Probability and Statistics(11/35 Axioms Bayes Theorem >The experiment of recording piston ring failure in compressors is an example of where Bayes'Theorem might be used,and we derive Bayes'Theorem using this example. Suppose our piston rings are purchased from two manufacturers: 60%from manufacturer A and 40%from manufacturer B.Let M denote the event that a part comes from manufacturer A,and Mg represent the event that a piston ring comes from manufacturer B P(MA)=0.6 P(MB)=0.4 日濟大学 AW TONGJI UNIVERSITY
Bayes Theorem ➢ The experiment of recording piston ring failure in compressors is an example of where Bayes’ Theorem might be used, and we derive Bayes’ Theorem using this example. Suppose our piston rings are purchased from two manufacturers: 60% from manufacturer A and 40% from manufacturer B. Let MA denote the event that a part comes from manufacturer A, and MB represent the event that a piston ring comes from manufacturer B. Probability and Statistics(11/35) Axioms
Probability and Statistics(12/35 Axiomsa Bayes'Theorem can be derived from the definition of conditional probability.Writing this in terms of our events,we are interested in the following probability: P(MAF)= P(MAOF) P(F) where P(MAF)represents the posterior probability that the part came from manufacturer A,and F is the event that the piston ring failed. @月停大学 TONGJI UNIVERSITY
➢ Bayes’ Theorem can be derived from the definition of conditional probability. Writing this in terms of our events, we are interested in the following probability: where represents the posterior probability that the part came from manufacturer A, and F is the event that the piston ring failed. Probability and Statistics(12/35) Axioms
Probability and Statistics(13/35 Axioms Using the Multiplication Rule,we can write in terms of event F and our prior probability that the part came from manufacturer A,as follows: P(MF)=P(MOF)=P(MA)P(EIMA) P(F) P(F) P(F)=P(MAOF)+P(MBOF) Applying the Multiplication Rule as before,we have: P(F)=P(MA)P(FMA)+P(MB)P(FMB) @日济大学 AW TONGJI UNIVERSITY
➢ Using the Multiplication Rule, we can write in terms of event F and our prior probability that the part came from manufacturer A, as follows: Applying the Multiplication Rule as before, we have: Probability and Statistics(13/35) Axioms