Chapter14

ProjectRiskandUncertaintyOriginofProjectRiskMethodsofDescribingProjectRiskProbabilityConceptsfroInvestmentDecisionsProbabilityDistributionofNPWDecisionTreesandSequentialInvestmentDecisions1(c)2019ContemporaryEngineeringEconomicsChapter14

ProjectRiskandUnOriginsofProjectRiskRisk:thepotentialforlossProjectRisk:variabilityinaproject’sNPWRiskAnalysis:Theassignmentofprobabilitiestothevariousoutcomesofaninvestmentproject2(c)2019ContemporaryEngineeringEconomicsOriginsofProjectRiskRisk:tExample14.1ImprovingtheOdds–AllItTakesis$7MillionandaDreamNumberofPrizesPrizeCategoryTotalAmount1FirstPrize$27,007,364228SecondPrizes($899each)204,97210,552ThirdPrizes($51each)538,152168,073FourthPrizes($1each)168,073TotalWinnings$27,918,56115263341393(c)2019ContemporaryEngineeringEconomicsExample14.1ImprovingtheOddNumberofPrizesPrizeCategoryWinningOdds1FirstPrize0.0000001416228SecondPrizes0.000032310,552ThirdPrizes0.00149168,073FourthPrizes0.023814(c)2019ContemporaryEngineeringEconomicsNumberofPrizesPrizeCategoryMethodsofDescribingProjectRiskSensitivityAnalysis:ameansofidentifyingtheprojectvariableswhich,whenvaried,havethegreatesteffectonprojectacceptability.Break-EvenAnalysis:ameansofidentifyingthevalueofaparticularprojectvariablethatcausestheprojecttoexactlybreakeven.ScenarioAnalysis:ameansofcomparinga“basecase”tooneormoreadditionalscenarios,suchasbestandworstcase,toidentifytheextremeandmostlikelyprojectoutcomes.5(c)2019ContemporaryEngineeringEconomicsMethodsofDescribingProjectExample14.2-After-taxCashFlowforBMC’sTransmission-HousingsProject–“BaseCase”012345Revenues:UnitPrice5050505050Demand(units)2,0002,0002,0002,0002,000Salesrevenue$100,000$100,000$100,000$100,000$100,000Expenses:Unitvariablecost$15$15$15$15$15Variablecost30,00030,00030,00030,00030,000Fixedcost10,00010,00010,00010,00010,000Depreciation17,86330,61321,86315,6135,575TaxableIncome$42,137$29,387$38,137$44,387$54,425Incometaxes(40%)16,85511,75515,25517,75521,770NetIncome$25,282$17,632$22,882$26,632$32,6556(c)2019ContemporaryEngineeringEconomicsExample14.2-After-taxCashCashFlowStatement012345OperatingactivitiesNetincome25,28217,63222,88226,63232,655Depreciation17,86330,61321,86315,6135,575InvestmentactivitiesInvestment125,000Salvage40,000Gainstax2,611Netcashflow$125,500$43,145$48,245$44,745$42,245$75,619(Example14.2,Continued)7(c)2019ContemporaryEngineeringEconomicsCashFlowStatement012345OperaExample14.2-SensitivityAnalysisforFiveKeyInputVariablesDeviation-20%-15%-10%-5%0%5%10%15%20%Unitprice$57$9,999$20,055$30,111$40,169$50,225$60,281$70,337$80,393Demand12,01019,04926,08833,13040,16947,20854,24761,28668,325Variablecost52,23649,21946,20243,18640,16937,15234,13531,11828,101Fixedcost44,19143,18542,17941,17540,16939,16338,15737,15136,145Salvagevalue37,78238,37838,97439,57340,16940,76541,36141,95742,553Base8(c)2019ContemporaryEngineeringEconomicsExample14.2-SensitivityAnaSensitivitygraph–BMC’stransmission-housingsproject(Example14.2)-20%-15%-10%-5%0%5%10%15%20%$100,00090,00080,00070,00060,00050,00040,00030,00020,00010,0000-10,000BaseUnitPriceDemandSalvagevalueFixedcostVariablecost9(c)2019ContemporaryEngineeringEconomicsSensitivitygraph–BMC’stranExample14.3-SensitivityAnalysisforMutuallyExclusiveAlternativesElectricalPowerLPGGasolineDieselFuelLifeexpectancy7year7years7years7yearsInitialcost$29,739$21,200$20,107$22,263Salvagevalue$3,000$2,000$2,000$2,200Maximumshiftsperyear260260260260Fuelconsumption/shift31.25kWh11gal11.1gal7.2galFuelcost/unit$0.05/kWh$1.02/gal$1.20/gal$1.13/galFuelcost/shift$1.56$11.22$13.32$8.14AnnualmaintenancecostFixedcost$500$1,000$1,000$1,000Variablecost/shift$4.5$7$7$710(c)2019ContemporaryEngineeringEconomicsExample14.3-SensitivityAnaOwnershipcost(capitalcost):

Electricalpower:CR(10%)=($29,739-$3,000)(A/P,10%,7)+(0.10)$3,000=$5,792

LPG: CR(10%)=($21,200-$2,000)(A/P,10%,7)+(0.10)$2,000=$4,144

Gasoline:CR(10%)=($20,107-$2,000)(A/P,10%,7)+(0.10)$2,000=$3,919

Dieselfuel:CR(10%)=($22,263-$2,200)(A/P,10%,7)+(0.1)$2,200=$4,34111(c)2019ContemporaryEngineeringEconomicsOwnershipcost(capitalcost):b)

Annualoperatingcostasafunctionofnumberofshiftsperyear(M):

Electric:$500+(1.56+4.5)M=$500+5.06M

LPG: $1,000+(11.22+7)M=$1,000+18.22M

Gasoline: $1,000+(13.32+7)M=$1,000+20.32M

Dieselfuel: $1,000+(8.14+7)M=$1,000+15.14M12(c)2019ContemporaryEngineeringEconomicsb)AnnualoperatingcostasaTotalequivalentannualcost:Electricalpower:

AE(10%)=6,292+5.06MLPG:

AE(10%)=5,144+18.22MGasoline:

AE(10%)=4,919+20.32MDieselfuel:

AE(10%)=5,341+15.14M13(c)2019ContemporaryEngineeringEconomicsTotalequivalentannualcost:SensitivityGraphforMultipleAlternatives020406080100120140160180200220240260ElectricforklifttruckispreferredGasolineforklifttruckispreferred12,00010,00080006000400020000GasolineLPGElectricPowerDieselAnnualEquivalentCost$Numberofshifts(M)14(c)2019ContemporaryEngineeringEconomicsSensitivityGraphforMultipleBreak-EvenAnalysis

withunknownSalesUnits(X)012345CashInflows:Netsalvage37,389X(1-0.4)($50)30X30X30X30X30X0.4(dep)7,14512,2458,7456,2452,230Cashoutflows:Investment-125,000-X(1-0.4)($15)-9X-9X-9X-9X-9X-(0.6)($10,000)-6,000-6,000-6,000-6,000-6,000NetCashFlow-125,00021X+1,14521X+6,24521X+2,74521X+24521X+33,61715(c)2019ContemporaryEngineeringEconomicsBreak-EvenAnalysis

withunknoBreak-EvenAnalysis

PWofcashinflows

PW(15%)Inflow=(PWofafter-taxnetrevenue) +(PWofnetsalvagevalue) +(PWoftaxsavingsfromdepreciation =30X(P/A,15%,5)+$37,389(P/F,15%,5) +$7,145(P/F,15%,1)+$12,245(P/F,15%,2) +$8,745(P/F,15%,3)+$6,245(P/F,15%,4) +$2,230(P/F,15%,5) =30X(P/A,15%,5)+$44,490 =100.5650X+$44,490 16(c)2019ContemporaryEngineeringEconomicsBreak-EvenAnalysisPWofcash2.PWofcashoutflows:

PW(15%)Outflow =(PWofcapitalexpenditure_ +(PW)ofafter-taxexpenses =$125,000+(9X+$6,000)(P/A,15%,5) =30.1694X+$145,1133.TheNPW: PW(15%) =100.5650X+$44,490 -(30.1694X+$145,113) =70.3956X-$100,623.4.Breakevenvolume: PW(15%) =70.3956X-$100,623=0

Xb =1,430units.17(c)2019ContemporaryEngineeringEconomics2.PWofcashoutflows:17(c)DemandPWofinflowPWofOutflowNPWX100.5650X-$44,49030.1694X+$145,11370.3956X-$100,6230$44,490$145,113100,62350094,773160,19865,4251000145,055175,28230,2271429188,197188,225281430188,298188,255431500195,338190,3674,9702000245,620205,45240,1682500295,903220,53775,36618(c)2019ContemporaryEngineeringEconomicsPWofPWofX100.5650X30.1694XOutflowBreak-EvenAnalysisChart030060090012001500180021002400$350,000300,000250,000200,000150,000100,00050,0000-50,000-100,000ProfitLossBreak-evenVolumeXb=1430AnnualSalesUnits(X)PW(15%)Inflow19(c)2019ContemporaryEngineeringEconomicsOutflowBreak-EvenAnalysisChaScenarioAnalysisVariableConsideredWorst-CaseScenarioMost-Likely-CaseScenarioBest-CaseScenarioUnitdemand1,6002,0002,400Unitprice($)485053Variablecost($)171512FixedCost($)11,00010,0008,000Salvagevalue($)30,00040,00050,000PW(15%)-$5,856$40,169$104,29520(c)2019ContemporaryEngineeringEconomicsScenarioAnalysisVariableWorsProbabilityConceptsforInvestmentDecisionsRandomvariable:variablethatcanhavemorethanonepossiblevalueDiscreterandomvariables:AnyrandomvariablesthattakeononlyisolatedvaluesContinuousrandomvariables:anyrandomvariablescanhaveanyvalueinacertainintervalProbabilitydistribution:theassessmentofprobabilityforeachrandomevent21(c)2019ContemporaryEngineeringEconomicsProbabilityConceptsforInvesTriangularProbabilityDistribution0L

Mo

H

x2(H-L)f(x)(a)ProbabilityfunctionMo-L(H-L)F(x)10L

Mo

H

x(b)Cumulativeprobabilitydistribution22(c)2019ContemporaryEngineeringEconomicsTriangularProbabilityDistribUniformProbabilityDistribution0L

H

x1(H-L)f(x)(a)ProbabilityfunctionF(x)10L

H

x(b)Cumulativeprobabilitydistribution23(c)2019ContemporaryEngineeringEconomicsUniformProbabilityDistributiProbabilityDistributionsforUnitDemand(X)andUnitPrice(Y)forBMC’sProjectProductDemand(X)UnitSalePrice(Y)Units(x)P(X=x)Unitprice(y)P(Y=y)1,6000.20$480.302,0000.60500.502,4000.20530.2024(c)2019ContemporaryEngineeringEconomicsProbabilityDistributionsforCumulativeDistribution(foradiscreterandomvariable)(foracontinuousrandomvariable)f(x)dx25(c)2019ContemporaryEngineeringEconomicsCumulativeDistribution(foraCumulativeProbabilityDistributionforXUnitDemand(X)ProbabilityP(X=x)1,6000.22,0000.62,4000.226(c)2019ContemporaryEngineeringEconomicsCumulativeProbabilityDistrib120014001600180020002200240026001.00.80.60.40.20AnnualSalesUnits(X)120014001600180020002200240026001.00.80.60.40.20AnnualSalesUnits(X)CumulativeProbabilityProbabilityProbabilitythatannualsalesunitswillbelessthanequaltox27(c)2019ContemporaryEngineeringEconomics12001400160018004042444648505254561.00.80.60.40.20UnitPrice($),Y1.00.80.60.40.20UnitPrice($),YCumulativeProbabilityProbabilityProbabilitythatannualsalesunitswillbelessthanequaltoy40424446485052545628(c)2019ContemporaryEngineeringEconomics40424446MeasureofExpectation(discretecase)(continuouscase)xf(x)dx29(c)2019ContemporaryEngineeringEconomicsMeasureofExpectation(discretMeasureofVariation30(c)2019ContemporaryEngineeringEconomicsMeasureofVariation30(c)2019XjPjXj(Pj)(Xj-E[X])(Xj-E[X])2(Pj)1,6000.20320(-400)232,0002,0000.601,200002,4000.20480(400)232,000E[X]=2,000Var[X]=64,000s=252,98YjPjYj(Pj)[Yj-E[Y]]2(Yj-E[Y])2(Pj)$480.30$14.40(-2)21.20500.5025.00(0)0530.2010.60(3)21.80E[Y]=50.00Var[Y]=3.00s=1.7331(c)2019ContemporaryEngineeringEconomicsXjPjXj(Pj)(Xj-E[X])(Xj-E[X])2JointandConditionalProbabilities32(c)2019ContemporaryEngineeringEconomicsJointandConditionalProbabilAssessmentsofConditionalandJointProbabilitiesUnitPriceYMarginalProbabilityConditionalUnitSalesXConditionalProbabilityJointProbability1,6000.100.03$480.302,0000.400.122,4000.500.151,6000.100.05500.502,0000.640.322,4000.260.131,6000.500.10530.202,0000.400.082,4000.100.0233(c)2019ContemporaryEngineeringEconomicsAssessmentsofConditionalandMarginalDistributionforXXj1,600P(1,600,$48)+P(1,600,$50)+P(1,600,$53)=0.182,000P(2,000,$48)+P(2,000,$50)+P(2,000,$53)=0.522,400P(2,400,$48)+P(2,400,$50)+P(2,400,$53)=0.3034(c)2019ContemporaryEngineeringEconomicsMarginalDistributionforX1,6After-TaxCashFlowasaFunctionofUnknownUnitDemand(X)andUnitPrice(Y)

Item012345Cashinflow:Netsalvage

X(1-0.4)Y0.6XY0.6XY0.6XY0.6XY0.6XY0.4(dep)7,14512,2458,7456,2452,230Cashoutflow:Investment-125,000-X(1-0.4)($15)-9X-9X-9X-9X-9X-(1-0.4)($10,000)-6,000-6,000-6,000-6,000-6,000NetCashFlow-125,0000.6X(Y-15)+1,1450.6X(Y-15)+6,2450.6X(Y-15)+2,7450.6X(Y-15)+2450.6X(Y-15)+33,61735(c)2019ContemporaryEngineeringEconomicsAfter-TaxCashFlowasaFunctNPWFunction1.CashInflow: PW(15%)=0.6XY(P/A,15%,5)+$44,490 =2.0113XY+$44,4902.CashOutflow: PW(15%)=$125,000+(9X+$6,000)(P/A,15%,5) =30.1694X+$145,113.3.

NetCashFlow: PW(15%)=2.0113X(Y-$15)-$100,62336(c)2019ContemporaryEngineeringEconomicsNPWFunction1.CashInflow:36(TheNPWProbabilityDistributionwithIndependentRandomVariablesEventNo.xyP[x,y]CumulativeJointProbabilityNPW11,600$48.000.060.06$5,57421,60050.000.100.1612,01031,60053.000.040.2021,66442,00048.000.180.3832,12352,00050.000.300.6840,16862,00053.000.120.8052,23672,40048.000.060.8658,67282,40050.000.100.9668,32692,40053.000.041.0082,80837(c)2019ContemporaryEngineeringEconomicsTheNPWProbabilityDistributiNPWprobabilitydistributions:WhenXandYareindependent0204060801000.50.40.30.20.10NPW(ThousandDollarsProbability38(c)2019ContemporaryEngineeringEconomicsNPWprobabilitydistributions:CalculationoftheMeanofNPWDistributionEventNo.xyP[x,y]CumulativeJointProbabilityNPWWeightedNPW11,600$48.000.060.06$5,574$33421,60050.000.100.1612,0101,20131,60053.000.040.2021,66486742,00048.000.180.3832,1235,78252,00050.000.300.6840,16812,05062,00053.000.120.8052,2366,26872,40048.000.060.8658,6723,52082,40050.000.100.9668,3266,83392,40053.000.041.0082,8083,312E[NPW]=$40,16839(c)2019ContemporaryEngineeringEconomicsCalculationoftheMeanofNPWCalculationoftheVarianceofNPWDistributionEventNo.xyP[x,y]NPW(NPW-E[NPW])2Weighted(NPW-E[NPW])11,600$48.000.06$5,5741,196,769,744$71,806,18521,60050.000.1012,010792,884,22779,228,42331,60053.000.0421,664342,396,53613,695,86142,00048.000.1832,12364,725,24311,650,54452,00050.000.3040,1680062,00053.000.1252,236145,631,79717,475,81672,40048.000.0658,672342,396,53620,543,79282,40050.000.1068,326792,884,22779,288,42392,40053.000.0482,8081,818,132,07772,725,283Var[NPW]=366,474,326s=$19,14440(c)2019ContemporaryEngineeringEconomicsCalculationoftheVarianceofComparingRiskyMutuallyExclusiveProjectsEvent(NPW)(unit:thousands)ProbabilitiesModel1Model2Model3Model41,0000.350.100.400.201,50000.4500.402,0000.4000.2502,50000.3500.303,0000.2000.2003,50000004,0000.0500.1504,50000.1000.1041(c)2019ContemporaryEngineeringEconomicsComparingRiskyMutuallyExcluComparisonRuleIfEA>EBandVA

VB,selectA.IfEA=EBandVA

VB,selectA.IfEA<EBandVA

VB,selectB.IfEA>EBandVA

>VB,Notclear.ModelTypeE(NPW)Var(NPW)Model1$1,950747,500Model22,100915,000Model32,1001,190,000Model42,0001,000,000Model2vs.Model3Model2>>>Model3Model2vs.Model4Model2>>>Model4Model2vs.Model1Can’tdecide42(c)2019ContemporaryEngineeringEconomicsComparisonRuleIfEA>EBandP{NPW1>NPW2}Model1Model2JointProbability$1,000Noevent(0.35)(0.00)=0.0002,0001,0001,500(0.40)(0.10)=0.040(0.40)(0.45)=0.1803,0001,0001,5002,500(0.20)(0.10)=0.020(0.20)(0.45)=0.190(0.20)(0.35)=0.0704,0001,0001,5002,500(0.05)(0.10)=0.005(0.05)(0.45)=0.230(0.05)(0.35)=0.0180.44543(c)2019ContemporaryEngineeringEconomicsP{NPW1>NPW2}Model1Model2JoDecisionTreeAnalysisAgraphicaltoolfordescribing(1)theactionsavailabletothedecision-maker,(2)theeventsthatcanoccur,and(3)therelationshipbetweentheactionsandevents.44(c)2019ContemporaryEngineeringEconomicsDecisionTreeAnalysisAgraphiStructureofaTypicalDecisionTreeSeekadvice(e)Donotseekadvice(e)NoprofessionalhelpStock(d)Band(d)(7.5%return)C,Low(-30%)B,Medium(9%)A,High(50%)o1U(e,d,A)12o1DecisionNodeChanceNodeConditionalProfitAddingcomplexitytothedecisionproblembyseekingadditionalinformation45(c)2019ContemporaryEngineeringEconomicsStructureofaTypicalDecisioBill’sInvestmentDecisionProblemOption1: 1) Period0:(-$50,000-$100)=-$50,100 Period1:(+$75,000-$100)=0.20($24,800)=$69,940 PW(5%)=-$50,100+$69,940(P/F,5%,1)=$16,510 2) Period0:(-$50,000-$100)=-$50,100 Period1:(+$54,000-$100)-(0.20)($4,200)=$16,510 PW(5%)=-$50,100+$53,540(P/F,5%,1)=$890 3) Period0:(-$50,000-$100)=-$50,100 Period1:(+$35,000-$100)–(0.20)(-$14,800)=$37,940 PW(5%)=-$50,100+$37,940(P/F,5%,1)=-$13,967Option2: Period0:(-$50,000-$150)=-$50,150 Period1:(+$53,750-$150)=$53,600 PW(5%)=-$50,150+$53,600(P/F,5%,1)=$898

EMV=$898Or,prioroptimaldecisionisOption246(c)2019ContemporaryEngineeringEconomicsBill’sInvestmentDecisionProDecisiontreeforBill’sinvestmentproblem$75,000-$100$54,500-$100$45,000-$100$53,750-$150$16,510$890$-13,967$898A,High(50%)B,Medium(9%)C,Low(-30%)(7.5%return)Stock(d)1Bond(d)2-$50,000-$100-$150(a)0.250.400.35ABCd1d2$898$898(b)Relevantcashflowsbeforetaxes(b)NetpresentworthforEachdecisionpath47(c)2019ContemporaryEngineeringEconomicsDecisiontreeforBill’sinvesExpectedValueofPerfectInformation(EVPI)WhatisEVPI?Thisisequivalenttoaskingyourselfhowmuchyoucanimproveyourdecisionifyouhadperfectinformation.MathematicalRelationship:EVPI=EPPI–EMV=EOL whereEPPI(Expectedprofitwithperfectinformation)istheexpectedprofityoucouldobtainifyouhadperfectinformation,andEMV(Expectedmonetaryvalue)istheexpectedprofityoucouldobtainbasedonyourownjudgment.Thisisequivalenttoexpectedopportunityloss(EOL).48(c)2019ContemporaryEngineeringEconomicsExpectedValueofPerfectInfoEOL=(0.25)($15,612)+(0.40)(0)+(0.35)(0)=$3,903ExpectedValueofPerfectInformationPotentialReturnLevelProbabilityDecisionOptionOptimalChoicewithPerfectInformationOpportunityLossAssociatedwithInvestinginBondsOption1:InvestinStockOption2:InvestinBondsHigh(A)0.25$16,510$898Stock$15,612Medium(B)0.40890898Bond0Low(C)0.35-13,967898Bond0EMV-$405$898

$3,903Prioroptimaldecision,Option2EPPI=(0.25)($16,510)+(0.40)($898)+(0.35)($898)=$4,801EVPI=EPPI–EV=$4,801-$898=$3,90349(c)2019ContemporaryEngineeringEconomicsEOL=(0.25)($15,612)ExpectedRevisedCashflowsafterPayingFeetoReceiveAdvice(Fee=$200)1.StockInvestmentOption

Period0:(-$50,000-$100-$200)=-$50,300 Period1:(+$75,000-$100)–(0.20)($25,000-$400)=$69,980 PW(5%)==$50,300+$69,980(P/F,5%,1)=$16,3482.BondInvestmentOption

Period0(-$50,000-$150-$200)=-$50,350 Period1:(+$53,750-$150)=$53,600 PW(5%)=-$50,350+$53,600(P/F,5%,1)=$69850(c)2019ContemporaryEngineeringEconomicsRevisedCashflowsafterPayinDecisiontreeforBill’sinvestmentproblemwithanoptionhavingaprofessionaladvice$16,510$890-$13,967$898ABC(d)1(d)2$16,348$728-$14,129$698ABC(d)1(d)2-$405-$898$16,348$728-$14,129$698ABC(d)1(d)2NoprofessionalhelpFavorable(F)Unfavorable(UF)DonotseekAdvice(e)SeekAdvice(e)1o-$89851(c)2019ContemporaryEngineeringEconomicsDecisiontreeforBill’sinvesWithImperfectSampleInformationGivenLevelofStockPerformanceWhattheReportWillSayHigh(A)Medium(B)Low(C)Favorable(F)0.800.650.20Unfavorable(UF)0.200.350.8052(c)2019ContemporaryEngineeringEconomicsWithImperfectSampleInformatNature’sTreeP(A)P(F/A)=0.20P(B)P(UF/B)=0.26FUFP(F/A)=0.80P(UF/A)=0.20P(A)P(F/A)=0.20P(B)P(UF/B)=0.14FUFP(F/B)=0.65P(UF/B)=0.35P(C)P(F/C)=0.07P(C)P(UF/C)=0.28FUFP(F/C)=0.20P(UF/C)=0.80ABCP(A)=0.25P(B)=0.40P(C)=0.35F=FavorableUF=UnfavorableP(F)=0.20+0.26+0.07=0.53P(UF)=1-P(F)=1-0.53=0.4753(c)2019ContemporaryEngineeringEconomicsNature’sTreeP(A)P(F/A)=0.20JointandMarginalProbabilitiesP(A,F)=P(F/A)P(A)=(0.80)(0.25)=0.20P(A,UF/A)P(A)=(0.20)(0.25)=0.05P(B,F)=P(F/B)P(B)=(0.65)(0.40)=0.26P(B,UF)=P(UF/B)P(B)=(0.35)(0.40)=0.1454(c)2019ContemporaryEngineeringEconomicsJointandMarginalProbabilitiJointasWellasMarginalProbabilitiesWhatReportWillSayJointProbabilitiesWhenPotentialLevelofReturnisGivenFavorable(F)Unfavorable(UF)MarginalProbabilitiesofReturnLevelHigh(A)0.200.050.25Medium(B)0.260.140.40Low(C)0.070.280.35MarginalProbabilities0.530.471.0055(c)2019ContemporaryEngineeringEconomicsJointasWellasMarginalProbDeterminingRevisedProbabilitiesP(A/F)=P(A,F)/P(F)=0.20/0.53=0.38P(B/F)=P(B,F)/P(F)=0.26/0.53=0.49P(C/F)=P(C,F)/P(F)=0.07/0.53=0.13P(A/UF)=P(A,UF)/P(UF)=0.05/0.47=0.30P(B/UF)+P(B,UF)/P(UF)=0.14/0.47=0.30P(C/UF)=P(C,UF)/P(UF)=0.28/0.47=0.5956(c)2019ContemporaryEngineeringEconomicsDeterminingRevisedProbabilitBill’sinvestmentproblemafterhavingprofessionaladvice$16,510$890-$1