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# Class Note for MATH 115A with Professor Dawson at UA

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Date Created: 02/06/15

Bayes 39 Math 1 1 EA Spring 2008 Dawson Definition Let the events Bl BZ B be non empty subsets of a sample space S for an experiment The 35 are a partition of S if B H Bk D unless ik u Bn s O BlLJ 32 U Example 1 Forty percent of the employees of a certain company are college graduates Sixty percent of those employees earn 35000 or more per year while only 10 percent of the employees who are not college graduates earn 35000 or more per year 0 Create a partition of the employees If the company has 100 employees how many employees are in each subset Example 1 o S employees of a company 0 100 employees B1 college graduates who earn 35000 or more 24 employees B college graduates who earn less than 55000 u 16 employees o B3 non college graduates who earn 35000 or more 0 6 employees B non college graduates who earn less than 555000 u 54 employees Example 1 What is the probability that an employee chosen at random is a college graduate who makes less than 35000 o What is the probability that an employee chosen at random is not a college graduate o What is the probability that the chosen employee makes less than 35000 if we know she is not a college graduate r Example 1 Earns V 60 35 000 or PCmM46Z 4 College 7 more 40 graduate V Earns less 40 than 35000 Chm 4 4 16 Employee 10 Earns 35000 60 39 or more Non college I graduate PNmM6106 quot Earns less r 90 than 35000 PNF L 6 9 54 0 college graduate N non college graduate M earns 35000 or more L earns less than 35000 7 lt Example 1 a What is the probability that an employee chosen at random is a college graduate who makes less than 35000 PC L 244 216 o What is the probability that an employee chosen at random is not a college graduate PN 60 o What is the probabilit that the chosen employee makes less than 35 00 if we know she is not a college graduate PLI N 90 Tree Diagrams o In tree diagrams each branch of the tree diagram represents the intersection of two events its probability can be computed by multiplying the probabilities bot regular and conditional that occur along the branch EX The tcgn branch represents C m M We know that PC m M PCPM 0406 024 As a check of our work the sum of the probabilities for all branches that go through C is 04 which is equal to the probability of C the sum of the probabilities that pass through N is 6 which is equal to the probability of N the four branches represent mutually exclusive events one of which must happen so the sum of their probabi ities must be equal to 1 Formulas 0 Suppose we have partitioned S into Bl BZ Bn and that A is an event then PAPA S PA BluBzuuBn PA BluA BzuuA Bn PA BlUA BZUUA Bn PAmBlPAmBZPAmBn PABlPBlPAIBZPBZPABnPBn iPAIBiPBi Formulas If Bl BZ Br partition 5 and A is any event then PltAgt PM I BiPltBi 11 Think of this as adding the probabilities of the branches that end in A or alternatively think of the following Venn diagram and recall that PA B PABPB Example 2 o A shop owner sells cameras produced by 3 factories 30 o come from Factory I 45 0 come from Factory II and 2500 come from Factor III It is also known that of all the cameras pro uced at Factory I 15 o are defective of those produced at Factories II and III 200 and 300 respectively are defective o What is the probability a camera purchased was made at Factory I and is defective o What is the probability a camera purchased was made at Factory III and is not defective o If we randomly chose a camera what are the chances that it is not defective r Example 2 015 Defective P1mD3O015AOQ452 Factory 1 quot 30 9852 Working PImW 3 0 9 85239955 02 Defective PII m D 45 02 009 Factor 11 5 98 Working PalmW 4598441 25 03 Defective PIII m D 25 03 0075 Factory III quot 97 Working PIIIrquot W 2597 2425 I Factory I II Factory II III Factory EW working D Defective Jv Example 2 What is the probability a camera purchased was made at Factory I and is defective PI m D 30 015 0045 What is the probability a camera purchased was made at Factory III and is not defective PIII NW 2597 2425 If we randomly chose a camera what are the chances that it is not defective PW PW IPIPW IIPIIPW IIIPIII 2955 441 2425 979 Main Theorem Often times we want to know PAB but we only know PBIA How can we reverse conditional probabilities o This can be done by something called Bayes Theorem Main Theorem 0 Simple case n Suppose B1 and 32 partition 5 and A is another event We want to use PBl PBZ PAlBl and PAIBZ to compute PBllA e We can derive the formula PAIBI PBI PBl IA PAIBlPBlPA3932 PBZ Note the numerator is just PA Bl and the denominator is just PA rewritten using the rules from above Example 3 Forty percent of the employees of a certain company are college graduates Sixty percent of those employees earn 35000 or more per year while only 10 percent of the employees who are not college graduates earn 35000 or more per year c What is the probability that a randomly selected employee makes less than 35000 if we know she is a college graduate o What is the probability that a randomly selected employee is a college graduate if we know she makes less than 35000 Example 3 What is the probability that a randomly selected employee makes less than 35000 if we know she is a college graduate PLC4O What is the probability that a randomly selected employee is a college graduate if we know she makes less than 35000 PL CAPC PLC CPLN C 404 16 z229 4 49 6 7 PCL r Example 3 Earns V 60 35 000 or PCmM46Z 4 College 7 more 40 graduate V Earns less 40 than 35000 Chm 4 4 16 Employee 10 Earns 35000 60 39 or more Non college I graduate PNmM6106 quot Earns less r 90 than 35000 PNF L 6 9 54 0 college graduate N non college graduate M earns 35000 or more L earns less than 35000 7 lt Main Theorem General case Suppose that Bl BZ Bn partition the outcomes of an experiment and that A is another event For any number k with lskSn we have the formula Bk A 2 n 2 PA I Bi PBi 11 This is Bayes Theorem Example 4 o All tractors made by a company are produced on one of three assembly lines named Red White and Blue The chances that a tractor will not start when it rolls off a line are 600 11 o and 800 for lines Red White and Blue respectively 4800 of the company s tractors are made on the Red line and 31 o are made on the Blue line 0 What is the probability that a random tractor chosen does not start 0 What is the probability that a tractor came off the Red line if it doesn t start Example 5 Thirty percent of the population have a certain disease Of those that have the disease 8900 will test positive for the disease Of those that do not have the disease 500 will test positive What is the probability that person has the disease given that they test positive for the disease

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