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This 8 page Class Notes was uploaded by Aileen Davis on Tuesday October 20, 2015. The Class Notes belongs to MATH0013 at Sierra College taught by JohnBurke in Fall. Since its upload, it has received 27 views. For similar materials see /class/225377/math0013-sierra-college in Mathematics (M) at Sierra College.
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Date Created: 10/20/15
Sierra College Math 13 Spring 2009 Class 1032 nstmemr Juhn Burk Erma juhniburkem nds ng eem Web Page mtg Urnth swerracuHegE EduSta Juhn urke Telephune am 33741425 omee huurs veam MWZ 3575 DEL M 2 4573 45 Emma SeetmnszzeSAJ 7 9 11 13 21 23 1 5 4771 3 5 7 9 13 1711 Next Sectmns 571572 Tuday Asswgnment 475 m 3 476 45 Mul a n Rule Complements and Condltional Probability The ump ement uf at east me 5 Name Tn me me prubabmty uf at least one uf sumetmng calculate the prubabmty uf nune then subtramthat resuxmemw Than Pateastune 1 immune 45 Complements and Conditional Probability Example Fwd the p rubabmty that 3 Emma Wm three thrEn has aHEastunE gm Conditional Probability Given the Multiplication Rule we can rewrite it as PB APAandB PA The conditional probability of event B occurring given that A has already occurred can be found by dividing the probability of events A and B both occun39ing by the probability of event A Testing for Independence Two events A and B are independent if PBA PB or PA and B PA PB Two events A and B are dependent if PBA PB or PA and B PA PB Example Men Women Boys Girls Totals Survived 332 318 29 27 706 Died 1360 104 35 18 Total 1692 422 64 56 Ifone passenger is selected what is the probability that this person survived given that the selected person is a m Example Men Women Boys Girls Totals Survived 332 318 29 Died 1360 104 35 18 Ifone passenger is selected what is the probability that this person is a man given that the selected person survive Practice Problems What ls the probability that a randomly selected student ls female Rated hIsher math ablllty as Just average and not rate hIsher math ablllty asjust average ls both male and rated hls math ablllt as Just average Rated hIsher math ablllty as eltherlust average or above average ls elther female or rated hIsher math ablllty as below average ls female glveh that heshe rated ther math ablllty as above average ls male glveh that heshe rated ther math ablllt as above average anr F P39FP P Rated her math ablllty asjust average glveh that she ls female When to treat dependent data as if it is independent It is a common practice to treat events as independent when small samples are dravm 39om large populations For example pollsters use this guideline to treat the members oftheir samples as independent If a sample size is no more than 5 of the size of the population treat the selections as being independent even if the selections are made without replacement so they are technically dependent Pollsters use this guideline when they survey roughly 1000 adults from a population ofmillions they assume independence even though they sample without replacement When to treat dependent data as if it is independent Example With one method ofthe procedure called acceptance sampling a sample of items is randomly selected without replacement and the entire batch is rejected if there is at least one defect Suppose the Medtyme Company hasjust manufactured 5000 blood pressure monitors and 4 are defective lf3 of them are selected and tested what is the probability that the entire batch will be rejected Calculate using replacement and without replacement 46 Probabilities Through Simulations A simulation of a procedure is a process that behaves the same way as the procedure so that similar results are produced Objectives What to considerwhen running a simulation Simulating the tossing ofa coin Excel Simulating the tossing oftwo dice TI83 Plus amp StatDisk Simulating the birthday problem TI83 Plus amp Excel Things to Consider When Running a Simulation Make sure your model is correct How many simulations are necessary to achieve reasonable results remember the Law of Large Numbers the more the better Simulating the Tossing of Two Dice 0 Simulated Tosses T1783 P u 39m39 39 39 s misi i i kulmn quot 7s n h Thefrequencyprubabmty eman 7 7 90500 78 The thenretma prubabmty eman 7 7 BBB m 7 m ng the Tossing of Two Dice Sim u Ia ti 10000 Simulated Tosses StatDisk The frequency prubabmty 0f m7th 7 7 1B731EI EIEIEI m 7 The heureuca prubabmty 0f rung 7 7 BSE m 7 5 Birthday Problem 3 minimum naxi i amtw mlanimlmm mmnmmmmvmw m 46 Simulations Class Participation Problem Page 179 15 47 Counting Fundamental Counting Rule Factorial Rule Permutations Rules Combinations Rule 47 Counting Fundamental Counting Rule For a sequence oftwo events in which the rst event can occur m ways and the secon event can occur n ways the events together can occur m n ways Note this extends to any number of events Example The typical home alarm system has a code that consists offour digits The digits 0 through 9 can be repeated and they must be entered in the correct order nd the correct one How many different codes are possible 47 Counting Factorial Rule A collection ofn different items can be arIanged in order n factorial different ways This re ects the fact that the rst item may be selected n different ways the second item may be selected n1 different ways etc nnn1n221 eg 5 5x4x3x2x1 120 Example Suppose you want to design a travel route that has you visit each state capital exactly once How many possible routes are there 47 Counting Permutations Rules Permutations Rule when items are all different The number ofpermutations or sequences of ritems from n available items without replacement is p i Mir Example You belong to a club with 20 members How many different slates of4 of cers are possible l l 20 2019i 17161167280 P 2 4 2074 47 Counting Permutations Rules Example You havejust been hired to determine the programming for the WB television network When selecting the shows to be shovm on Monday night you nd that you have 27 shows available and you must select 4 ofthem ecause ofleadin effects the order ofthe shows is Important How many different sequences of4 shows are possible when there are 27 shows available 47 Counting Combinations Rule The number ofcombinations ofritems selected from n different items is mam Example Suppose you have a 20member club From the members you need to establish a committee offour How many ways is this possible l l C 20 7 2019181716 5193174845 2 4 20744 16l4l