Probability Translation Guide

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Nicholas Willis
5 years ago
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1 Quick Guid to Translation for th inbuilt SWARM Calculator If you s information looking lik this: Us this statmnt or any variant* (not th backticks) And this is what you ll s whn you prss Calculat Th chancs that a valv will xprinc dfcts within a givn month is 12% Whn a station A snds a signal to station B, thr is only a 65% chanc of th signal bing rcivd by B. Prior to knowing anything xcpt background information, th initial probability that a trrorist bombing would occur in Blushistad in any givn month is 10%. positiv rat (rporting that an activity occurrd, whn in fact non took plac) is 10%. `chanc of ValvDfct is 12%` chanc of ValvDfct is 12% `chanc of B givn A is 65%` chanc of B givn A is 65% `chanc of Bombing is 10%` chanc of Bombing is 10% `chanc of Rport givn no Activity is 10%` chanc of Rport givn no Activity is 10% Ngativ Rat (rporting that activity did not occur, whn in fact it did tak plac): 10% `chanc of Rport givn Activity is 90%` Not: hr w convrt to tru positiv rat, which is 1 minus th fals ngativ rat, bcaus dclarations must b th chanc of. chanc of Rport givn Activity is 90% Miss rat is 10% `chanc of Rport givn Activity is 90%` Miss = Ngativ chanc of Rport givn Activity is 90% You ar initially quit confidnt that th substanc is not radioactiv i.. you think thr is only a 20% chanc it is radioactiv. `chanc of Radioactiv is 20%` chanc of Radioactiv is 20% Thr is a 75% chanc that th movmnt dtctor signal is valid (hit rat) and only a 5% chang that it is falsly rports som activity (fals positiv rat). Tst A is highly rliabl, with only a small (15%) chanc of making a mistak. `chanc of Dtctor givn Activity is 75%` `chanc of Dtctor givn not Activity is 5%` `chanc of TstPositiv givn ConditionPrsnt is 85%` `chanc of TstPositiv givn not ConditionPrsnt is 15%` chanc of Dtctor givn Activity is 75% chanc of Dtctor givn not Activity is 5% chanc of TstPositiv givn ConditionPrsnt is 85% chanc of TstPositiv givn not ConditionPrsnt is 15% Th hatch activators fail in 1% of cass. `chanc of ActivatorFail is 1%` or `chanc of ActivatorWorks is 99%` chanc of ActivatorFail is 1% chanc of ActivatorWorks is 99% or If th hatch activators fail, th hatch is 85% likly to b jammd, rgardlss of whthr th manual rlas is opratd. `chanc of HatchJammd givn ActivatorFail is 85%` chanc of HatchJammd givn ActivatorFail is 85% If th ladr is killd but his dputy is not killd, th dputy is 75% likly to tak control. `chanc DputyControl givn LadrKilld and not DputyKilld is 75%` chanc DputyControl givn LadrKilld and not DputyKilld is 75% * You can us any variant accptabl to th probability calculator which xprsss th sam information. Accptabl variants includ: `probability of ValvDfct = 12%` `pr ValvDfct = 12%` `pr(valvdfct) = 12%` `prob ValvDfct is 12%` ** 12% can also b 0.12, or.12, or 12/100, 0.24/2, tc.

2 In Dpth Guid Problm Dscription Dpndncy Diagram SWARM Calculator mnt Probability tabl for Bays Nt approach (UnBBays prfrs 12% b writtn as 0.12.) Nots Bas Rat: Th chancs that a valv will xprinc dfcts within a givn month is 12%. `chanc of ValvDfct is 12%` Variations: `probability of ValvDfct = 12%` `pr ValvDfct = 12%` `pr(valvdfct) = 12%` 12% can also b 0.12, or.12, or 12/100, tc. Probabilitis for: ValvDfct 0.12 (0.88)* *Parnthss indicat 0.88 was calculatd from th providd information in th othr row i.. 1 minus 0.12 This is th chanc w would assign bfor sing any cas spcific vidnc. For xampl, w would giv a coin a 50% chanc of hads. Somtims (lik hr) it is basd on data. Somtims it is a judgmnt, such as Th chanc North Kora will tst an ICBM this Jun. Signals (1/2): Du to intrfrnc, whn a station A snds a signal to station B, thr is only a 65% chanc of th signal bing rcivd by B. `chanc of B givn A is 65%` You can also us : for givn : `chanc of B:A is 65%`. Conditional Probabilitis for: B If A = If A = Rcivd 0.65 Not spcifid This xampl says nothing about B in th cas whr A dos not snd. S Signals (2/2).. Not Rcivd (0.35) Not spcifid Signals (2/2):...continuing... B only transmits rcivd signals, and B rcivs signals only from A. W chang our intrprtation of B from simply rciv to transmit. W could also writ: `chanc of B givn not A = 0%` But Calculator assums 0% by dfault, so w could omit this. Conditional Probabilitis for: B If A = If A = Transmit [0.65, from bfor] 0.0 S Gnric Dtctor for an xampl with nonzro fals alarm rat. Silnt [0.35, from bfor] 1.0 Gnric Dtctors Witnsss, mdical tsts, and othr dtctors rport on For som Evnt TBD. In all cass w omit th probabilitis for th Evnt to focus on modling th Rport.

3 an vnt. Thr ar svral ways to rport dtctor rliability. a) Hit & Alarm: Th hit rat is 85%. Th fals alarm rat is 10%. `chanc of Rport givn Evnt is 85%` `chanc of Rport givn no Evnt is 10%` If Evnt = If Evnt = Th Hit Rat is th proportion of positiv cass which ar rportd as positiv, i.. th proportion of s rportd as. Th Alarm Rat is th proportion of ngativs rportd as positiv, i.. th proportion of s rportd as. (0.15) (0.90) b) Snsitivity & Spcificity: Th snsitivity is 85%. Th spcificity is 70%. `chanc of Rport givn Evnt is 85%` Eithr : `chanc of no Rport givn no Evnt is 70%` Or: `chanc of Rport givn no Evnt is 30%` If Evnt = If Evnt = 0.85 (0.30) (0.15) 0.70 Snsitivity is Hit rat, for xampl th proportion of s rportd as. Spcificity is th proportion of ngativs rportd as ngativ, such as th proportion of s rportd as. (It is 1 minus th Alarm Rat.) Ths trms ar common in mdical diagnosis. Rdundant Systms: Mains powr is 99% rliabl. Battry powr is 90% rliabl. Th computr rmains on providd it has ithr mains or battry powr. `chanc of Mains = 99%` `chanc of Battry = 90%` `chanc of Computr givn Mains = 1` `chanc of Computr givn Battry = 1` Probabilitis for: Mains 0.99 (0.01) Probabilitis for: Battry 0.90 (0.10) This xampl rally shows off th Calculator s dfault modl. In this cas w hav: (1) a causal ntwork whr th causs ar indpndnt, so you don t hav to list thir combinations, and (2) jointly ncssary, so th ffct cannot happn without at last on caus happning, so w don t hav to spcify th chanc of Computr whn causs ar. Not: Gnralizs to thr or mor causs. Backup or rdundant systms ar indpndnt. S latr for intractions. Conditional Probabilitis for: Computr If Battry = = = If Battry = = = Ths dfaults mak causal modls vry compact to writ. But as w hav sn abov, you can ovrrid thm

(0) (0) 0 1 Multi Dtctor: A radar dtcts rgular plans 95% of th tim. It dtcts stalth plans 20% of th tim. It has a 1% fals alarm rat.

01 Calculator automatically computs chanc of Radar givn Stalth & Rgular, by assuming indpndnt dtction failurs. For th tabl, w nd to do that calculation ourslvs*.

99) *Ntica supports: P(Radar Rgular, Stalth) = NoisyOrDist(Radar,.01, Rgular,.95, Stalth,.

4 (0) (0) 0 1 Multi Dtctor: A radar dtcts rgular plans 95% of th tim. It dtcts stalth plans 20% of th tim. It has a 1% fals alarm rat. `chanc of Radar givn Rgular = 95%` `chanc of Radar givn Stalth = 20%` `chanc of Radar givn no Rgular and no Stalth = 1%` Conditional Probabilitis for: Radar If Stalth = If Stalth = (0.96) Calculator automatically computs chanc of Radar givn Stalth & Rgular, by assuming indpndnt dtction failurs. For th tabl, w nd to do that calculation ourslvs*. Of 100 vnts with both plans, w miss 5 of th rgular Plans; of thos 5 w dtct 1 (20%) bcaus of th Stalth plans. So w miss 4 and dtct 96, pr 100. (0.04) (0.80) (0.05) (0.99) *Ntica supports: P(Radar Rgular, Stalth) = NoisyOrDist(Radar,.01, Rgular,.95, Stalth,.20) Mutually Exclusiv I Whn you s It is saf to assum that xactly on of ths vnts happns, or that sms th natural rading. For xampl: On if by land and two if by sa suggsts ithr or. Th ky is to do: And not : All variabls in Calculator ar /. So: `chanc of ByLand = 50%` `chanc of On givn ByLand = 100%` `chanc of On givn ByLand = 0%` Bcaus w can assum thy ar mutually xclusiv, ByLand mans BySa. Probabilitis for: Attack Chanc Conditional Probabilitis for: Rport ByLand 0.5 If Attack = ByLand BySa (0.5) On 1 (0) Two (0) 1 If Attack = BySa Mutually xclusiv and xhaustiv altrnativs ar stats of a singl variabl. Only us multipl variabls whn you also nd to modl both and nithr, as in Multi Dtctor abov. Th probabilitis for Attack do not mattr for our purposs. W usd 50%. Not that th Bays nt allows maningful stat nams, whr Calculator has only /. Mutually Exclusiv II Mor than two vnts, assum xactly on happns. On if by land and two if by sa and thr if both ls non. Sam as bfor. Just mor stats in Attack. N/A: Calculator is /. Modling multi stat rquirs constraint variabls and binary logic. As a hack, you could crat this modl: givn Attack =.. Nith r Land Sa Both Non Should not happn in CREATE tsting this yar. If it dos, us Bays nts or othr flxibl mthods. As you can s, th Calculator is not wll suitd. W do not show th probabilitis for Attack bcaus thy ar straightforward and do not rally mattr hr. On Two but rally...: Thr

Multipl Rports Thr ar 95 grn drons and 5 nw blu ons. Sarah and Sam both say thy saw a blu dron. Sam gts 60% of blu drons and 80% of grn drons right. Sarah is right on 80% of all hr calls.

5 Multipl Rports Thr ar 95 grn drons and 5 nw blu ons. Sarah and Sam both say thy saw a blu dron. Sam gts 60% of blu drons and 80% of grn drons right. Sarah is right on 80% of all hr calls. What color was th dron? `chanc Blu = 5%` `chanc SamBlu givn Blu = 60%` `chanc SamBlu givn Blu = `chanc SarahBlu givn Blu = `chanc SarahBlu givn Blu = Th Qury `chanc Blu givn (SamBlu & SarahBlu)` Probabilitis for: Blu Chanc Conditional Probabilitis for: SamBlu if Blu is: (0.2) (0.4) 0.8 Conditional Probabilitis for: SarahBlu if Blu is: This is anothr variant on th Grn/Blu cab problm. Hr w r mostly concrnd with modling multipl rports vrsus vidnc chains. But s th hlp docs for mor discussion of th GrnCo and Blu Dynamics dron problm. Hr w mad two changs: Far fwr Blu drons. A scond witnss, with diffrnt rliabilitis. What diffrnc dos th scond rport mak? Th proportion of Blu drons? What if th rports disagr? 0.8 (0.2) (0.2) 0.8

u x v x dx u x v x v x u x dx d u x v x u x v x dx u x v x dx Integration by Parts Formula

u x v x dx u x v x v x u x dx d u x v x u x v x dx u x v x dx Integration by Parts Formula 7. Intgration by Parts Each drivativ formula givs ris to a corrsponding intgral formula, as w v sn many tims. Th drivativ product rul yilds a vry usful intgration tchniqu calld intgration by parts. Starting