stat400lec7 - woman with this cancer, there are about 16%...

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Statistics 400 Lecture 7 Bayes’ Theorem Review: Conditional Probability: The goal is to find the probability of a certain event given that a specific condition is satisfied. Assume P (A) > 0 P(B|A) = P(A and B) P (A) P (A and B) = P(B|A) * P (A) =P(A|B)*P(B) Thomas Bayes (c. 1702 April 17 , 1761 ) Ping Ma Lecture 7 Fall 2005 - 1 -
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If A and B are any two events, A=(B A) h (B’ A) Venn Diagram P(A)=P(B A)+ P(B’ A) =P(B)*P(A|B)+ P(B’)*P(A|B’) We have Bayes’ Theorem P(B|A)= ) ( ) ( A P A B P = ) B' | P(A * ) P(B' B) | P(A * P(B) ) | ( * ) ( + B A P B P Terminology: P(B) is called prior probability P(B|A) is called posterior probability Ping Ma Lecture 7 Fall 2005 - 2 -
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Generalization: Let B1, B2,…Bm be mutually exclusive and exhaustive events. If A is an event, then we have A=(B1 A) h (B2 A) h h (Bm A) Thus, P(A)=P(B1 A)+ P(B2 A)+…+P(Bm A) =P(B1)*P(A|B1)+ P(B2)*P(A|B2)+…+ P(Bm)*P(A|Bm) Then we have Bayes’ Theorem P(B1|A)= ) ( ) 1 ( A P A B P = Bm) | P(A * P(Bm) B2) | P(A * P(B2) B1) | P(A * P(B1) ) 1 | ( * ) 1 ( + + + B A P B P Ping Ma Lecture 7 Fall 2005 - 3 -
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Example: A pap smear is a screening procedure used to detect cervical cancer. For
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Unformatted text preview: woman with this cancer, there are about 16% false negative; that is P(test negative | cancer)=0.16 For woman without this cancer, there are about 19% false postive; that is P(test positive | no cancer)=0.16 In the US, there are about 8 woman in 100,000 who have this cancer; that is P(cancer)=0.00008 What is the posterior probability of P(cancer | test positive) ? Ping Ma Lecture 7 Fall 2005- 4 -Example: An insurance company knows that the following probabilities relating to automobile accidents: Age of driver Prob. of accident fraction of companys insured driver 16-25 0.05 0.10 26-50 0.02 0.55 51-65 0.03 0.20 66-90 0.04 0.15 A random selected driver from the companys insured drivers has accident. What is the conditional probability that the driver is in 16-25 age group? Ping Ma Lecture 7 Fall 2005- 5 -...
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stat400lec7 - woman with this cancer, there are about 16%...

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