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In probability theory and applications, Bayes' theorem shows the relation between two conditional probabilities which are the reverse of each other. This theorem is named for Thomas Bayes (pronounced /ˈbeɪz/ or "bays") and often called Bayes' law or Bayes' rule. Bayes' theorem expresses the conditional probability, or "posterior probability", of a hypothesis H (i.e. its probability after evidence E is observed) in terms of the "prior probability" of H, the prior probability of E, and the conditional probability of E given H. It implies that evidence has a stronger confirming effect if it was more unlikely before being observed.^{[1]} Bayes' theorem is valid in all common interpretations of probability, and it is commonly applied in science and engineering.^{[2]} However, there is disagreement among statisticians regarding its proper implementation.
The key idea is that the probability of an event A given an event B (e.g., the probability that one has breast cancer given that one has tested positive in a mammogram) depends not only on the relationship between events A and B (i.e., the accuracy of mammograms) but on the marginal probability (or "simple probability") of occurrence of each event. For instance, if mammograms are known to be 95% accurate, this could be due to 5.0% false positives, 5.0% false negatives (missed cases), or a mix of false positives and false negatives. Bayes' theorem allows one to calculate the conditional probability of having breast cancer, given a positive mammogram, for any of these three cases. The probability of a positive mammogram will be different for each of these cases. In the example at hand, there is a point of great practical importance that is worth noting: if the prevalence of mammograms resulting positive for cancer is, say, 5.0%, then the conditional probability that an individual with a positive result actually does have cancer is rather small, since the marginal probability of this type of cancer is closer to 1.0%. The probability of a positive result is therefore five times more likely than the probability of the cancer itself. This shows the value of correctly understanding and applying Bayes' mathematical theorem.
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