ON THE WORTH OF GROUNDWATER CONTAMINANT SAMPLING

Transcription

1 ON THE WORTH OF GROUNDWATER CONTAMINANT SAMPLING 1. No-spatial case Lars Rosé Departmet of Geology Chalmers Uiversity of Techology SE Gotheburg, SWEDEN Tommy Norberg Departmet of Mathematical Statistics Chalmers Uiversity of Techology SE Gotheburg, SWEDEN

2 2 Abstract This is the first i a series of three papers studyig the ecoomical worth of groudwater cotamiat samplig i applicatios of icreased complexity. This paper presets the theoretical basis for Bayesia data-worth aalysis ad discusses the o-spatial case with oe hydraulically isolated cell of uiform ad isotropic coditios. The two subsequet papers will discuss data worth i two oe-dimesioal models ad i a two-dimesioal spatial settig. The mai cotributio of this paper is that it provides a ew theory for estimatig the worth of simultaeous samplig i more tha oe locatio, which is a especially importat aspect i cases of large samplig errors. Data worth depeds directly o the prior probability of cotamiatio ad we compare a case i which this probability is completely kow with a case i which it is kow with give ucertaity. I the latter case, we show that the data worth is sesitive with respect to the selected desig criteria. Itroductio Cotamiated soil ad groudwater is a problem of growig cocer i our society ad is today a major issue i lad use plaig ad maagemet, real estate assessmet, ad property sellig. Ivestigatio ad remediatio of cotamiated areas are ofte associated with high costs. As a example, the Swedish Evirometal Protectio Agecy (EPA) provides more tha the equivalet of $50 millio aually of govermetal resources to the Swedish couty authorities for ivestigatio ad remediatio of cotamiated areas where o resposible part ca be foud. Still, the bulk of ecessary ivestmets for ivestigatio ad remediatio come from resposible ladowers ad operators. The Swedish EPA (1999)

3 3 estimates that there are cotamiated sites i Swede, of which approximately 4000 are i eed of remediatio. Similar, or eve more severe, situatios are preset throughout Europe ad North America. The Polluter Pays Priciple, which is the regulatory philosophy i the Europea Uio, puts large fiacial pressure o resposible compaies ad authorities. The regulatory framework is therefore a strog icetive for both the public ad private compaies for applyig costefficiet ivestigatio ad remediatio strategies. I additio, the evirometal legislatio i may coutries, e.g. i Swede, state that the evirometal value of remediatio must be i proportio to the remediatio costs i order to provide proper prioritizatio of resources ad to achieve sustaiable use of lad ad water. Due to a combiatio of complex geological, hydrogeological, ad geochemical coditios at cotamiated sites ad high ivestigatio costs, it is usually ot possible to obtai complete iformatio or characterize a site with a high degree of certaity. Ivestigatios of cotamiated areas are therefore typically associated with large ucertaities regardig e.g. type ad extet of cotamiatio ad possible future cotamiat spreadig. These ucertaities trasform ito substatial ecoomical risks i the remediatio process. To hadle these risks properly ad to achieve a cost-efficiet maagemet of cotamiated sites, it is therefore of primary importace to desig ivestigatio strategies cosiderig both the samplig costs ad the ecoomical risks associated with makig icorrect decisios regardig remediatio.

4 4 The umber ad type of ew data i cotamiatio studies are ofte estimated with respect to the potetial for reductio of ucertaity. However, i order to be cost-effective the worth of ew data should be evaluated with respect to the reductio of the risk-cost of makig icorrect decisios, rather tha ucertaity. I a cost-beefit perspective, the worth of samplig depeds o the samplig cost ad the risk-cost reductio it provides with respect to failure of successfully solvig the specific problem at had. The risk-cost must the icorporate all expected costs for the decisio-maker related to improper evaluatio of the situatio, e.g. chage of remediatio strategy, prologed clea-up times, restrictios o lad-use, ad loss of evirometal values. A tool for achievig cost-effective ivestigatio is Bayesia data-worth aalysis, evolvig from a ecoomical decisio aalysis framework. Bayesia data-worth aalysis i hydrogeology has bee described by e.g. Freeze et al (1990; 1992), James & Freeze (1993) ad James & Gorelick (1994). The basic philosophy behid this approach i remediatio projects is to itegrate the cost of samplig, the reductio of the risk-cost of failure, ad the cost ad efficiecy of the remedial desig i order to fid the best samplig strategy amog a give set of alterative strategies. The data-worth aalysis is made iteratively accordig to Figure 1 util o further samplig is cosidered justifiable. Our work will be preseted i a series of three papers of which this is the first. The mai purpose is two-fold: (1) to aalyze ad describe how ecoomical factors, samplig ucertaity, ad prior assumptios regardig cotamiatio affect cost-efficiecy of the samplig strategy ad (2) to preset a spatial statistical model for Bayesia data-worth aalysis. We focus o the o-spatial aspects of data-worth i the first paper, 1-dimesioal aspects i the secod paper, ad 2-dimesioal aspects i the third paper. The fist two papers i this series are directed at aalyzig ecoomical ad statistical implicatios of data-worth

5 5 aalysis i a sigle cell 1 ad alog a lie of several cells. We preset coclusios ad give recommedatios regardig a umber of strategic decisios to be take whe applyig Bayesia data-worth aalysis. I the third paper we preset a 2D spatial statistical data-worth model, based o Mote Carlo Markov Chai (MCMC) aalysis. To fully uderstad ad appreciate the capabilities ad limitatios of Bayesia data-worth aalysis, implicatios o lower dimesioalities eed to be studied ad evaluated before full 2D ad/or 3D applicatios ca be made correctly. We believe that the work preseted i these three papers provide importat kowledge for a icreased cost-efficiecy i projects o ivestigatio ad remediatio of cotamiated soil ad groudwater. Geeral descriptio of Bayesia data-worth aalysis We state that the worth of samplig data ca be best evaluated by the use of Bayesia decisio aalysis, which has bee applied for cotamiated soil ad groudwater by several authors, e.g. Massma & Freeze (1987), Mari et al (1989), Freeze et al (1990, 1992), Massma et al (1991) James & Freeze (1993), James & Gorelick (1994), James et al (1996), Rosé & Wladis (1998), Wladis et al (1999), Norrma (2001), Back (2001) ad Rosé (2001). A special applicatio of Bayesia decisio aalysis is referred to as Bayesia dataworth aalysis, which is described below, primarily based o the works by Freeze et al (1992) ad James & Freeze (1993). We describe the methodology, assumig a simple case ivolvig a decisio of whether to remediate or ot, that this decisio depeds o the cotamiat levels with respect to a defied actio level, ad that failure occurs if o actio is take whe cotamiatio occurs. 1 A cell is a homogeeous part of the cotamiated site

6 6 Let C deote the evet of cotamiatio above a specific actio level i a hypothetical cell, which is hydraulically isolated, ad let C c deote the complemetary evet that cotamiatio is ot preset. The probability Pr(C) is the the prior probability of cotamiatio for this cell. Typically, Pr(C) is ot kow with certaity. I early stages, a estimate may therefore be based o professioal judgmet usig existig site iformatio ad experieces from other sites. Give that remediatio has to be made if cotamiatio occurs, the prior risk-cost is defied as: R = k Pr( ) (1) 0 F C where k F deotes the moetary cosequeces of failure. Thus, R 0 is the expected cost of o actio. Let k R deote the remediatio cost. Aother alterative is to take a sample to see if ay cotamiat is preset ad to act accordigly. Let D ad D c deote the evets that the cotamiat is detected ad ot detected, respectively, give that the cell is sampled. Bayes theorem gives us the pre-posterior probability of failure to remediate, give that o cotamiat is detected: Pr ( C c D Pr ( C) Pr ( D c C) ) = (2) c c c c Pr ( C) Pr ( D C) + Pr ( C ) Pr ( D C ) ad the pre-posterior risk-cost is: c R = k Pr( C D ) (3) 1 F

7 7 Let k M deote the samplig cost. The expected risk-cost of takig a sample is: c k R Pr( D) + mi[ k R, R1 ]Pr( D ) (4) sice the cell is remediated if cotamiatio is detected, ad if cotamiatio is ot detected the cell is remediated oly if the remediatio cost is less tha the pre-posterior risk-cost. Followig Freeze et al (1992) we defie the data worth as: c W = mi[ k R, R0 ] { k R Pr( D) + mi[ k R, R1 ]Pr( D )} (5) Sice the quatities ivolved i the calculatio of the worth W typically are kow oly approximately, oe may require that W be sigificatly larger tha the samplig cost i order to justify samplig. This aalysis applies to all cells of the regio of iterest ad the most elemetary ad aive way to proceed would be to say that the cells that should be aalyzed further are those with W sufficietly larger tha the samplig cost. However, such a aalysis does ot take ito accout the obvious spatial depedece that exists i ay remediatio site. The effects of spatial depedece are further described i the secod ad third papers i this series. Below, we preset a aalysis of ecoomical ad statistical implicatios o the dataworth without spatial cosideratios.

8 8 The worth of takig additioal samples Now cosider a cell to be ivestigated with respect to groudwater cotamiatio. Prior to aalyzig the value of additioal samples, we have two alterative optios: (1) to remediate the cell to the cost of k R ; (2) to do othig with a remaiig risk-cost of R 0 =k F Pr(C). The decisio rule is to do othig if Pr(C) k R /k F ad remediate if Pr(C) > k R /k F. A third alterative is to take ew samples ad act accordig to the outcome of these (see also Figure 1). A objective fuctio, Φ(), is defied for each samplig alterative i which the umber of ew samples is =1,2. The objective fuctio of the optimal actio is a priori, before cosiderig ay ew samples: Φ ( 0) = mi( k, k Pr( C)) (6) R F We will ow cosider the possibilities of takig oe or several imperfect samples with the coditioal detectio probabilities Pr(D C) (which may be much less tha 1) ad Pr(D C c ) (which may be cosiderably larger tha 0). Let k M deote the cost of oe ew sample ad k M () deote the cost of simultaeously takig ew samples. A sample cost fuctio that we have used i our work is: 1 k M ( ) = k M r, = 1, 2,... (7) where r 1 is a discout factor. The objective fuctio of takig ew samples is:

9 9 Φ( ) = k = k M M ( ) + k = 0 mi( k ( ) + E[mi( k R R, k, k F F Pr( C D Pr( C D ))] = k)) Pr( D = k) (8) where D (=1, 2, ) is the total umber of cotamiat detectios i idepedet samples. The coditioal probabilities Pr(C D = k) ca be show, by meas of Bayes rule, to be: Pr( C D Pr( C) Pr( D = k C) = k) = (9a) Pr( D = k) where c c Pr( D = k) = Pr( C) Pr( D = k C) + Pr( C ) Pr( D = k C ) (9b) Pr(D = k C) is the biomial probability of gettig k successes i idepedet trials with success probability Pr(D C) (a aalogous statemet holds for Pr(D = k C c )). The optimal umber of samples is: opt = arg mi Φ( ) 0 (10) Note that Φ = Φ ) is the expected cost of the optimal actio, which is do othig if opt ( opt opt = 0 ad Pr(C) < k R /k F, take opt samples ad act accordigly if opt 1, ad remediate if opt = 0 ad Pr(C) k R /k F.

10 10 The Expected Net Value (ENV) of takig samples is: ENV ( ) = Φ(0) Φ( ), = 1, 2,... (11) This is related to the Expected Value of Sample Iformatio (EVSI) discussed i Bruce & Freeze (1993) ad Bruce & Gorelick (1994). It is ecessary to iclude the sample cost i ENV, sice it depeds o the umber of samples, while it is ot icluded i the EVSI, which therefore is positive for all values of Pr(C). Aother differece of our approach to evaluatig the worth of additioal samples is that Bruce & Freeze (1993) ad Bruce & Gorelick (1994) discuss the value of takig oe additioal sample, whereas we are ope to the possibility that it may be cost-effective to simultaeously take several additioal samples. Obviously, it is cost-effective to take samples oly if: ENV = max ENV( ) > 0 1 Examples igorig ucertaity i Pr(C) estimates Havig described the geeral theory of Bayesia data-worth aalysis, we ow preset two hypothetical cases o usig the described methodology for evaluatio of the data-worth of additioal samples. I this first presetatio of cases, the purpose is to illustrate the applicatio of the data-worth assessmet approach described i this paper. I the subsequet

11 11 sectio, we itroduce ucertaity assessmet of iput variables ad ivestigate the impact of this o decisios regardig the desig of the samplig program. Case 1 Cosider a hydraulically isolated cell of uiform ad isotropic coditios. The cost fuctios ad the coditioal detectio probabilities, which reflect the errors of the chose samplig techique, are show i Table 1. As see from the table, we assume relatively large samplig errors. We further assume that the samplig cost is small compared to the remediatio cost, ad that the failure cost is sigificatly larger tha the remediatio cost. Note that we used o discout factor i the sample cost fuctio, k M, i this example. Usig the procedure for calculatig the optimal umber of samples, opt, described above, the optimum umber of samples for differet Pr(C) was calculated ad is show i Figure 2. Figure 3 shows the prior expected cost (Φ(0)) ad the optimal cost (Φ opt ) for differet Pr(C). Figure 4 shows the ENV for differet values of Pr(C). Cosiderig the results show i these figures, we take Pr(C) = k R /k F = 0.4 prior to ay samples, sice this represets the maximum decisio ucertaity. As ca be see from Figure 2, the optimal umber of samples for Pr(C) = 0.4 is 3. The four possible outcomes of takig three samples, D 3, are show i Table 2. Give the iformatio about the cell ad the data-worth aalysis, we thus coclude that three additioal samples represet the most cost-effective sample alterative. The example also illustrates that eve if cotamiatio was to be detected i oe sample (D 3 = 1), the decisio would still be to do othig, due to the large samplig error associated with the sample techique that we have chose. I real world applicatios the samplig error is ofte igored.

12 12 Case 2 The coditios of the secod hypothetical case, also assumig a cell of uiform ad isotropic coditios, are show i Table 3. Also here, we let Pr(C) = k R /k F = 0.4 prior to ay samples. I this case we estimate the samplig error to be smaller tha i the first example ad we use a discout factor for the k M cost fuctio. Figure 5 shows the prior expected cost (Φ(0)) ad the optimal cost (Φ opt ) for differet values of Pr(C). Figure 6 shows the ENV for differet Pr(C). Calculatig the optimum umber of additioal samples yields opt = 1. The two possible outcomes of takig oe sample, D 1, are show i Table 4. If the sigle ew sample is made the decisio would be do othig if cotamiatio was ot detected ad remediate if cotamiatio was detected. This example illustrates the situatio of a expesive, but also reasoably accurate samplig techique, which makes remediatio cost-effective whe cotamiatio is detected i a sigle sample. Examples icludig ucertaity i P(C) estimates I the two hypothetical cases just studied, Pr(C) is take to equal k R /k F sice for this value, the expected et value (ENV) is maximal. However, it is typical that oe has some idea about the true Pr(C), but that the estimate is associated with some ucertaity. A evaluator may therefore wat to express the estimatio as Pr(C) 0.65 ad 0.5 Pr(C) 0.8. This iformatio may be tured ito a prior distributio of Pr(C). The most obvious prior is the triagular distributio with mode 0.65 ad support (0.5, 0.8). This prior distributio gives mass 1 to the iterval (0.5, 0.8). Thus, it does ot accout for the fact that the expert specificatio 0.5 Pr(C) 0.8 might be wrog.

13 13 A more realistic prior is therefore the beta distributio, havig desity fuctio: Γ( α + β ) Γ( α) Γ( β ) α 1 β 1 f ( θ ) = θ (1 θ ) (12) for 0 < θ < 1 (i order to avoid a cumbersome otatio, we ow write θ istead of Pr(C)). The parameters α ad β cocretize the expert opiio ad Γ deotes the Euler gamma fuctio. A beta prior allows for wrog expert estimates. For example, ote that Pr{θ < 0.5} = Pr{θ > 0.8} = 0.05 if α = 16.4, β =8.6. This beta distributio is plotted i Figure 7. This is oe strategy to cocretize ad quatify the reliability of the experts. Aother strategy would be to specify the ucertaity usig a estimate of the most likely value, θ, ad a umber of equivalet samples represetig the reliability of this estimate. The appropriate beta distributio the is the oe with α = θ + 1, β = (1 θ ) + 1. This follows from the wellkow fact that the beta distributio is cojugate to the biomial (Bickel & Doksum, 1977). Providig θ (=Pr(C)) with a beta prior forces us to redefie opt. Firstly, we specify a desig criterio such as the expected value or a suitable percetile i the distributio of Φ() for various umber of samples. Secodly, if the expected value is take, the opt = arg mi E[ Φ( )] 0 Similarly, if the 100p th percetile is chose, the

14 14 opt = arg mi G 0 1 ( p) where G 1 ( p) deotes the percetile fuctio of the distributio fuctio G ( x) = Pr{ Φ( ) x}. We ow retur to the two hypothetical cases. Case 1 The coditios for this case are idetical to the first case i the previous sectio (see Table 1), except that we ow itroduce ucertaity i the θ (=Pr(C)) estimate. Let us assume that the experts agree o the followig: θ 0.45 ad the ucertaity (or reliability) of this statemet is equivalet to the oe achieved after takig 8 idepedet samples. A appropriate parameterizatio of the prior beta distributio is the α = 4.6 ad β = 5.4. This beta distributio is plotted i Figure 8. From Figure 2 it is see that the optimal umber of samples is opt = 3 if θ = However, if the ucertaity is take ito accout, the optimal umber of samples is 2 if the desig criterio is the expected value, i.e. = arg mi E[ Φ( )]. I Figure 9 the desity of Φ(2) is plotted. Notice that it has a discrete compoet: Pr{Φ(2) < 11} = ad Pr{Φ(2) = 11} = Note also that if the ucertaity i θ istead correspods to a larger umber of idepedet samples, e.g. 50, the opt =3 if the desig criterio is the expected value. opt 0 Case 2 I the secod hypothetical case, opt = 1 for Pr(C) 0.627, 2 for Pr(C) ad 0 otherwise. This meas that if it is estimated that θ 0.4, the opt = 1 also if ay

15 15 reasoable ucertaity is take ito accout. The situatio chages, however, ad becomes similar to what we observed i Case 1, if θ Assume that the expert opiio is 0.50 θ Assume also that the experts are regarded as highly qualified ad reliable by the resposible decisio-maker. A quatificatio of the reliability of the expert opiios may therefore be expressed as: Pr{θ < 0.5} = Pr{θ > 0.75} = This holds for α = ad β = I this case mi E[Φ()] is obtaied for both = 1 ad = 2 if the desig criterio is expectatio. This holds true also if the media is chose. However, if the desig criterio is the 75 th percetile, opt = 2. The outcome would differ if the decisio-maker would be less willig to deped o the expert opiio. If the reliability of the expert opiio was istead expressed as: Pr{θ < 0.5} = Pr{θ > 0.75} = 0.3, this would correspod to α = 2.70 ad β = I this case, the optimal umber of samples would be 1 if the desig criterio is expectatio, 2 if the media is the desig criterio ad 1 if the criterio is the 75 th percetile. Coclusios The followig coclusios were made with respect to the o-spatial case of Bayesia dataworth aalysis: 1. Give the expected costs for samplig, remediatio ad failure, the probability of cotamiatio, ad the sample error, the optimal umber of samples to be take simultaeously i the ext ivestigatio phase ca be calculated.

16 16 2. Bayesia data-worth aalysis is directed at miimizig the sum of ivestmet ad risk cost. The samplig cost is compared to the reductio of the ecoomical risk of failig to meet set up maagemet goals of the site, implyig that the risk reductio should exceed the samplig cost i order to make samplig cost-effective. This is i cotrast to more commoly applied variace reductio approaches, which are directed at reachig a level of acceptable ucertaity, rather tha lookig at the cost-efficiecy of the samplig program. 3. Bayesia data-worth aalysis provides a possibility for formal icorporatio of the samplig error i the valuatio of the cost-efficiecy of the samplig program. As show i the 1 st hypothetical case, that eve if cotamiatio is detected remediatio may ot be ecoomically motivated if the samplig error is large. 4. Differet results ca be obtaied, depedig o whether ucertaity i the prior estimate of the probability of cotamiatio is icluded or ot. 5. Differet results ca be obtaied, depedig o the strategy for hadlig the ucertaity of the prior estimatio of the probability of cotamiatio. If the ucertaity of the prior estimate is take ito accout, the decisio is sesitive to the selected desig criteria, e.g. expected value, media or aother percetile. We preset a approach to model this type of ucertaity usig a beta-distributio. Also remediatio costs ad failure costs may be associated with substatial ucertaities, which may have large impact o the fial decisio. We do ot preset specific approaches for hadlig these ucertaities, but recommed that they are icluded i the same maer as the cotamiatio ucertaity. Fially, the Bayesia data-worth aalysis approach preseted i this paper icorporates the possibility of takig several ew samples simultaeously, ad therefore differs to some extet

17 17 to the previous works by e.g. James & Freeze (1993) ad James & Gorelick (1994). We also illustrate the importace for hadlig ucertaities i the estimates of the key variables of the data-worth aalysis. We do this without the complexities of spatial statistical models. This will be further explored i the 2 d ad 3 rd papers i this series. Ackowledgmets Refereces Back, P-E, Samplig strategies ad data-worth aalysis for cotamiated lad a literature review. Departmet of Geology, Chalmers Uiversity of Techology. Publ B 486 ad SGI Varia 500. Bickel, P.J. & K.A. Doksum, Mathematical Statistics. Basic Ideas ad Selected Topics. Holde-Day, Ic. Freeze, R. A., Massma, J., Smith, J. L., Sperlig, T. & B. R. James, Hydrogeological decisio aalysis: 1. A Framework. Groud Water. Vol. 28, o. 5, pp Freeze, R. A., Massma, J., Smith, J. L., Sperlig, T. & B. R. James, Hydrogeological decisio aalysis: 4. The cocept of data worth ad its use i the developmet of site ivestigatio strategies. Groud Water. Vol. 28, o. 5, pp James, B. R. & S. M. Gorelick, Whe eough is eough: The worth of moitorig data i aquifer remediatio desig. Water Resources Research. Vol. 30, o. 12, pp

18 18 James, B. R., Huff, D. D., Trabalka, J. R., Ketelle, R. H. & C. T. Rightmire, Allocatio of evirometal fuds usig ecoomic risk-cost-beefit aalysis: A case study. Groud Water Remedaitio ad Moitorig. Fall 1996, pp James, B.R. & Freeze, R.A., 1993: The worth of data i predictig aquitard cotiuity i hydrogeological desig. Water Resources Research, Vol 29, No. 7, pp Mari, C. M., Media, M. A. & J. B. Butcher, Mote Carlo aalysis ad Bayesia decisio theory for assessig the effects of waste sites o groudwater, I: Theory. Joural of Cotamiat Hydrology. No. 5, pp Massma, J. & R. A. Freeze, Groudwater cotamiatio from waste sites: The iteractio betwee risk-based egieerig desig ad regulatory policy, 1: Methodology. Water Resources Research. Vol. 23, o. 2, pp Massma, J., Freeze, R. A., Smith, L., Sperlig, T. & B. James, Hydrogeological Decisio Aalysis: 2.Applicatios to Groud-Water Cotamiatio. Groud Water. Vol. 29, o. 4, pp Norrma, J, Decisio Aalysis uder Risk ad Ucertaity at Cotamiated Sites A literature review. Departmet of Geology, Chalmers Uiversity of Techology. Publ B 486 ad SGI Varia 500. Rosé, L. ad Wladis, D., 1998: Hydrogeological Decisio Aalysis for Moetary Evaluatio of Nitrate Reductio Measures i Europe. Proceedigs of the Iteratioal Associatio of Hydrogeologists XXVII Cogress. Wladis, D., Rosé, L. ad Kros, J., 1999: Risk-Based Decisio Aalysis of Emissio Alteratives to Reduce Groudwater Acidificatio o the Europea Scale. Groud Water, Vol. 37, No. 6, pp

19 19 Prior aalysis Use kow iformatio to choose best alterative Data-worth aalysis Is ew iformatio likely to be cost-effective? Yes Take sample No Stop Posterior aalysis Use best ew iformatio to choose best alterative Figure 1. Schematic descriptio of Bayesia data-worth aalysis (after James et al, 1996).

20 20 opt P C Figure 2. Plot of opt vs Pr(C) for the 1 st hypothetical case. 0, opt P C Figure 3. Plot of Φ(0) (dashed) ad Φ opt vs Pr(C) for the 1 st hypothetical case.

21 21 0, opt P C Figure 4. Plot of Φ(0) (dashed) ad Φ opt vs Pr(C) for the 2 d hypothetical case. ENV P C Figure 5. Plot of ENV vs Pr(C) for the 1 st hypothetical case.

22 22 ENV P C -1-2 Figure 6. Plot of ENV vs Pr(C) for the 2 d hypothetical case Figure 7. The beta(16.4, 8.6) desity fuctio.

23 Figure 8. The beta(4.6, 5.4) desity fuctio Figure 9. The distributio of Φ(2) for the 1 st hypothetical case for a beta(4.6, 5.4) prior o θ = Pr(C).

24 24 Table 1. Costs ad probabilities for 1 st hypothetical case. k M () k R k F Pr(D C) Pr(D C c ) Table 2. The four possible outcomes of takig three additioal samples, D 3. k Pr(D 3 = k) Pr(C D 3 = k) k F Pr(C D 3 = k) Decisio do othig do othig remediate remediate Table 3. Costs ad probabilities for 2 d hypothetical case. k M () k R k F Pr(D C) Pr(D C c ) Table 4. The two possible outcomes of takig oe additioal sample, D 1. k Pr(D 1 = k) Pr(C D 1 = k) k F Pr(C D 1 = k) Decisio do othig remediate