Hazard, Risk and Vulnerability

Reading6.3.pdf

Home >English homework help >Hazard, Risk and Vulnerability

303

11 Probability Review

11.1 Introduction

Risk. analysis. is. decision. making. under. uncertainty.. Resolving. and. addressing. that. uncertainty.is.a.primary.reason.for.risk.assessment.. Probability.is.the.language.we. use.to.express.our.uncertainty..Someone.has.to.understand.probability.if.risk.assess- ment. is. going. to. address. the. knowledge. uncertainty. and. natural. variability. in. our. decision.problems..Quantitative.risk.assessment.uses.the.language.of.probability,.and. it.is.essential.to.know.the.structure.of.any.language.so.we.do.not.misuse.it..Learning. that.basic.structure.is.the.purpose.of.this.chapter.

Risk. has. been. described. as. the. product. of. a. consequence. and. its. probability.. Assessors. need. to. understand. probabilities. to. assess. them. honestly.. Managers. need. to. understand. probability. to. manage. risks. effectively.. Risk. communicators. have. to. understand. probability. to. explain. it. to. others.. Not. everyone. needs. to. know. how. to. do.good.probabilistic.risk.assessment,.but.everyone.does.have.to.be.conversant.and. knowledgeable.about.some.very.basic.facts.about.probability.

One.problem.with.probability.is.that.it.does.not.lend.itself.well.to.intuition..If.there. is.a.10%.chance.of.rain.and.we.get.caught.in.a.downpour,.we.are.more.inclined.to. think.the.forecast.was.wrong.than.we.are.to.think.the.rain.was.an.event.that.had.only. a.10%.chance.

To.further.illustrate.the.nonintuitive.nature.of.probability,.consider.the.so-called. Monty. Hall. problem,. which. I. have. used. in. risk. assessment. training. for. years.. A. letter. writer. to. Parade. magazine. (Whitaker. 1990). posed. the. following. question.. “Suppose.you’re.on.a.game.show,.and.you’re.given.the.choice.of.three.doors:.Behind. one.door.is.a.car;.behind.the.others,.goats..You.pick.a.door,.say.No. 1,.and.the.host,. who. knows. what’s. behind. the. doors,. opens. another. door,. say. No. 3,. which. has. a. goat..He.then.says.to.you,.‘Do.you.want.to.pick.door.No. 2?’.Is.it.to.your.advantage. to.switch.your.choice?”

The.answer.is.yes,.you.win.twice.as.often.if.you.switch..This.just.does.not.make. sense.to.most.people..The.usual.logic.goes.something.like.this..When.presented.with. the.original.choice.of.one.door.from.among.three,.we.have.a.1/3.chance.of.winning. the.car..Once.there.are.only.two.doors.left,.most.people.tend.to.believe.each.one.now. has.a.½.chance.of.hiding.the.car,.and.so.one.perceives.that.each.door.has.a.½.chance. of.winning..This.is.not.so.

There.was.a.1/3.chance.the.car.was.behind.the.contestant’s.door.and.a.2/3.chance. it.was.behind.one.of.the.other.two..When.the.host.opens.one.of.these.other.two.doors. he.is.providing.the.contestant.with.new.information,.the.car.is.not.behind.door.3..The. observant.contestant.now.knows.that.2/3.chance.all.resides.in.door.2..Given.a.choice.

304 Principles of Risk Analysis: Decision Making Under Uncertainty

between.a.1/3.chance.of.winning.and.a.2/3.chance.of.winning,.who.would.not.prefer. the.better.odds?.So.you.should.always.switch.in.such.a.game.

Here.is.how.it.works..Monty.Hall,.the.game.show.host,.always.knows.where.the.car. is,.and.this.is.critical..Let.the.car.be.behind.door.1..If.you.correctly.chose.the.door. with. the. car. (door. 1),. he. can. open. either. of. the. other. doors. to. reveal. a. goat.. If. you. decide.to.switch.when.you.pick.door.1.you.will.give.up.the.car.and.end.up.with.a.goat.. Switching.is.a.losing.strategy.in.this.case..The.score.is.switch.0.and.stay.1.

Suppose,.now,.you.choose.door.2,.which.is.hiding.a.goat..Monty,.knowing.the.car. is.behind.door.1,.opens.the.third.door.to.show.you.a.goat..This.time.switching.doors. wins.for.you..The.score.is.now.switch.1.and.stay.1..Now.imagine.you.choose.door.3.. Monty.now.opens.door.2.to.show.you.the.goat..Once.again,.switching.doors.means. you.trade.a.goat.for.a.car.and.win..The.score.is.now.switch.2.and.stay.1..No.matter. which.door.the.car.is.behind,.the.logic.here.always.leads.to.a.score.of.switch.winning. twice.and.stay.winning.once..If.you.switch,.you.will.win.2/3.of.the.time;.if.you.stay. with.your.original.choice.you.win.1/3.of.the.time..You.are.not.guaranteed.a.win.if. you.switch,.but.you.will.win.twice.as.often.as.you.lose.in.the.long.run,.and.that.is.not. intuitively.obvious.to.many.people.

This. chapter. provides. a. review. of. probability. concepts. that. you’ll. need. to. under- stand.to.do.basic.quantitative.risk.assessment..It.is.also.a.good.review.for.managers. and.communicators.as.well..The.chapter.does.not.attempt.a.rigorous.theoretical.devel- opment. or. treatment. of. probability.. What. it. does. do. is. offer. a. survey. review. of. the. kinds.of.concepts.anyone.working.in.risk.analysis.is.likely.to.need.

11.2 Two Schools of Thought

Probability. is. the. chance. that. something. will. or. will. not. happen.. There. are. two. schools.of.thought.on.the.nature.of.probability..They.go.by.many.names,.but.we’ll.call. them.the.frequentist.and.subjectivist.schools..The.frequentist.approach.to.probability. is.based.loosely.on.the.notion.that.true.probability.values.are.“out.there”.and.that.we. can.discover.them.through.data..Specifically,.we.can.calculate.the.long-run.expected. frequency.of.occurrence..The.probability.of.an.event.A,.P(A),.is.equal.to.n/N,.where. n.is.the.number.of.times.event.A.occurs.in.N.opportunities..It.is.the.frequency.with. which.A.occurs.out.of.the.number.of.times.it.could.occur..So.the.annual.probability. of.a.hurricane.hitting.your.community.is.estimated.by.the.number.of.years.a.hurri- cane.strikes.out.of.the.number.of.years.observed..The.probability.of.an.accident.per. vehicle.mile.is.the.number.of.accidents.divided.by.the.number.of.vehicle.miles..The. frequentist.view.of.probability.works.quite.well.with.repeatable.events.

There. is. also. a. subjective. or. degree-of-belief. view. of. probability.. This. is. based. loosely.on.the.notion.that.probability.is.an.intrinsic.phenomenon,.i.e.,.it.is.a.device.we. humans.use.to.explain.and.deal.with.natural.variability.and.knowledge.uncertainty.. Probability.is.not.out.there;.it.is.in.us..Probability.is.a.measure.of.the.plausibility.of.an. event.given.our.incomplete.knowledge.

Some.events.have.a.uniqueness.about.them.that.denies.the.existence.of.frequency.. There.are.many.things.that.have.never.happened.before.or.things.that.may.only.hap- pen.once,.so.we.cannot.estimate.their.probabilities.with.frequencies.of.occurrence..A.

Probability Review 305

subjectivist.view.of.probability.is.especially.useful.for.these.unique.events..Bayesians. favor.this.view.of.probability.

What.is.the.probability.I.will.obtain.a.head.before.I.toss.a.fair.coin?.It.is..5;.this.is.a. frequency..We.can.verify.this.probability.by.observing.a.great.many.flips.and.count- ing.the.proportion.of.heads..Once.I.flip.the.coin,.what.is.the.probability.it.is.a.head?. The.true.probability.is.now.either.0.if.it.is.a.tail.or.1.if.it.is.a.head..The.true.probability. is.uncertain,.and.now.a.degree-of-belief.view.of.probability.applies..With.this.simple. example. we. have.natural. variation. in.the.large,. and. this. is. amenable. to.the.relative. frequency.view..We.also.have.uncertainty.in.the.small.(a.single.completed.toss)..Thus,. there.is.room.for.both.schools.of.thought.in.my.own.approach.to.risk.assessment..You. will.have.to.make.your.own.choice!

Two.people.can.have.different.degrees.of.belief.about.the.probability.of.the.same. uncertain.event.and.both.can.be.right..If.I.believe.there.is.an.80%.chance.of.rain.today. and.you.believe.there.is.a.50%.chance.of.rain,.we.are.both.right.if.it.rains..In.fact.we. are.both.right.if.it.does.not.rain..If.reading.that.hurts.your.head.it.may.be.because.you. are.beginning.to.understand..Probability.is.not.easily.intuited.

The.mathematics.of.probability.are.pretty.well.settled..The.two.schools.of.thought. pretty.much.agree.on.these.mathematical.matters..It.is.the.philosophy.of.probability. that.causes.the.problems..Discussions.of.this.philosophy.can.bring.out.an.intensity.of. emotions.unrivaled.by.any.other.topic.in.mathematics,.an.intensity.often.reserved.for. politics.and.religion..If.you.want.to.be.a.good.risk.manager,.never.seat.a.frequentist. next.to.a.subjectivist.at.a.dinner.party.

11.3 Probability Essentials

Probability. is. measured. as. a. number. between. zero. and. one.. Zero. means. there. is. no. chance.the.event.will.occur,.i.e.,.it.is.impossible..Numbers.close.to.zero.are,.therefore. describing.events.that.are.close.to.impossible..One.means.the.event.has.happened.or.is. sure.to.happen..Numbers.close.to.one.describe.events.that.are.almost.certain.to.occur.

What. then. is.the. most. uncertain.number. of. all?. Is. it. your.chance. of. winning. the. Powerball.grand.prize,.which.is.1.in.195,249,054.(5.×.10−9)?.Surely.not,.as.that.prob- ability.is.pretty.definitive;.winning.that.prize.is.as.close.to.impossible.as.you.are.likely. to.get!.Small.probabilities.mean.unlikely.events;.they.do.not.convey.great.uncertainty,. however..The.most.uncertain.probability.of.all.is..5..If.the.probability.of.rain.is.50%,.it. is.as.likely.to.rain.as.to.not.rain..Once.the.probability.shades.a.little.one.way,..500001,. or. the. other,. .499999,. we. can. see. it. is. slightly. more. like. to. rain. than. not. rain. or. to. not.rain.than.rain,.and.the.uncertainty.slowly.begins.to.resolve.as.probabilities.move. toward.zero.or.one.

FREQUENTIST AND BAYESIAN MEANS

A. frequentist. believes. a. population. mean. is. real. but. unknown. and. often. unknowable..It.can.only.be.estimated.from.the.data.using.confidence.intervals.. A.Bayesian.believes.the.population.mean.is.an.abstraction..Only.the.data.are. real.

306 Principles of Risk Analysis: Decision Making Under Uncertainty

The.“sample.space”.is.one.probability.theory.concept.it.would.be.unwise.to.over- look..Let’s.start.with.something.simple.like.tossing.two.dice,.one.red.and.one.white.. There.are.exactly.36.possible.outcomes,.no.more.and.no.less..Table 11.1.enumerates. the.possibilities..The.sample.space.is.the.set.of.all.the.possible.outcomes..The.sum. of. the. probabilities. of. all. the. possible. outcomes. must. equal. one.. This. is. equivalent. to.saying.that.one.and.only.one.of.these.dice.rolls.must.happen.if.we.roll.the.dice..A. sample.space.must.be.mutually.exclusive.(only.one.of.them.can.happen.at.a.time).and. collectively.exhaustive.(there.is.no.result.that.is.not.included.in.the.sample.space).

There. are. 36. possibilities.. Each. outcome. in. the. space. has. an. equal. chance,. 1/n,. where.n.is.the.number.of.possible.outcomes..In.this.case.the.probability.of.any.one. outcome. is. 1/36.. We. can. define. an. event. as. rolling. a. six,. and. using. a. frequentist. approach.we.can.see.five.different.ways.of.doing.this,.each.with.a.probability.of.1/36.. So.the.probability.of.this.event.is.5/36.

Now.let’s.use.an.event.tree.to.identify.a.sample.space..Figure 11.1.shows.an.example. for.tossing.three.coins.in.sequence..The.figure.shows.how.the.sample.space.(circled). is.derived.through.the.various.pathways..All.possible.outcomes.are.shown;.there.are. no.other.possibilities..The.probability.of.each.triplet.outcome.is.equally.likely,.1/n.. This.time,.n.is.8,.and.the.probability.of.three.heads.(HHH).is.12.5%.or.1/8..We.can. again.count.up.different.outcomes.to.define.events.such.as.getting.two.heads..There. are.three.ways.to.do.that,.each.with.a.probability.of.12.5%,.so.the.probability.of.the. event.two.heads.and.one.tail.is.37.5%.or.3/8.

Event.trees.are.handy.risk.models.for.a.wide.variety.of.problems..The.endpoints.of. such.a.model.define.the.sample.space.for.the.risk.problem..The.endpoints.need.not.be. equally.likely.in.a.risk.problem,.but.the.sum.of.the.probabilities.of.all.endpoints.must. sum.to.one,.just.as.they.do.here.

Note. that. Figure. 11.1. shows. a. sample. space. constructed. of. numerous. pathways. comprising. nodes. and. branches.. When. we. arrive. at. a. node,. something. has. to. hap- pen..That.is,.we.move.forward.on.one.branch.or.another..Thus,.all.branches.must.be. mutually.exclusive.and.collectively.exhaustive..The.probabilities.of.all.the.branches. coming.out.of.a.single.node.must.also.sum.to.one,.because.something.has.to.happen.at. each.node,.and.the.branches.define.all.the.possibilities..So.an.event.tree,.in.this.sense,. has.sample.spaces.within.sample.spaces..The.endpoints.of.each.sample.space.must. have.probabilities.that.sum.to.one.

When.using.tree.models.to.represent.risks,.the.same.simple.probability.rules.hold.. Identifying. the. mutually. exclusive. and. collectively. exhaustive. set. of. endpoints. that. will. define. the.sample. space. is. not. often. a. simple. exercise.in.counting. as. it.is.with.

TABLE 11.1

Sample.Space.of.Outcomes.for.Tosses.of.One.Red.and.One.White.Die

Red White Red White Red White Red White Red White Red White

1 1 2 1 3 1 4 1 5 1 6 1

1 2 2 2 3 2 4 2 5 2 6 2

1 3 2 3 3 3 4 3 5 3 6 3

1 4 2 4 3 4 4 4 5 4 6 4

1 5 2 5 3 5 4 5 5 5 6 5

1 6 2 6 3 6 4 6 5 6 6 6

Probability Review 307

dice.and.coins..Nonetheless,.it.is.important.to.understand.that.such.models.are.just. defining.a.sample.space.of.outcomes.of.interest.to.risk.managers..These.models.must. obey. the. mathematical. laws. that. order. probability.. It. is. not. okay. to. simply. build. a. model.and.start.filling.in.numbers..The.numbers.must.behave.properly.and.follow.the. laws.of.probability.

Figure 11.2. shows. a. simplified. and. hypothetical. earthquake. risk. for. a. structure.. There.are.four.possible.endpoints.for.this.simple.risk.model..The.probabilities.of.these. four.endpoints.must.sum.to.one.

Each.node.in.an.event.tree.identifies.a.new.event,.and.each.has.its.own.unique.prob- ability,.as.seen.in.Figure 11.2..The.first.event.is.that.the.soil.does.(30%).or.does.not.(70%). liquefy.as.the.result.of.an.earthquake..If.it.does,.the.second.event.is.that.the.structure.does. (80%).or.does.not.(20%).crack..If.the.result.of.the.first.event.is.no.liquefied.soil,.then.the. probability.of.cracking.(60%).and.not.cracking.(40%).change.because.of.different.prec- edent.conditions..Unlike.the.coin.example,.where.each.event.is.independent.of.every.other. event.and.probabilities.are.constant,.the.world.of.risk.is.full.of.dependencies..Preceding. events.and.conditions.usually.influence.the.probability.and.nature.of.antecedent.events.. This.leads.to.a.sample.space.where.the.outcomes.are.not.all.equally.likely.

The.most.likely.outcome.in.the.sample.space.is.structure.cracking.with.no.liquefi- able.soil..The.least.likely.outcome.is.no.cracking.when.the.soil.does.liquefy..There. are.principles.and.facts.that.underlie.the.probability.values.in.a.risk.model.like.the. event.tree.of.Figure 11.2.

Coin 3

50.0%

12.5% HHH

12.5% HHT

Head

Tail

Coin 3

50.0%

12.5% HTH

12.5% HTT

Head

Tail

Coin 2

Coin 1

50.0%

Head

Tail

Head

Tail

Coin 3

50.0%

12.5% THH

12.5% THT

Head

Tail

Coin 3

50.0%

12.5% TTH

12.5% TTT

Head

Tail

Coin 2

50.0%

Head

Tail

3 Coin Toss

FIGURE 11.1 Event-tree.sample.space.for.a.three.coin.toss.experiment.

308 Principles of Risk Analysis: Decision Making Under Uncertainty

Probabilities. can. be. expressed. as. a. decimal,. percentage,. fraction,. or. odds.. Using. the.model.in.Figure 11.2,.the.probability.the.soil.will.liquefy.and.the.structure.will. crack.can.be.expressed.as.follows:

•. Decimal.=..24

•. Percentage.=.24%

•. Fraction.=.24/100.=.6/25

•. Odds.=.6:19.(x:y.based.on.x/[x.+.y])

In. the. United. States,. the. most. common. way. of. expressing. probabilities. may. be. odds..Games.of.chance.and.gambling.are.at.the.root.of.much.of.our.knowledge.about. probability,.and.gamblers.understand.odds..In.risk.analysis.the.other.three.forms.are. more.commonly.used.

Decimals.are.sometimes.preferred.when.speaking.about.the.probability.of.a.single. event,.e.g.,.the.probability.of.a.single.egg.containing.Salmonella.Enteritidis.is.about.5. ×.10−5.(FSIS.1998)..Percentages.are.sometimes.preferred.to.convey.information.about. a. population,. e.g.,. 0.005%. of. all. eggs. contain. Salmonella. Enteritidis.. Fractions. are. another.popular.alternative.for.conveying.risk.information.to.the.public,.e.g.,.about.1. in.20,000.eggs.contain.Salmonella.Enteritidis.

11.4 How Do We Get Probabilities?

How.do.we.manage.to.identify.these.numbers.between.zero.and.one?.Where.do.they. come.from?.There.are.three.basic.ways.to.estimate.probabilities,.although.these.three. ways.really.just.reflect.the.two.schools.of.thought.on.probability.mentioned.earlier.

Cracking

80.0%

20.0%

24.0%

6.0%

42.0%

28.0%

Yes

Cracking 0

60.0%

40.0%

Yes

Liquefiable Soil

30.0%

70.0%

Yes

Earthquake Mode

FIGURE 11.2 Event-tree.sample.space.for.structure.cracking.after.an.earthquake.

Probability Review 309

Classical.or.analytical.probabilities.are.mathematical.calculations.that.are.used.by. both.schools..I.like.to.describe.analytical.probabilities. as.the.kinds.of.probabilities. that.“smart.people”.can.do.with.pencil.and.paper.or.a.calculator..When.you.open.a. new.deck.of.cards.and.find.the.odds.of.being.dealt.different.poker.hands.(see.text.box). before.the.draw,.these.probabilities.have.been.calculated.analytically.

Analytical. probabilities. rely. on. identifying. sample. spaces. and. taking. subsets. of. them.. Combinatorics. like. the. factorial. rule. of. counting,. permutations. (n!/[n. −. r]!),. combinations.(n!/(r![n.−.r]!),.and.other.counting.techniques.are.used.to.estimate.prob- abilities.of.specific.events,.like.a.full.house..Unfortunately,.risk.assessors.do.not.too. often.get.the.opportunity.to.work.with.analytical.probabilities,.although.random.pro- cesses. like. the. binomial,. Poisson,. and. hypergeometric. processes. do. provide. oppor- tunities. to. calculate. these. analytical. probabilities. at. times.. So,. although. analytical. probabilities. occupy. a. great. deal. of. our. higher. education. exposure. to. probabilities,. they.do.not.come.up.as.often.as.we.would.like.when.doing.risk.assessment.

A. second. source. of. probabilities. is. empirical. or. frequentist. probabilities.. These. are. based. on. observation.. How. many. times. did. the. event. of. interest. happen. out. of. the.number.of.times.it.could.have.happened?.Think.of.a.traffic.light.near.your.home.. What.is.the.probability.you.will.catch.it.red.the.next.time.you.encounter.it?.All.we. need.to.do.is.keep.a.little.pad.of.paper.on.the.seat.of.the.car..Make.a.strike.mark.under. “red”. each. time. it. is. red. and. under. “not. red”. when. it. is. not.. After. a. hundred. or. so. observations,.we.can.calculate.the.relative.frequency.of.a.red.light..This.is.now.our. approximation. of. the. true. probability. of. catching. the. light. red.. As. we. record. more. observations.over.the.years,.our.estimate.gets.better..A.relative.frequency.is.nothing. but.an.estimate.of.the.true.probability.for.a.frequentist..It.is.but.data.to.a.Bayesian.

Empirical. probabilities. are. especially. useful. when. the. process. of. interest. is. repeated.many.times.under.the.same.circumstances..Empirical.probabilities.are.good. for.estimating.the.reliability.of.electrical.components,.stream.flows,.causes.of.death,. probabilities.of.cancer.from.animal.toxicity.studies,.and.the.like..This.makes.relative. frequencies.or.frequencies.one.of.the.most.common.sources.of.probabilistic.informa- tion.for.risk.assessment.

The. third. source. of. probability. estimates. is. subjective. probabilities.. Subjective. probability.is.based.on.evidence.and.the.experience.of.the.estimator..It.relies.heavily.

RANK AND ODDS OF POKER HANDS

. . . Royal.flush. . 1:649,739

. . . Straight.flush. . 1:64,973

. . . 4.of.a.kind. . 1:4,164

. . . Full.house. . 1:693

. . . Flush. . . 1:508

. . . Straight. . . 1:254

. . . 3.of.a.kind. . 1:46

. . . 2.pair. . . 1:20

. . . 1.pair. . . 1:1.37

. . . High.card. . 1:1

310 Principles of Risk Analysis: Decision Making Under Uncertainty

on.expert.opinion..It.is.most.useful.when.we.deal.with.the.uncertainty.that.surrounds. events.that.will.occur.once.or.that.have.not.yet.occurred.

Subjective.probability.estimates.are.especially.useful.for.filling.in.data.gaps.and. supplementing. data. with. experience. and. judgment.. Risk. assessors. must. deal. with. different. kinds. of. events. as. well. as. different. levels. of. data. and. information. about. these. events.. Subjective. probabilities. are. most. useful. for. those. unique. events. for. which.there.are.no.relative.frequency.data.and.for.which.analytical.calculations.are. not.possible.

Suppose. we. want. to. estimate. the. probability. that. the. channel. bottom. is. 30%. or. more.rock,.or.that.there.will.be.structural.damage.to.a.building.if.an.earthquake.less. than.6.2.on.the.Richter.scale.occurs?.We.might.want.to.estimate.the.probability.of.a. fatal.accident.on.a.dangerous.curve.if.the.curve.is.redesigned.and.eased..We.might. need. to. estimate. the. probability. of. illness. from. a. low-dose. exposure. to. pathogenic. bacteria,.and.so.on..These.lend.themselves.well.to.subjective.probability.estimates.

11.5 Working with Probabilities

If.it.was.as.simple.as.the.previous.material.might.suggest,.anyone.could.work.with. probabilities.. It. is. not. that. simple.. There. are. rules. and. theories. that. govern. our. use. of.probabilities..Estimating.probabilities.of.real.situations.requires.us.to.think.about. complex.events.and.to.apply.these.rules.carefully..Most.of.us.do.not.naturally.assess. probabilities.well..Hence,.it.is.critical.to.good.quantitative.risk.assessment.that.you. have. people. who. can. work. effectively. with. probabilities.. Some. of. the. fundamental. axioms.and.rules.of.probabilities.are.described.below.

11.5.1 Axioms

Probability. density. functions. and. cumulative. probability. distribution. functions. (these. ideas. are. discussed. at. length. in. Chapter. 12). and. their. properties. are. all. essentially. developed. from. three. fundamental. axioms. of. probability.. An. event,. Ei,.is.anything.for.which.you.want.to.know.the.probability..S.is.the.sample.space. that. includes. all. events. of. interest.. The. probability. of. an. event,. P(Ei),. is. defined. such.that:

. 1.. 0.≤.P(Ei).≤.1

. 2.. P(S).=.1

. 3.. If.A.and.B.are.mutually.exclusive.events,.then.P(A.or.B).=.P(A).+.P(B)

The. first. axiom. means. the. probability. of. an. event. is. a. nonnegative. real. number. between.0.and.1..The.second.axiom.says.the.probability.that.some.event.in.the.sample. space. will. occur. is. 1,. and. there. are. no. events. that. can. occur. that. are. outside. the. sample.space..The.third.axiom.says.the.probability.of.two.mutually.exclusive.events. occurring.together.is.the.sum.of.their.individual.event.probabilities..P(A.or.B).is.the. probability.that.A.or.B.occurs.

Probability Review 311

11.5.2 Propositions and Rules

In.addition.to.these.axioms,.there.are.some.consequences.of.these.axioms.that.prove. to.be.essential.for.quantitative.risk.assessment.

11.5.2.1 Marginal Probability

A.marginal.probability.is.the.probability.of.a.single.event,.P(A).(read,.the.probability. of.A),.where.A.stands.for.any.event.whose.probability.we.want.to.know..To.illustrate. this.and.the.following.concepts,.we’ll.use.the.data.in.Table 11.2..The.hypothetical.data. shows.ownership.of.the.levees.in.a.region.in.the.rows.and.the.maintenance.condition. of.those.levees.in.the.columns.

Two.examples.of.marginal.probabilities.are.shown.as.follows:

. P(Private).=.100/300.=..333

. P(Adequate).=.160/300.=..533

11.5.2.2 Complementarity

The.rule.of.complementarity.ensures.that.the.probability.of.an.event.and.its.comple- ment.in.the.sample.space.sum.to.one..The.probability.that.event.A.does.not.happen. (~A.means.not.A).is:

. P(~A).=.1.−.P(A). (11.1)

Two.examples.follow:

. P(~Private).=.1.−..333.=..667

. P(~Adequate).=.1.−..533.=..467

11.5.2.3 Addition Rules

There.are.many.times.when.assessors.will.be.interested.in.more.than.one.event.occur- ring..For.two.different.events,.A.and.B,.the.probability.that.A.or.B.or.both.occur.is. defined.by.the.addition.rule..The.form.of.the.rule.depends.on.whether.A.and.B.are. mutually. exclusive. or. not.. When. events. are. mutually. exclusive. they. cannot. occur. simultaneously..When.they.are.not.they.may.both.occur.

TABLE 11.2

Levee.Ownership.and.Maintenance.Condition

Inadequate Maintenance

Adequate Maintenance Total

Private . 80 . 20 100

Locally.constructed . 50 . 50 100

Federal.construction . 10 . 90 100

Total 140 160 300

312 Principles of Risk Analysis: Decision Making Under Uncertainty

The.addition.rule.for.mutually.exclusive.events.is:

. P(A.or.B).=.P(A).+.P(B). (11.2)

An.example.is:

. P(Private.and.Local).=.100/300.+.100/300.=.200/300.=..667

The.rule.works.the.same.for.N.mutually.exclusive.events.

. P(A1.or.A2.or.….AN).=.P(A1).+.P(A2).+.….+.P(AN). (11.3)

The.addition.rule.for.nonmutually.exclusive.events.is:

. P(A.or.B).=.P(A).+.P(B).–.P(A.and.B). (11.4)

Because.the.events.can.occur.at.the.same.time.we.must.avoid.double.counting.out- comes..For.example,.consider.the.probability.a.levee.is.both.private.and.inadequately. maintained.. There. are. 100. private. levees. and. 140. inadequately. maintained. levees.. There.are.eighty.levees.that.are.both,.so.they.are.counted.twice.and.we.must.subtract. out.these.joint.events.

An.example.is:

. P(Private.and.Inadequate).=.100/300.+.140/300.–.80/300.=.160/300.=..533

The.rule.works.for.N.nonmutually.exclusive.events,.but.all.multiple.counts.of.the. same.elements.must.be.accounted.for.

11.5.2.4 Multiplication Rules

Multiplication.rules.apply.when.we.are.interested.in.the.probability.that.two.things. occur.together..The.proper.formula.depends.on.whether.the.events.A.and.B.are.inde- pendent.or.dependent.events.

Two.events,.A.and.B,.are.independent.if.the.fact.that.A.occurs.does.not.affect.the. probability.of.B.occurring..When.we.have.independent.events,.the.probability.of.both. occurring,.i.e.,.the.multiplication.rule,.is:

. P(A.and.B).=.P(A)P(B). (11.5)

The.probabilities.in.Table 11.2.are.dependent,.so.we.must.vary.the.example.for.a. moment,.and.then.we.will.return.to.the.table..In.Figure 11.3.let.a.head.for.coin.1.be. event.A.and.a.head.for.coin.2.be.event.B..They.are.independent.events..The.result.of. the.first.coin.toss.has.no.effect.on.the.probability.of.the.outcome.of.the.second.coin. toss..Note.that.the.probability.of.a.head.is.the.same.regardless.of.what.the.result.of.the. first.coin.toss.was..In.this.example.of.independent.events:

. P(A.and.B).=.P(coin.1.is.a.head.and.coin.2.is.a.head).=.(.5)(.5).=..25.or.25%.

Two.events,.A.and.B,.are.dependent.if.the.outcome.or.occurrence.of.A.affects.the. outcome.or.occurrence.of.B.so.that.the.probability.of.B.is.changed..The.earthquake. event.tree.of.Figure 11.2.is.an.example.of.dependent.events..We.saw.that.liquefiable. soil.made.structure.cracking.more.likely..If.A.and.B.are.dependent.events,.then.the. probability.of.both.occurring.is:

. P(A.and.B).=.P(A)P(B.after.A). (11.6)

Probability Review 313

P(B. after. A). can. also. be. written. as. P(B|A). (read,. the. probability. of. B. given. that. A. has.happened),.and.then.the.multiplication.rule.for.dependent.events.is.rewritten.as:

. P(A.and.B).=.P(A)P(B|A). (11.7)

We. can. now. return. to. our. example. using. Table 11.2.. However,. let’s. display. it. as. the. probability. tree. in. Figure 11.4.. Note. that. the. probability. of. inadequate. maintenance.varies.depending.on.ownership..If.ownership.and.maintenance.were. independent,. the. probability. of. inadequate. maintenance. would. be. the. same. for. each.kind.of.ownership.and.it.would.equal.the.marginal.probability.P(Inadequate). =..467.

Coin Two

50.0%

25.0%

Head

Tail

Coin Two

50.0%

Head

Tail

Coin One

50.0%

Head

Tail

Two Coin Toss

FIGURE 11.3 Probability.tree.for.a.two.coin.toss.experiment.

Maintenance Condition

80.0%

20.0%

26.6667%

6.6667%

16.6667%

Inadequate

Adequate

Maintenance Condition

50.0%

Ownership

33.3333%

Private

Local

Levee Condition

Inadequate

Adequate

3.3333%

30.0%

Maintenance Condition

10.0%

90.0%

33.3333%

Inadequate

Adequate

Federal

FIGURE 11.4 Probability.tree.for.levee.ownership.and.maintenance.condition.

314 Principles of Risk Analysis: Decision Making Under Uncertainty

The.P(private.and.inadequate).can.be.read.directly.from.the.tree:.it.is..267..If.we. look.at.the.table,.it.is.even.easier.to.find..There.are.80.levees.that.are.both.private.and. inadequate.out.of.a.total.of.300.levees,.80/300.=..267.

Unfortunately,. most. probability. problems. do. not. come. with. a. neatly. worked. out. table. or. probability. tree.. That. is. when. the. formula. becomes. important.. The. term. P(B|A). in. equation. (11.7). is. called. a. conditional. probability.. It. is. the. subject. of. the. next.section.

11.5.2.5 Conditional Probability

Information. changes. probabilities.. We. see. that. from. the. previous. discussion.. The. P(Inadequate).=..467,.but.when.provided.with.information.about.ownership,.the.probabil- ity.that.a.levee.is.inadequate.changes..To.reflect.this.fact.we.use.conditional.probability.

The.probability.that.event.A.happens,.given.the.condition.that.event.B.happens,.is. written.P(A|B)..In.our.examples.above.we.could.say:

. P(Inadequate|Private).=..8

. P(Inadequate|Local).=..5

. P(Inadequate|Federal).=..1

Let’s. revisit. the. multiplication. rule. for. dependent. events. using. the. conditions. of. inadequate.maintenance.and.private.ownership.

. P(Inadequate.and.Private).=.P(Inadequate)P(Private|Inadequate)

. . . . . . . . . . . .=.(140/300)(80/140).=.80/300.=..267

P(Private|Inadequate).is.a.conditional.probability,.and.it.is.a.switch.of.the.earlier. conditional.probability.P(Inadequate|Private).shown.previously..Using.Table 11.2,.we. can.see.the.condition.that.the.levee.is.inadequately.maintained.in.the.second.column.. This.tells.us.the.levee.of.interest.is.one.of.the.140.levees.in.that.column..We.are.no.lon- ger.dealing.with.all.300.levees..Thus,.the.conditional.information.changed.the.prob- ability.of.a.private.levee.from.100/300.to.80/140..Information.changes.probabilities.

Conditional. probabilities. can. also. be. defined. with. a. formula.. Starting. from. the. multiplication.rule.for.dependent.events.we.have:

. P(A.and.B).=.P(A)P(B|A)

Rearranging.we.obtain

P B A P A B

P A | =

( and ) ( )

( ) . (11.8)

Thus,.substituting.into.equation.(11.8),.we.get:

. P(Private|Inadequate).=.P(Inadequate.and.Private)/P(Inadequate)

. . . . . . . ..=.(80/300)/(140/300).=.80/140.=..571

Probability Review 315

11.5.2.6 Bayes’ Theorem

Building.on.the.notion.that.information.can.change.probabilities,.we.introduce.Bayes’. theorem,. which. is. useful. for. updating. probabilities. on. the. basis. of. newly. obtained. information.. Often. we. begin. with. an. initial. or. prior. probability. that. an. event. will. occur.. Then,. as. uncertainty. is. reduced. or. new. information. comes. in,. we. revise. the. probability.to.what.we.call.the.posterior.probability..This.revision.can.be.done.using. Bayes’.theorem.

Bayes’.theorem.is

P A B P B A P A

P B P A B

P B | =

( | ) ( ) ( )

= ( and )

( ) (( )) . (11.9)

To.illustrate.the.theorem,.let’s.vary.the.example.and.consider.the.hypothetical.event. that.a.randomly.selected.crate.of.imported.produce.has.some.form.of.pathogenic.E. coli.to.be..001..Thus,

. P(Ec).=..001.and.P(~Ec).=..999

Suppose. a. diagnostic. test. can. accurately. detect. E. coli. 99%. of. the. time.. Further. assume.that.5%.of.crates.that.do.not.have.E. coli.will.test.positive..That.is,

. P(+|Ec).=..99.and.P(+|~Ec).=..05

Now. suppose. the. test. is. administered. to. a. randomly. selected. crate. of. imported. produce,.which.may.or.may.not.have.pathogenic.E. coli.on.it,.and.the.test.is.positive.. What. is. the. probability. that. the. crate. has. pathogenic. E. coli?. Using. the. concepts. we’ve.developed,.we.are.looking.for.the.probability,.P(Ec|+).

This.is.sometimes.called.the.Bayesian.flip.because.it.is.the.opposite.of.the.known. probability.P(+|Ec)..Note.also.that.we.know.P(Ec),.which.is.the.prior.probability.that. is.being.updated.with.new.information,.i.e.,.that.the.crate.tested.positive..Substituting. into.Bayes’.theorem.in.equation.(11.9).we.are.calculating

P Ec P Ec P Ec

P P Ec

P ( | +) =

(+ | ) ( ) (+)

= ( and+)

(+) . (11.10)

Substituting.the.available.values.we.get

P Ec

P ( | +) =

(.99)(.001) (+)

So.far.we.do.not.know.P(+)..There.are.two.ways.a.crate.can.test.positive:.if.it.has.E. coli.and.tests.positive.or.if.it.has.no.E. coli.and.tests.positive..These.two.possibilities. are.defined.as:

. P(+).=.P(Ec.and.+).+.P(~Ec.and.+). (11.11)

Recall.from.equation.(11.7).that.P(A.and.B).=.P(A)P(B|A),.enabling.us.to.rewrite. this.as

. P(+).=.P(Ec)P(+|Ec).+.P(~Ec)P(+|~Ec). (11.12)

316 Principles of Risk Analysis: Decision Making Under Uncertainty

Substituting,.we.have:

P Ec

P Ec P Ec P Ec P ( | +) =

(.99)(.001) ( ) (+ | ) + (~ ) (+ |~ EEc)

P Ec( | +) =

(.99)(.001) (.001)(.99) + (.05)(.999)

= ..019

Although.the.prior.P(Ec).=..001,.we.now.have.an.updated.probability,.conditioned. on.the.knowledge.that.this.crate.has.tested.positive,.and.we.see.the.probability.that. it.is.actually.contaminated.is.only..019..This.is.the.posterior.of.Ec.given.the.positive. test.result..This.is.essentially.telling.us.that.if.all.crates.were.tested,.only.1.9%.of. those.that.tested.positive.would.actually.be.contaminated..That.means.that.98.1%.of. all.positive.testing.crates.would.actually.be.free.of.pathogenic.Ec..This.somewhat. surprising. result. is. because. so. few. crates. are. actually. contaminated.. Most. crates. are.free.of.the.organism,.and.these.yield.false.positives.at.a.low.rate.but.in.rather. large.numbers.

To.see.this,.suppose.we.import.1,000,000.crates..At.a.rate.of..001,.we.have.1,000. that.are.contaminated,.and.our.test.picks.up.99%.of.them..So.we.have.990.true.posi- tives..But.999,000.are.free.of.contamination.and.5%.of.them.show.up.positive:.That. is.49,950.false.positives..There.is.a.total.of.50,940.positive.tests,.but.only.990.or.1.9%. of.them.are.actually.contaminated..This.is.a.powerful.argument.against.100%.inspec- tion..We.would.be.destroying.a.lot.of.good.product.

There. is. an. entire. body. of. statistics. based. on. Bayes’. theorem.. It. is. sometimes. controversial. if. the. prior. probability. is. based. on. subjective. considerations.. This. example.began.with.a.probability.imbued.with.some.credibility..When.there.is.little. to.no.data.to.support.a.prior.probability,.some.people.become.uncomfortable.with. this.approach.

11.6 Why You Need to Know This

You.can’t.build.valid.models.unless.you.know.and.follow.the.axioms.and.propositions. of. probability.. It. is. essential. to. understand. and. obey. the. rules. of. probability. when. building.models..It.is.not.acceptable.to.simply.apply.probabilities.in.an.uninformed. fashion..It.is.not.possible.to.do.credible.probabilistic.risk.assessment.without.a.careful. knowledge.of.and.adherence.to.the.rules.of.probability.

Consider.the.levee.ownership.model.once.more,.this.time.as.seen.in.Figure 11.5.. Every.rule.discussed.here.is.employed.to.build.this.simple.model..Complementarity. must.be.observed.on.the.branches.from.a.node.and.in.the.model.endpoints..Probabilities. of.endpoints.are.determined.via.multiplication..Probabilities.of.dependent.events.rely. on. conditional. probabilities.. Probabilities. throughout. the. model,. but. especially. for. endpoints,.can.be.added.to.obtain.other.probabilities.of.interest..And.this.is.a.simple. model..A.risk.assessor.cannot.know.too.much.about.probability.

Probability Review 317

11.7 Summary and Look Forward

Probability.is.the.language.of.uncertainty..It.is.used.to.characterize.our.knowledge. uncertainty. and. to. describe. the. natural. variability. in. the. universe.. Someone. on. the. risk.assessment.team.has.to.understand.the.basic.laws,.axioms,.propositions,.and.rules. of.probability..A.brief.review.of.these.essentials.is.presented.here.

Chapter.12.is.devoted.to.helping.the.self-taught.or.trained-on-the-job.risk.assessor. to.accomplish.one.of.most.challenging.tasks.in.quantitative.risk.assessment:.choosing. the.right.probability.distribution..This.is.a.subject.that.is.not.treated.very.comprehen- sively.in.the.literature..A.nine-step.process.is.offered.and.illustrated.in.Chapter.12.. Chapter.13.follows.with.a.discussion.of.subjective.probability.elicitation.

REFERENCES FSIS.. 1998.. Department. of. Agriculture.. Salmonella. Enteritidis. Risk. Assessment. Team..

Salmonella. Enteritidis risk assessment: Shell eggs and egg products. Washington,. DC:.Food.Safety.Inspection.Service.

Whitaker,.Craig.F..1990..Letter.in.“Ask.Marilyn”.column..Parade Magazine.

Ownership

57.1429%

7.1429%

35.7143%

26.67%

16.67%

Private

Local

3.33%

Maintenance Condition

Complementarity Rule e.g., 46.67% + 53.33 = 100%

Addition Rule e.g., 3.33% + 30.0% = 33.33%

Multiplication Rule e.g., 53.33% × 56.25% = 30.0%

Conditional Probability e.g., 35.71% vs. 31.25%

46.6667%

53.3333%

Inadequate Maintenance

Adequate Maintenance

Levee Condition

Federal

Ownership

12.5%

56.25%

31.25%

6.67%

16.67%

Private

Local

30.0%Federal

FIGURE 11.5 Levee-condition.event.tree.showing.application.of.rules.of.probability.calculation.