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# Class Note for EMSE 269 with Professor Dorp at GW (10)

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Date Created: 02/07/15

Chapter 9 Theoretical Probability Models Decisions R T Clemen T Reilly Draft Version 1 E Making Hard Decisions Chapter 9 Theoretical Probability Models Slide 1 of 47 a i came Rail Le uveNmesbv JR vanDamandlA Mazzuchl COPVRlGHl 2EIEI5 Niplvwwvseasngeduldamll hvevvu Theoretical Models Applied Draft Version 1 Theoretical Probability Models may be used when they describe the physical model quotadequatelyquot Examples 1 2 3 The outcome of an IQ test Normal Distribution The lifetime of a component exhibiting aging Weibull Distribution The length of a telephone call Exponential distribution The time between two people arriving at a post of ce Exponential distribution The number of people arriving at a post office in one hour Poisson Distribution The number of defectives in releasing a batch of fixed size Binomial distribution Making Hard Decisions r clemenr RellW Slide 2 of 47 conRchr zuus vaWU Chapter 9 Theoretical Probability Models Le uveNmeshv J R Mazzuchl Niplvwwvse The Binomial Distribution Assumptions 1 Afixed number oftrials say N 2 Each trial results in a Success or Failure 3 Each Trial has the same probability of success p 4 Different Trials are independent Define X Successes in a sequence of N trials XBNpltgt N XBNPltgtPIXXNP X jpx 117 Draft Version 1 x01N Making Hard Decisions Chapter 9 Theoretical Probability Models Slide 3 of 47 r cwemem Reilly Le meNmesbv J p dTA Mamcm copvmewezuus Niplvwwv dudavPrv bv cvvu The Binomial Distribution N N Kx xN x NNN 1N 2N 34321 N x 1 of ways you can choose x from a group of N EX1 Np VarX NID1ID Draft Version 1 Making Hard Decisions Chapter 9 Theoretical Probability Models Slide 4 of 47 r Clemen l Reilly Le uveNmesbv m Mazzucm conRleHrezuus Niplvwwvse bvovvu The Binomial Distribution DUAL RANDOM VARIABLE OF X X Successes in a sequence of N trials Y Failures in a sequence of N trials 1 YNX PIYyN1 pPrN XyNPPrXN ylNJ Draft Version 1 2 lt B 2 lt A p A B V 2 5 lt A p A B V lt B 2 Making Hard Decisions Chapter 9 Theoretical Probability Models Slide 5 of 47 r Clemen l Reilly Le uveNmesbv m Mazzucm covwewmuua Niplvwwvse bvevvu The Binomial Distribution Conclusion X BNP ltgt Y BN1P Pretzel Example You are planning to sell a new pretzel and you want to know whether it will be a success or not Initially you are 50 certain that it will be a Hit Thus Pr Hit Pr Flop 05 If your pretzel is a HIT you expect to gain 30 of the market Let X be the number of people out of a group of N that buy your pretzel Draft Version 1 Making Hard Decisions Chapter 9 Theoretical Probability Models Slide 6 of 47 Mazzucm r cwemem Remy Le meNmesbv m copvmewezuus Niplvwwvse bvevvu Pretzel Example The Binomial Distribution Assumption N PrXx N pretzel is a Hitquot x 0 07 Assumption N x Nix PrXx N pretzel is a Flopquot x 01 09 You decide to investigate the market for your pretzel and on a trial day it appeared that 5 OUT OF 20 PEOPLE bought your pretzel Draft Version 1 What do you think now of your chances of the pretzel being a Hit or a Flop Making Hard Decisions r memen r ReiiW Chapter 9 Theoretical Probability Models Slide 7 of 47 Mazzucm Le meNmesbv J R conRicHr 2uu5 Niplvwwvse bvcvvu Pretzel Example The Binomial Distribution Notation Data 205 Calculation PrquotHitquot Data 2 PrData quotHit quotPrquotHit PrData quotHit quotPrquotHit PrData quotFlop quotPrquotFlop quot 20W PrData HztquotK 50350715 0179 Table Page 686 Draft Version 1 Making Hard Decisions Chapter 9 Theoretical Probability Models Slide 8 of 47 r Ciememi Remy Le meNmesbv m van dYA Menu2m copvmewezuus Niplvwwvseasg u m y bvevvu Pretzel Example The Binomial Distribution 5 15 PrData quotFlopquot K 5 01 09 0032 Table Page 686 Conclusion Pr quot Hitquot Data 0848 017905003205 Further Development of selling Pretzel may be warranted Draft Version 1 Making Hard Decisions Chapter 9 Theoretical Probability Models Slide 9 of 47 r Ciemen i Remy Le meNmesbv m van dYA Menu2m copvmewezuus Niplvwwvseasg u m y bvevvu The Poisson Process and Distribution Consider a particular event eg a customer arriving at a bank Assumptions 1 The events can occur at any point in time 2 The arrival rate per hour is constant eg customers per hour 3 The number of customers arriving in disjoint time intervals are independent of each other eg the number of customers in the rst hour day and the number of customers in the second hour ofthe day Draft Version 1 Define Xt of such events in the time interval 0t Making Hard Decisions Chapter 9 Theoretical Probability Models Slide 10 of 47 R r Clemen l Reilly Le uveNmesbv J zucm conRicHrezuus Niplvwwv bvcvvu The Poisson Process and Distribution Xt Poissonmt t 1 PrX k m0 1 mk39 e mquot nk I PrY k I n Friquot is called the Poisson distribution EY n VarY n 3 EX mt E VarX mt Making Hard Decisions Chapter Theoretical Probability Models Slide 11 of47 R r clemen r Reilly Le meNmesbv A Mazzucm copvmew 2uus hvGWU Pretzel Example The Poisson Process Based on your previous market research you decide to invest in a pretzel stand Now you just need to select a good location You consider your location to be good bad or dismal if you sell 20 10 or 6 respectively per hour You assume that customers arrive according to a Poisson Process Xt of customer in the interval 0t PrXtk Good 201quot 7201 e k Jr E prXtkuBadu 10 0 e710 k 61quot PTXtk Dismal k Making Hard Decisions Chapter 9 Theoretical Probability Models Slide 12 of 47 r clemem Reilly Le meNmesbv J p dlA Mazzucm conRiGHrezuus Niplvwwv by cvvu Pretzel Example The Poisson Process Pr GOOD 070 Pr BAD 020 Pr DSMAL 010 You give yourself one week for people to get to know you at this location The second week you open your stand in the morning and in the first half hour 7 people bought your pretzel Hmmm You want to reevaluate your location SHOULD YOU RELOCATE Notation Data 7 005 We want to know Draft Version 1 Prquot Goodquot Data 2 Making Hard Decisions Chapter 9 Theoretical Probability Models Slide 13 of 47 v Ciemen Remy Le uveNmesbv m Mazzucm conRichezuus mplvwwvse bvcvvu Draft Version 1 Making Hard Decisions Pretzel Example The Poisson Process Calculation PrquotGoo Data PrData quotGoodquot PI Good PrData quotGoodquot Pr Good PrDam Bad Pr Bad PrDam Dismal Pr Divmal 20 057 372005 7 009 Table Page 7700 PrData Bad PrX057 Bad 103905 1005 7 0104 Table Page 698 PrData DismalquotPrX057 DismaI 6057 3605 7 0022 Table Page 698 PrDataquotGoodquotPrX057quotGoodquot Slide 14 of47 copvmew e was hvGWU r memem REHW Chapter 9 Theoretical Probability Models Le uveNmeshv J R Mazzucm mplvwwvse Pretzel Example The Poisson Process Draft Version 1 Pr Goodquot 070 Pr Badquot 020 Pr Dismalquot 010 009070 PrquotG00d Data 2 00907001040200022010 20733 Simlarly Pr BadquotData 0242 Pr DismalData 0025 Conclusion In light of the new data you decide that the chances ofthis being a Dismal location for the pretzel stand is remote and your chance for this being a Good location has slightly improved You decide to stay Making Hard Decisions Slide 15 of47 conRicHr 0 2005 hvGWU Chapter 9 Theoretical Probability Models Le uveNmeshv J R Mazzuchi Niplvwwvse r clemem ReilW The Exponential Distribution Consider a particular event eg a customer arriving at a bank Now consider the length of time between two consecutive events eg the time between two customers arriving Alternative assumptions for Poisson process 1 The arrival rate per hour is constant eg m customers per hour 2 Interarrival Times are exponentially distributed with parameter m 3 Customers arrive independently from each other Define Draft Version 1 T Time between two consecutive customers arrivingquot 39 Making Hard Decisions Chapter 9 Theoretical Probability Models Slide 16 of 47 R r clemem Reilly Le meNmesbv m vanDamandlA Mazzucm conRchrezuus Niplvwwvseasngedul39vdamn bvcvvu The Exponential Distribution T Exponentialm FTtmPrTStm1 e quotquot FT I ll Cumulative Distribution Function of TCDF The density function follows from dF t m T mt fTtm me t dt E E g a a 6 Making Hard Decisions Chapter 9 Theoretical Probability Models Slide 17 of 47 R v Ciemen i Remy Le uveNmeshv J zucm copvmew zuus mp NW bv ewu The Exponential Distribution Probability Density Function Exp2 frtl2 25 2 15 1 g 000 050 100 150 200 250 300 350 400 E g a e t l a Making Hard Decisions Chapter 9 Theoretical Probability Models Slide 18 of47 R r mm W onlywa m MW Mam cowwem mPlvwwwseasngeduldamlv hvevvu The Exponential Distribution PrTSa m 1 e m39 PrTgtam1 PrTSamequotquot39a PrbltTSamPrTSam PrTSbm 1ema emb eimb ema E E ET VarT 2 g m m a Lquot Q Making Hard Decisions Chapter 9 Theoretical Probability Models Slide 19 of 47 R r Ciemen i Remy Le meNmesbv A Mazzucm copvmew 2uu5 hvGWU Pretzel Example Exponential Distribution You want to provide fast service for your customers and you are wandering whether you can in your current setup of your stand It takes approximately 35 minutes to cook a pretzel What is the probability that the next customer arrives before the pretzel is finished You recall your initials assumptions ie You assume that customers arrive according to a Poisson Process and you consider your location good bad or dismal if you sell 20 10 or 6 respectively per hour Initially you belief that for your rst location Pr Good 070 Pr Bad 020 Pr Dismal 010 Draft Version 1 3 Making Hard Decisions Chapter 9 Theoretical Probability Models Slide 20 of47 a r clenenr Rellly Le uveNmeshv l in am Mazzuchl conRchrezuus Niplvwwv duldaml l by cvvu Draft Version 1 Pretzel Example Exponential Distribution Calculation PrT gt 35Min PrT gt 35Min quotGoodquot Prquot Goo quot PrT gt 35Min quot Badquot Prquot Badquot PrT gt 3 SMin quot Dismalquot Prquot Dismalquot PrT gt 35Minl quotGoodquot PrT gt 2mm 20 60 PrT gt 00583 20 62000583 03114 PrT gt 35Minl quotBad quot PrT gt 2mm 10 60 PrT gt 00583 10 61000583 05580 Making Hard Decisions Slide 21 of47 copvmew 2005 hvGWU Chapter 9 Theoretical Probability Models m Mazzucm r Ciemen l Remy Le meNmesbv m Pretzel Example Exponential Distribution 35 PrT gt 35Mml quotDismal quot PrT gt h0urs 6 60 PrT gt 00583 6 e 6390390583 07047 Pr Good 070 Pr Bad 020 Pr Disma 010 PrT gt 35Min 0311407 0558002 0704701 040 Or in other words PrT S 35Min 060 Conclusion Draft Version 1 60 of your customers will have to wait until the pretzel is ready Making Hard Decisions Chapter 9 Theoretical Probability Models Slide 22 of 47 7 Ciemen Remy Le uveNmesbv m van mm Mazzuchi conRieHrezuus hvGWU Pretzel Example Exponential Distribution You realize that customers prefer hot pretzels and you are not to concerned about this number However you decide to reevaluate after one week of operation The second week you open your stand in the morning and in the rst half an hour 7 people brought your pretzel What do you think now of is the percentage of people waiting for a pretzel Notation Data 7 005 Draft Version 1 3 Making Hard Decisions Chapter 9 Theoretical Probability Models Slide 23 of47 a i cleneni Reilly Le uveNmesbv J in am Mazzuchl COPVRlGHl 2EIEI5 Niplvwwv duldaml by cvvu Pretzel Example Exponential Distribution PrT gt 35Min Data 2 PrT gt 35Min G00dquotData PrquotG00dquot Data PrT gt 35Min Ba quot Data Prquot Badquot Data PrT gt 35Min quotDismalquot Data Prquot Dismalquot Data PrT gt 35Min quotGoodquotData PrT gt 35Min quotGoodquot 03114 PrT gt 35Min quotBad quotData 2 PrT gt 35Min quotBadquot 2 05580 PrT gt 35Min quotDismalquotData PrT gt 35Min quotDismalquot 07047 Draft Version 1 Making Hard Decisions Chapter 9 Theoretical Probability Models Slide 24 of 47 v memem Remy Le uveNmesbv m m Menu2m covvRieHvezuus Niplvwwvsea m y bvovvu Pretzel Example Exponential Distribution PrquotG00 quot Data 0733 Prquot Badquot Data 0242 PrquotDismal Data 2 0025 Hence PrT gt 35Min Data 2 03114 0733 05580 0242 07047 0025 03809 Or in other words PrT S 35Min Data 062 Conclusion 62 of your customers will have to wait until the pretzel is ready which increased from the previous 60 You are concerned about the chance of customers waiting increasing You decide to continue to monitor this percentage and may consider investing in another pretzel oven Draft Version 1 3 Making Hard Decisions Chapter 9 Theoretical Probability Models Slide 25 of47 Mam a i cleneni Reilly Le uveNmesbv JR COPVRlGHl 2EIEI5 mullvwwvse hvcvvu The Normal Distribution Consider the production of men shoes You want to offer these shoes in many different sizes However you need to decide the percentage of shoes to produce in each size Let Y be the length of men s feet Many biological phenomena height weight length follow a bellshaped curve that can be represented by a normal distribution YNM6I 1 xauf 039 2 e 202 EYu g VarY62 Making Hard Decisions Chapter 9 Theoretical Probability Models Slide 26 of 47 r clemem Remy Le meNmesbv J p diA Mazzucm copvmewezuus Niplvwwv dummy by cvvu The Normal Distribution Some handy rules ofthumb Pry UltYltyaz068 Pry 20ltYlty20z095 Pry 3UltY lty30z099 Draft Version 1 Making Hard Decisions Chapter 9 Theoretical Probability Models Slide 27 of 47 R v Clemen l Remy Le uveNmeshv 1 new conRleHvozuus Niplvwwv bvovvu The Normal Distribution Probability Density Function N205 Draft Version 1 Making Hard Decisions Chapter 9 Theoretical Probability Models Slide 28 of 47 r Clemen l Reilly Le meNmesbv m vanDamandlA Menu2m copvmewezuus Niplvwwvseasngedul39vdamn bvevvu The Normal Distribution Define Z Standard Normal Distributed ltgt Z N01 The Standard Normal CDF is available in Table Format Normal distribution is symmetric around its mean PrY lt y0PrY gty0ltgt PrZ lt z PrZ gt2 Draft Version 1 Making Hard Decisions Chapter 9 Theoretical Probability Models Slide 29 of 47 r clemen r Reilly Le meNmesbv J p dYA Mazzucm conRiGHrezuus Niplvwwv dudavmv bv ewu The Normal Distribution How do we calculate Pra lt Y S b lua if only the CDF for Z is available in Table format Convert to a Standard Normal Distribution PraltYSbly0 PraltYSb uy039 O O O PraltZSb uPrZSb PIZSa U U U U Draft Version 1 Making Hard Decisions Chapter 9 Theoretical Probability Models Slide 30 of 47 r Clemen l Remy Le uveNmesbv m Mazzucm covleeHrezuus Niplvwwvse bvovvu The Normal Distribution Example Probability Density Function N205 09 08 07 06 05 04 03 02 01 0 000 050 100 150 200 250 300 350 400 7 lt gt Pr125 lt Y s 225 205 125 225 a 125 2 225 2 3 1 Pr lt23 Pr lt23 05 05 2 2 1 Making Hard Decisions Chapter 9 Theoretical Probability Models Slide 31 of47 The Normal Distribution Probability Density Function N01 r i i 2 g 250 200 1 3950 100 050 000 0 0 100 150 200 if i a 12547150 2154050 1 Making Hard Decisions Chapter 9 Theoretical Probability Models Slide 32 of47 R i cm W WWst m MWA Mam comwwm Niplvwwvseasngedudamiv bvevvu Draft Version 1 The Normal Distribution Probability Density Function N01 250 200 150 100 050 000 050 100 150 200 PrZ 3206915 il 50 See Table Page 708 Making Hard Decisions r cwemem REHW Chapter 9 Theoretical Probability Models Le meNmesbv m vanDamandlA Mazzucm mp lvwwvseas ng eduldavmn Slide 33 of 47 copvmew e was hvGWU The Normal Distribution Probability Density Function N01 250 200 150 100 050 000 050 100 150 200 1257 2 225 2 150 05 05 PrZ S 00668 See Table Page 707 050 Draft Version 1 Making Hard Decisions Chapter 9 Theoretical Probability Models Slide 34 of 47 r clemem Reilly Le meNmesbv m vanDamandiA Mazzucm copvmwmuus Niplvwwvseasngedul39vdamn bvevvu The Normal Distribution Conclusion 3 1 1 3 Pr ltZS PrZS PrZS 2 2 2 2 06915 00668 2 06247 QUALITY CONTROL EXAMPLE You are the producer of hard drives for personal computers One of your machines that produces a part is used in the final assembly of the disk drive The width of this part is important for the proper functioning ofthe hard drive Ifthe width falls below 3995mm or the width falls above 4005mm the hard drive will not function properly If the disk drive does not work it must be repaired at a cost of 1 040 Draft Version 1 3 Making Hard Decisions Chapter 9 Theoretical Probability Models Slide 35 of47 Mam a i cleneni Reilly Le uveNmeshv JR COPVRlGHl 2EIEI5 Niplvwwvse hvcvvu QC Example The Normal Distribution The machine can be set a width of 4mm but it is not perfectly accurate The production speed ofthe machine can be set high or low However the higher production speed result in lower accuracy In fact if W is the width ofthe part W High Production Speed N4 00026 W Low Production Speed N4 00019 Of course at a higher production speed more hard drives are produced and the cost per hard drive is 2045 At the lower production speed the cost per hard drive is 2075 Should you turn at high production speed or low production speed Draft Version 1 Making Hard Decisions Chapter 9 Theoretical Probability Models Slide 36 of 47 r Clemen Reilly Le uveNmesbv m Mazzuchl conRchrezuus Niplvwwvse by cvvu Draft Version 1 QC Example The Normal Distribution Calculation Production At Low Speed PrDefective Low Speed 1 PrN0t Defectivel Low Speed 1 Pr3995 lt W S 4005 u 4 039 00019 2 3995 4 lt 00019 W 4 lt4005 4 1 Pr 00019 00019 y40 00019 1 Pr 263 lt Z s 263 1 PrZ s 263 PrZ s 263 1 09957 00043 21 09914 00086 See Table Page 709707 Slide 37 of 47 copvmew e was hvGWU Chapter 9 Theoretical Probability Models m Mazzucm Le meNmesbv J R Draft Version 1 QC Example The Normal Distribution Calculation Production at High Speed PrDefective High Speed 1 PrNot Defectivel High Speed 1 Pr3995 ltW g 4005 y 40 00026 3995 4 lt 1Pr W 4 lt4005 4 00026 00026 00026 m 40 00026 1 Pr 192 lt Z 192 1 PrZ Sl92 PrZ s 192 1 09726 00274 21 09452 2 00548 See Table Page 709707 Slide 38 of 47 copvmew e was hvGWU Chapter 9 Theoretical Probability Models m Mazzucm Le meNmesbv J R QC Example The Normal Distribution Min Cost EMV Defective 00086 3115 2084 Low Speed EMV 2075 Not Defective 09914 2084 2075 0 EMV Defective 00548 30 85 2102 1040 39 High Speed 2045 Not Defective 09452 2045 0 Draft Version 1 Conclusion Run at a slower speed Increased cost from slow speedare offset by the increased precision 3 Making Hard Decisions Chapter 9 Theoretical Probability Models Slide 39 0f47 R r em Kelli Le uveNmesbv m vanDamandYA Mazzuchl conRlGHrezuus mpwwseasngeduldamlv hvevvu The Beta Distribution Suppose you are interested in the proportion of voters in your town that will vote for the next republican president This proportion is uncertain and may range from 0 to 1 Let Q be that proportion and assume QBetanp TO fQ61 JW 61 11 61 10 lt 1 lt1 I n n 1n 1n 2321n 123 Draft Version 1 3 Making Hard Decisions Chapter 9 Theoretical Probability Models Slide 40 of47 a i am Ml Mum Le uveNmeshv J a COPVRlGHY ozuus Niplvwwvse hvevvu The Beta Distribution SYMMETRIC BETA DISTRIBUTIONS fQlI l quota r 400 350 300 250 200 150 100 050 000 000 020 040 060 Draft Version 1 Making Hard Decisions Chapter 9 Theoretical Probability Models Slide 41 of 47 r clemem Rellly Le meNmesbv m vanDamandYA Menu2m copvmwmuus Niplvwwvseasngedul39vdamlvl hvevvu The Beta Distribution ASYMMETRIC BETA DISTRIBUTIONS fQlI l quota r l E E u g 040 060 080 6 gt C1 Making Hard Decisions Chapter 9 Theoretical Probability Models Slide 42 of 47 r Clemen Y Rellly Le uveNmesbv m vanDavpandlA Menu2m conRleHrezuus Niplvwwvseasngedul39vdamlvl hvevvu The Beta Distribution EQ 5 VarltQgt 7 n n2 n 1 Elicitation Of Parameters Using Informal Parameter Interpretation n Number of Trialsquot r Number of Successesquot EXAMPLE You rst guess for the preference of the Republican Candidate is that 4 out of 10 people would vote for the Republican Candidate You set n10 r4 Draft Version 1 Note this coincides with an expected proportion of 40 Making Hard Decisions Chapter 9 Theoretical Probability Models Slide 43 of 47 r Clemen l Reilly Le uveNmeshv m Mazzucm conRchrezuus Niplvwwvse bvcvvu The Beta Distribution After talking to people on the street you reevaluate your beliefs and estimate that 40 out of 100 people would vote for the Republican Candidate You set n 100 r 40 Note that this still also coincides with an expected proportion of 40 What is the difference with the previous estimate First Estimate StDevQ Ml4m V10 101 Second Estimate g StDevQ w 49 V100 1001 3 Making Hard Decisions Chapter 9 Theoretical Probability Models Slide 44 of47 a r mm W Mam Le uveNmesbv J a COPleGHl ozuus mPlvwwvse hvcvvu Pretzel Example The Beta Distribution You want to re evaluate your decision to invest in a pretzel stand Sales have been okay in the first week but not too great You are wandering whether you should proceed You estimate at this point that you are 50 sure that your market share is less than 20 and your 75 sure that your market share is less than 38 Let Q be the proportion of the market You decide to model your uncertainty in Q as a beta distribution and using the table on page 711 that PrQ 020 n 4r 1 049 Draft Version 1 PrQ 3 038m 4r 1 076 3 Making Hard Decisions Chapter 9 Theoretical Probability Models Slide 45 of47 Mam R r ciemenr Reilly Le uveNmesbv m conRlGHiezuus mPlvwwvse by ewu Pretzel Example The Beta Distribution You decide that that is close enough and proceed with the analysis You estimate that the total monthly market is 100000 pretzels Your price for a pretzel is set at 050 and it costs you 010 to produce a pretzel You estimate 8000 of monthy fixed cost for your pretzel stand and some overhead Given the market share Q you calculate for your net monthly profit Net Profit Revenue Cost 100000Q050 100000Q0108000 40000Q8000 However Q is uncertain so you decide to calculate your expected profit Draft Version 1 E Making Hard Decisions Chapter 9 Theoretical Probability Models Slide 46 of47 Mum R r Cierllerl i Rellly Le uveNmesbv m conRchrezuus mPlvwwvse bvcvvu Pretzel Example The Beta Distribution EPro t E40000Q8000 40000EQ 8000 EQl25 n 4 EPro t 40000Xi 8000 2000 You start to be more comfortable with your decision to start a pretzel career but careful as you are you decide to evaluate your chances of loosing money PrNet Profit 3 O PrQ s 020 n4 r1 049 See Table Page 711 Conclusion There is approximately 50 chance of loosing money Are you willing to continue to take this RISK Draft Version 1 E Making Hard Decisions Chapter 9 Theoretical Probability Models Slide 47 of47 a r creme My Mum Le uveNmesbv J a conRlGHr 2005 Niplvwwvse hvcvvu

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