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From the 5 users, 0.99 × 5 ≈ 5 true positives are expected.Out of 15 positive results, only 5, about 33%, are genuine.
When applied, the probabilities involved in Bayes' theorem may have different probability interpretations.The importance of specificity in this example can be seen by calculating that even if sensitivity is raised to 100% and specificity remains at 99% then the probability of the person being a drug user only rises from 33.2% to 33.4%, but if the sensitivity is held at 99% and the specificity is increased to 99.5% then the probability of the person being a drug user rises to about 49.9%.The entire output of a factory is produced on three machines.For example, if 100,000 items are produced by the factory, 20,000 will be produced by Machine A, 30,000 by Machine B, and 50,000 by Machine C.Machine A will produce 1000 defective items, Machine B 900, and Machine C 500.To use concrete numbers, if 1000 individuals are tested, there are expected to be 995 non-users and 5 users.
From the 995 non-users, 0.01 × 995 ≃ 10 false positives are expected.
The role of Bayes’ theorem is best visualized with tree diagrams, as shown to the right.
The two diagrams partition the same outcomes by A and B in opposite orders, to obtain the inverse probabilities.
The three machines account for different amounts of the factory output, namely 20%, 30%, and 50%.
The fraction of defective items produced is this: for the first machine, 5%; for the second machine, 3%; for the third machine, 1%.
Bayes’ theorem serves as the link between these different partitionings. R, C, P and P bar are the events representing rare, common, pattern and no pattern. Note that three independent values are given, so it is possible to calculate the inverse tree (see figure above).