Probability Review Risk Analysis for Water Resources Planning and Management Institute for Water Resources 2008.

Probability Review

Risk Analysis for Water Resources Planning and Management

Institute for Water Resources

2008

Learning ObjectivesAt the end of this session participants will understand:

The definition of probability.Where probabilities come from.There are basic laws of probability.The difference between discrete and continuous random variables.The significance of learning about populations.

Probability Is Not Intuitive

Pick a door.What is the probability you picked the winning door?What is the probability you did not?

Suppose you picked door #2

Should you switch doors or stay with your original choice if your goal is to win the game?

It’s True

Your original choice had a 1/3 chance of winning.It still does. Switching now has the 2/3 chance of winning.See exe

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Definition

Probability => Chance something will or will not happen.

A state of belief.

What’s the probability of….A damaging flood this year? A 100% increase in steel prices?A valve failure at lock in your District?A collision between two vessels?A lock stall?More than 30% rock in the channel bottom?Levee overtopping?Gas > $5/gal?

ProbabilityHuman construct to understand chance events and uncertaintyA number between 0 and 1* 0 is impossible* 1 is certain* 0.5 is the most uncertain of all

ProbabilityOne of our identified possibilities has to occur or we have not identified all the possibilitiesSomething has to happenThe sum of the probability of all our possibilities equals one

Probability of all branches from a node =1

Expressing ProbabilityDecimal = 0.6Percentage = 60%Fraction = 6/10 = 3/5Odds = 3:2 (x:y based on x/(x + y))

Where Do We Get ProbabilitiesClassical/analytical probabilitiesEmpirical/frequentist probabilitiesSubjective probabilities

Analytical ProbabilitiesEqually likely events (1/n)

Chance of a 1 on a die = 1/6Chance of head on coin toss = ½

CombinatoricsFactorial rule of countingPermutations (n!/(n - r)!)Combinations (n!/(r!(n - r)!)

Empirical ProbabilitiesObservationHow many times the event of interest happens out of the number of times it could have happened

P(light near your house is red when you drive through)

Subjective ProbabilityEvidence/experience basedExpert opinion

Working With ProbabilitiesIf it were that simple anyone could do itIt ain’t that simpleThere are rules and theories that govern our use of probabilitiesEstimating probabilities of real situations requires us to think about complex events

Contingency Table

Casualties Safe Transits

Total

Towboats 270 31,256 31,526

Deep Draft 29 2,178 2,207

Recreation Craft

134 3,421 3,555

Total 433 36,855 37,288

Marginal ProbabilitiesMarginal Probability => Probability of a single event P(A)P(Towboat Casualty) = 270/31526=0.0086

Casualties Safe Transits

Total

Towboats 270 31,256 31,526

Deep Draft 29 2,178 2,207

Recreation Craft

134 3,421 3,555

Total 433 36,855 37,288

ComplemntarityP(Towboat) = 0.0086P(Towboat’) = 1 – 0.0086 = .9914

Casualties Safe Transits

Total

Towboats 270 31,256 31,526

Deep Draft 29 2,178 2,207

Recreation Craft

134 3,421 3,555

Total 433 36,855 37,288

General Rule of Addition For two events A & B

* P(A or B) = P(A) + P(B) - P(A and B)

* P(Towboat or Safe)=P(T)+P(S)-P(T and S)

* 31526/37288 + 36855/37288 -31256/37288 = 37125/37288 = 0.9956

Casualties Safe Transits

Total

Towboats 270 31,256 31,526

Deep Draft 29 2,178 2,207

Recreation Craft

134 3,421 3,555

Total 433 36,855 37,288

Addition Rules* For mutually exclusive events P(A and B) is zero

* P(A and B) is a joint probability* P(Towboat and Deep) = 0

* For events not mutually exclusive P(A and B) can be non-zero and positive

Multiplication Rules of ProbabilityIndependent Events* P(A and B) = P(A) x P(B)

* Dependent Events* P(A and B) depends on nature of the dependency* General rule of multiplication

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

Conditional ProbabilitiesInformation can change probabilitiesP(A|B) is not same as P(A) if A and B are dependentP(A|B) = P(A and B)/P(B)P(Casualty|Deep)=29/2207=0.0131P(Casualty)= 433/37288=0.0116

Casualties Safe Transits

Total

Towboats 270 31,256 31,526

Deep Draft 29 2,178 2,207

Recreation Craft

134 3,421 3,555

Total 433 36,855 37,288

Marginal=>P(contains oil)

Additive=>This times this times

this time this equals this

Conditional probability=>P(D>CD|Oil)

Conditional probability=>P(D>CD| No Oil)

Probabilities on branchesconditional on whathappened before

Probability --Language of Variability & Uncertainty

Addresses likelihood of chance eventsAllows us to bound what we don’t know* Know nothing* Know little* Some theory

Probability—Language of Random Variables

ConstantVariables

Some things vary predictablySome things vary unpredictably

Random variablesIt can be something known but not known by us

Types of Random VariablesDiscrete

Given any interval on a number line only some of the values in that interval are possible

ContinuousGiven any interval on a number line any value in that interval is possible

Discrete VariablesBarges in a tow Houses in floodplainPeople at a meetingResults of a diagnostic testCasualties per yearRelocations and acquisitions

Continuous VariablesAverage number of barges per towWeight of an adult striped bassSensitivity or specificity of a diagnostic testTransit timeExpected annual damagesDuration of a stormShoreline erodedSediment loads

Effectively One or the OtherEffectively discrete

Weight of grain exported (tons)Levee length (yards)

Effectively continuousDollar amounts

Populations & SamplesPopulationAll of the things we are interested inNumerical characteristics called parametersThey are constants

SamplePart of a populationMany kinds of sample, many ways to take oneNumerical characteristics called statistics (sample statistics)They are variablesPopulation

Sample

We’d really like our samples to be representative of the population from which they are taken.

Numerical CharacteristicsMinimumMaximumFifthSecond largestMeanMode

Standard deviationRangeVariance27th percentileInterquartile rangeAnd so on

Take Away PointsProbability is human construct, number [0,1]Estimates are analytical, frequency, subjectiveThere are laws that govern calculationsIt is language of variability and uncertaintyLearning about populations is important function of probability