Lecture Notes on Probability

# Lecture Notes on Probability - Chapter 7 1 Section 7.3 2...

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Chapter 7 1

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Section 7.3 2
3 Section Summary y Bayes’ Theorem y Generalized Bayes’ Theorem y Bayesian Spam Filters

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4 Motivation for Bayes’ Theorem y Bayes’ theorem allows us to use probability to answer questions such as the following: y Given that someone tests positive for having a particular disease, what is the probability that they actually do have the disease? y Given that someone tests negative for the disease, what is the probability, that in fact they do have the disease? y Bayes’ theorem has applications to medicine, law, artificial intelligence, engineering, and many diverse other areas.
5 Bayes’ Theorem Bayes’ Theorem : Suppose that E and F are events from a sample space S such that p ( E ) ് 0 and p ( F ) ് 0 . Then: Example : We have two boxes. The first box contains two green balls and seven red balls. The second contains four green balls and three red balls. Bob selects one of the boxes at random. Then he selects a ball from that box at random. If he has a red ball, what is the probability that he selected a ball from the first box. y Let E be the event that Bob has chosen a red ball and F be the event that Bob has chosen the first box. y By Bayes’ theorem the probability that Bob has picked the first box is:

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6 Derivation of Bayes’ Theorem y Recall the definition of the conditional probability p ( E |F): y From this definition, it follows that: , continued ՜
7 Derivation of Bayes’ Theorem On the last slide we showed that continued ՜ , , Solving for p ( E | F ) and for p ( F | E ) tells us that Equating the two formulas for p(E|F) shows that

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8 Derivation of Bayes’ Theorem On the last slide we showed that: Note that Hence, since because and By the definition of conditional probability,
9 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. y Should someone who tests positive be worried?

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10 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 ( ).
11 Applying Bayes’ Theorem y What if the result is negative?

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