UA MATH 115A - Bayes’ Theorem - D994945

Home> Schools> University of Arizona> Mathematics Main (MATH) > MATH 115A> Bayes’ Theorem

UA MATH 115A - Bayes’ Theorem

Course Math 115a- Business Mathematics I

Pages 25

Download Save

Unformatted text preview:

Bayes’ TheoremExampleSlide 3Slide 4Venn DiagramSlide 6Finding ProbabilitiesComputing ProbabilitiesSlide 9Where are we going with this?Finding our ProbabilityArithmetic!Another Example—Tree DiagramsDefine EventsExtracting the InformationWhat are we trying to find?Tree DiagramSlide 18Follow the Branch?The Path Less Traveled?Slide 21Putting It All TogetherBonus Question:I thought this was called Bayes’ Theorem?In English Please?Bayes’ TheoremExampleThree jars contain colored balls as described in the table below.One jar is chosen at random and a ball is selected. If the ball is red, what is the probability that it came from the 2nd jar?Jar # Red White Blue1 3 4 12 1 2 33 4 3 2ExampleWe will define the following events:J1 is the event that first jar is chosenJ2 is the event that second jar is chosenJ3 is the event that third jar is chosenR is the event that a red ball is selectedExampleThe events J1 , J2 , and J3 mutually exclusive Why?You can’t chose two different jars at the same timeBecause of this, our sample space has been divided or partitioned along these three eventsVenn DiagramLet’s look at the Venn DiagramVenn DiagramAll of the red balls are in the first, second, and third jar so their set overlaps all three sets of our partitionFinding ProbabilitiesWhat are the probabilities for each of the events in our sample space?How do we find them?     BPBAPBAP |Computing ProbabilitiesSimilar calculations show:     813183|111 JPJRPRJP          2743194|1813161|333222JPJRPRJPJPJRPRJPVenn DiagramUpdating our Venn Diagram with these probabilities:Where are we going with this?Our original problem was:One jar is chosen at random and a ball is selected. If the ball is red, what is the probability that it came from the 2nd jar?In terms of the events we’ve defined we want:   RPRJPRJP22|Finding our Probability         RJPRJPRJPRJPRPRJPRJP321222|We already know what the numerator portion is from our Venn DiagramWhat is the denominator portion?Arithmetic!Plugging in the appropriate values:       17.0711227418181181|32122RJPRJPRJPRJPRJPAnother Example—Tree DiagramsAll tractors made by a company are producedon one of three assembly lines, named Red,White, and Blue. The chances that a tractorwill not start when it rolls off of a line are 6%,11%, and 8% for lines Red, White, and Blue,respectively. 48% of the company’s tractorsare made on the Red line and 31% are madeon the Blue line. What fraction of the company’stractors do not start when they roll off of anassembly line?Define EventsLet R be the event that the tractor was made by the red companyLet W be the event that the tractor was made by the white companyLet B be the event that the tractor was made by the blue companyLet D be the event that the tractor won’t startExtracting the InformationIn terms of probabilities for the events we’ve defined, this what we know:      08.0|11.0|06.0|31.021.048.0BDPWDPRDPBPWPRPWhat are we trying to find?Our problem asked for us to find:The fraction of the company’s tractors that do not start when rolled off the assembly line?In other words: DPTree DiagramBecause there are three companies producing tractors we will divide or partition our sample space along those events only this time we’ll be using a tree diagramTree DiagramFollow the Branch?There are three ways for a tractor to be defective:It was made by the Red CompanyIt was made by the White CompanyIt was made by the Blue CompanyTo find all the defective ones, we need to know how many were:Defective and made by the Red Company?Defective and made by the White Company?Defective and made by the Blue Company?The Path Less Traveled?In terms of probabilities, we want:   DBPDWPDRPComputing ProbabilitiesTo find each of these probabilities we simply need to multiply the probabilities along each branchDoing this we find               BPBDPDBPWPWDPDWPRPRDPDRP|||Putting It All TogetherBecause each of these events represents an instance where a tractor is defective to find the total probability that a tractor is defective, we simply add up all our probabilities:             BPBDPWPWDPRPRDPDP ||| Bonus Question:What is the probability that a tractor came from the red company given that it was defective?   DPDRPDRP|I thought this was called Bayes’ Theorem?Bayes’ TheoremSuppose that B1, B2, B3,. . . , Bn partition the outcomes of an experiment and that A is another event. For any number, k, with 1  k  n, we have the formula:niiikkkBPBAPBPBAPABP1)()|()()|()|(In English Please?What does Bayes’ Formula helps to find?Helps us to find:By having already known: ABP | BAP

View Full Document


School:
Email:
New Password:
Confirm Password:

UA MATH 115A - Bayes’ Theorem

Sign up for free to view:

Please select your school