6 3 Bayes Theorem We can use Bayes
6. 3 Bayes Theorem
We can use Bayes Theorem… • …when we know some conditional probabilities, but wish to know others. • For example: We know P(test positive|have disease), and we wish to know P(have disease|test positive)
Ex. 1 (book ex 2 - p. 419) • Suppose that one person in 100, 000 has a particular rare disease for which there is a fairly accurate diagnostic test. This test is correct 99% of the times for someone who has the disease and 99. 5%% of the time for someone who does not.
Define E, F, E’, F’ Let F=event one has the disease E=event one tests positive We know that P(F) = 1/100, 000 =. 00001 P(E|F)= P(positive|disease) =. 99 and P(E’ |F’ ) = P(negative| don’t have disease) =. 995 Determine P(F|E) = P(has disease|test positive) = ___ P ( F’ |E’ )= P(does not have disease |test negative)= ___ and
Draw tree diagram starting with F, F’ •
Find P(F|E) and P(F’| E’) • P(F|E) = = 0. 002 • P(F’ | E’)= = 0. 9999999
Ex. 2: F=studied for final, E=passed class Assume: P(F) = P(studied)=. 8 P(E|F)= P(passed|studied)=. 9 and P(E|F ’ ) = P(passed|didn’t study)=. 2 Find P(F|E) = P(studied|passed)= ___ P (F’ | E’ )= P(didn’t study | failed) = ___
Tree diagram, starting with F, F’ •
Spam filters Ex. 3: Spam filters Idea: spam has words like “offer”, “special”, “opportunity”, “Rolex”, … Non-spam has words like “mom”, “lunch”, … False negatives: when we fail to detect spam False positives: when non-spam is seen as spam Let S=spam E=has a certain word Assume P(S)=0. 5
Tree diagram starting with S, S’ •
Given a message says “Rolex”, find probability it is spam Consider that “Rolex” occurs in 250/2000=. 125 spam messages and in 5/1000=. 005 non-spam messages. Assume P(S)=0. 5 Ex: P(S|uses word “Rolex”) = =. 962
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