result of the test), when you know (or can estimate):(1)The probability of C in the absence of any evidence:(2)The evidence of C(3)The reliability of the evidence (i.e., the probability that the evidence is correct. In this example, the probability in (1) is 0.01, theevidence in (2) is that the test came out positive, and the probability in (3) has to be computed from the 75% figure given. All three pieces of information are highly relevant, and to evaluate the probability that you have the cancer you have to combine them I the right manner. Bayes' theorem allows us to do this.To simplify the illustration, assume a population of 10,000, since we are only interested in percentages, the reduction in population size will not affect the outcome. Thus, in a population of 10,000, 100 will have cancer and 9,900 will not. Bayes method, as previously mentioned, is about improving an initial estimate after you have obtained the new evidence. In the absence of the test, all you could say about the likelihood of you having the cancer is there is a 1% chance that you do. Then you take the test, it shows positive. How do you revise the probability that you have the cancer?There are, we know, 100 people in the population that do have the cancer, and for all of them, the test will show a positive result. But what of the 9,900 people that do not have the cancer, for 25% of them, the test will incorrectly give a positive result, thereby identifying 9900 X .025 = 2,475 people as having the cancer when they actually do not. Thus, overall, the test identifies a total of 100 + 2,475 = 2,575 people as having the cancer. Having tested positive, you are among that group (evidence tells you this). The question is, are you in the group that has the cancer or the group that does not but tested as if they did (false positive). Of the 2,575 that tested positive, 100 of them really do have cancer. To calculate the probability that you really have cancer...
