when we incorrectly switch those probability values or think that they are equal. In this section we extend the discussion of conditional probability to include applications of Bayes’ theorem (or Bayes’ rule), which we use for revising a probability value based on additional information that is later obtained. Let’s consider a study showing that physicians often give very misleading information when they experience confusion of the inverse. They tended to confuse P(cancer positive test result) with P(positive test result cancer). (A positive test result indicates the patient has cancer; a negative test result indicates the patient is cancer-free.) About 95% of physicians estimated P(cancer positive test result) to be about 10 times too high, with the result that patients were given diagnoses that were very misleading, and patients were unnecessarily distressed by the incorrect information. Let’s take a closer look at this example, and let’s hope that we can give physicians information in a better format that is easy to understand. PART 3 Bayes’ Theorem Interpreting Medical Test Results EXAMPLE 4 Assume cancer has a 1% prevalence rate, meaning that 1% of the population has cancer. Denoting the event of having cancer by C, we have P1C2 = 0.01 for a subject randomly selected from the population. This result is included with the following performance characteristics of the test for cancer (based on Probabilistic Reasoning in Clinical Medicine by David Eddy, Cambridge University Press). ■ P1C2 = 0.01 (There is a 1% prevalence rate of the cancer.) ■ The false positive rate is 10%. That is, P(positive test result given that cancer is not present) = 0.10. ■ The true positive rate is 80%. That is, P(positive test result given that cancer is present) = 0.80. Find P(C positive test result). That is, find the probability that a subject actually has cancer given that he or she has a positive test result. SOLUTION Using the given information, we can construct a hypothetical population with the above characteristics. We can find the entries in Table 4-2 on the next page, as follows. ■ Assume that we have 1000 subjects. With a 1% prevalence rate, 10 of the subjects are expected to have cancer. The sum of the entries in the first row of values is therefore 10. ■ The other 990 subjects do not have cancer. The sum of the entries in the second row of values is therefore 990. continued . Bayesian Statistics for Saving Lives A key characteristic of Bayesian statistics is that probabilities can be updated as additional information is acquired. The U.S. Coast Guard has been using methods of Bayesian statistics since the 1970s. In one case, it was reported that a fisherman fell from his boat sometime between 9:00 PM and 6:00 AM the next morning. The Coast Guard began to enter data into their Search and Rescue Optimal Planning System (SAROPS). Additional information, such as directions of currents, was acquired and entered into SAROPS. The search area could then be narrowed. A helicopter rescued the fisherman who was clinging to two buoys after having been in the water for 12 hours.
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