Lecture Notes on Probability


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Unformatted text preview: vation of Bayes’ Theorem On the last slide we showed that: Note that since because and By the definition of conditional probability, Hence, 8 Applying Bayes’ Theorem Example: Suppose that one person in 100,000 has a particular disease. There is a test for the disease that gives a positive result 99% of the time when given to someone with the disease. When given to someone without the disease, 99.5% of the time it gives a negative result. Find a) the probability that a person who test positive has the disease. b) the probability that a person who test negative does not have the disease. Should someone who tests positive be worried? 9 Applying Bayes’ Theorem Solution: Let D be the event that the person has the disease, and E be the event that this person tests positive. We need to compute p(D|E) from p(D), p(E|D), p( E | ), p( ). 10 Applying Bayes’ Theorem What if the result is negative? So, the probability you have the disease if you test negative is So, it is extremely unlikely you have the disease if you test negative. 11 Generalized Bayes’ Theorem Generalized Bayes’ Theorem: Suppose that E is an event from a sample space S and that F1, F2, …, Fn are mutually exclusive events such that Assume that p(E) ≠ 0 for i = 1, 2, …, n. Then 12 Bayesian Spam Filters How do we develop a tool for determining whether an email is likely to be spam? If we have an initial set B of spam messages and set G of non‐spam messages. We can use this information along with Bayes’ law to predict the probability that a new emai...
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