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ECE 307 – Techniques for Engineering Decisions Basic Probability: Case Studies George Gross Department of Electrical and Computer Engineering University of Illinois at Urbana-Champaign © 2006 – 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved OIL WILDCATTING: SITE DATA ‰ We consider two possible exploratory well sites  site 1: fairly uncertain  site 2: fairly certain for a low production level ‰ Geological fact: If the rock strata underlying site are characterized by a “dome” structure, there are better chances of finding oil than if no dome structure exists © 2006 – 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved OIL WILDCATTING: SITE DATA state site with $ 100k drilling costs payoffs dry site with $ 200k drilling costs probability payoffs – 100k 0.2 – 200k low production 150k 0.8 50k high production 500k – © 2006 – 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved MODELING OF SITE ⎧dome structure ⎪ S = structure r.v = ⎨  ⎪⎩other with prob 0.6 with prob 0.4 conditioning on the event { S = dome}  dry P { state = x S = dome}   0.60 low production 0.25 high production 0.15 state x (r.v outcome) © 2006 – 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved SITE 1: NO DOME conditioning on the event { S = no dome}  state outcome x  P { state = x S = no dome}   dry 0.850 low production 0.125 high production 0.025 © 2006 – 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved DECISION TREE DIAGRAM payoffs dry – 100 low prod te i s high prod dry 0.2 si t e2 low prod 0.8 © 2006 – 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved 150 500 – 200 50 COMPUTATION OF PROBABILITIES OF STATES : SITE P {dry} = P { state of site = dry}  = P { state = dry S = dome}   P { S = dome } +  P { state = dry S = no dome} P{ S = no dome }    = (0.6)(0.6) + (0.85)(0.4) = 0.7 © 2006 – 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved COMPUTATION OF PROBABILITIES OF STATES : SITE P {low prod } = P { state of site = low prod }  = P { state = low prod S = dome}   P { S = dome} +  P { state = low prod S = no dome} P { S = no dome}    = (0.25)(0.6) + (0.125)(0.4) = 0.2 © 2006 – 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved CONFIGURATION OF PROBABILITIES OF STATES : SITE P {high prod } = P { state of site = high prod }  = P { state = high prod S = dome}   P{ S = dome} +  P { state = high prod S = no dome} P{ S = no dome}    = (0.15)(0.6) + (0.025)(0.4) = 0.1 © 2006 – 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved DECISION DIAGRAM WITH PROBABILITIES payoffs dry low prod te i s high prod dry si t e2 (0.7) – 100 (0.2) 150 (0.1) 500 (0.2) – 200 low prod (0.8) © 2006 – 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved 50 10 DECISION ANALYSIS MONTHLY PROBLEM: MAY DATA May subscription data expiring subscriptions (%) renewal ratio (%) gift subscriptions 70 75 promotional subscriptions 20 50 previous subscribers 10 10 total 100 © 2006 – 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved 19 DECISION ANALYSIS MONTHLY PROBLEM: JUNE DATA June subscription data expiring subscriptions (%) renewal ratio (%) gift subscriptions 45 85 promotional subscriptions 10 60 previous subscribers 45 20 total 100 © 2006 – 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved 20 DECISION ANALYSIS MONTHLY PROBLEM: SUBSCRIPTIONS DATA ‰ The overall proportion of renewals had dropped from May to June ‰ Figures indicate that the proportion of renewals had increased in each category ‰ We need to analyze the data in a meaningful fashion and interpret it © 2006 – 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved 21 DECISION ANALYSIS MONTHLY PROBLEM ‰ We can view the data in the two tables as providing probabilities for the renewal r.v ⎧ renewal R = ⎨  ⎩ no renewal ‰ However, the information is given as conditional probabilities with the conditioning on the subscription type with r.v S  ⎧ gift ⎪ S = ⎨ promotional  ⎪ previous ⎩ © 2006 – 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved 22 DECISION ANALYSIS MONTHLY PROBLEM ‰ We use the May and June data and compute: P { R = renewal} = P { R = renewal S = gift} • P { S = gift} +     P { R = renewal S = promo}   • P { S = promo} +  P { R = renewal S = previous} • P { S = previous}    ‰ The renewal probabilities are computed for each month © 2006 – 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved 23 DECISION ANALYSIS MONTHLY PROBLEM P{ R May = renewal } = (0.75)(0.7) + (0.5)(0.2) + (0.1)(0.1)  = 0.635 P{ R June = renewal } = (0.85)(0.45) + (0.6)(0.1) + (0.2)(0.45)  = 0.5325 ‰ Due to the change of the mix, P{ R June = renewal } < P{ R May = renewal }   even though the renewal proportion increased in each category © 2006 – 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved 24 DISCRIMINATION CASE STUDY ‰ We explore the relationship between the race of convicted defendants in murder trials and the imposition of the death penalty in these trials on the defendants ‰ This is a good example to illustrate the care required in correctly interpreting data © 2006 – 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved 25 DISCRIMINATION CASE STUDY: DATA yes no total defendants white 19 141 160 black 17 149 166 36 290 326 defendants race death penalty imposed total © 2006 – 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved 26 DISCRIMINATION CASE STUDY: USING THE DATA ‰ We define the r.v.s ⎧⎪ D = death penalty = ⎨  ⎪⎩ ⎧⎪ white R = race = ⎨  ⎪⎩ black death penalty is imposed otherwise defendant is white defendant is black ‰ We use data of the table to determine P { D = R = white} and P { D = R = black} 27  © 2006 – 2009 George  Gross, University of Illinois at Urbana-Champaign,  All Rights Reserved  DISCRIMINATION CASE STUDY: USING THE DATA ‰ The table provides values 19 P { D = R = white} = = 0.119   160 17 = 0.102 P { D = R = black} =   166 ‰ These two probabilities indicate little difference between the treatment of the two races ‰ We use additional data to probe deeper © 2006 – 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved 28 DISCRIMINATION CASE STUDY: USING MORE DATA race of victim white black death penalty imposed race of total defendant defendants yes no white 19 132 151 black 11 52 63 total 30 184 214 white 9 black 97 103 total 106 112 36 290 326 total for all cases © 2006 – 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved 29 DISCRIMINATION CASE STUDY: USING MORE DATA ‰ Next, we bring in the race of the victim by defining the r.v ⎧⎪ white V =⎨  ⎪black ⎩ victim is white victim is black ‰ We have the following probabilities 19 P { D = R = white , V = white} = = 0.126    151 11 P { D = R = black , V = white} = = 0.175    63 © 2006 – 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved 30 DISCRIMINATION CASE STUDY: USING MORE DATA =0 P { D = R = white , V = black} =    = 0.058 P { D = R = black , V = black} =    103 ‰ Data disaggregation on the basis of conditioning also on V shows that blacks appear to get the  death penalty more frequently, about % more than whites independent of the race of the victim © 2006 – 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved 31 APPARENT PARADOX ‰ No difference between the overall imposition of death penalty and the race of the convicted murderers in the aggregated data case ‰ Clear difference in the disaggregated data case where the race of the victim is explicitly considered: blacks appear to get the penalty with % higher incidence than whites ‰ The classification of the victim’s race allows the distinct differentiation of the R = white from the  R = black cases  © 2006 – 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved 32 KEY ISSUE ‰ Since the number of black victims for R = white  cases is 0, the result is a rate of death penalty, making no contribution to the overall rate for the R = white cases  ‰ In addition, the many black victims for the R = black cases results in the relatively low death  penalty rate for black defendant / black victim cases and brings down the overall death penalty rate for black victims © 2006 – 2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved 33 ... penalty imposed race of total defendant defendants yes no white 19 13 2 15 1 black 11 52 63 total 30 18 4 214 white 9 black 97 10 3 total 10 6 11 2 36 290 326 total for all cases © 2006 – 2009 George Gross,... Urbana-Champaign, All Rights Reserved 11 VARIANCE EVALUATION ‰ Site evaluation: σ = 0.7[ − 10 0 − 10 ] + 0.2 [15 0 − 10 ] + 0 .1[ 500 − 10 ] = 36,400 ( k$ ) 2 and so σ = 19 0.8 k$ ‰ Site evaluation: σ 2 =... DISCRIMINATION CASE STUDY: USING THE DATA ‰ The table provides values 19 P { D = R = white} = = 0 .11 9   16 0 17 = 0 .10 2 P { D = R = black} =   16 6 ‰ These two probabilities indicate little difference between