1 CPGomes – AEM 03 Electronic Markets Combinatorial Auctions Notes by Prof. Carla Gomes

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Probability ReviewRisk Analysis for Water Resources Planning and ManagementInstitute for Water Resources2008

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

Suppose you picked door #2Should you switch doors or stay with your original choice if your goal is to win the game?

Its TrueYour original choice had a 1/3 chance of winning.It still does. Switching now has the 2/3 chance of winning.See exercise 67

DefinitionProbability => Chance something will or will not happen.

A state of belief.

Whats 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 10 is impossible1 is certain0.5 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 oneProbability 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 = 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 happenedP(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 aint 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 & BP(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 RulesFor mutually exclusive events P(A and B) is zeroP(A and B) is a joint probabilityP(Towboat and Deep) = 0

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

Multiplication Rules of ProbabilityIndependent EventsP(A and B) = P(A) x P(B)Dependent EventsP(A and B) depends on nature of the dependencyGeneral rule of multiplicationP(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 timesthis 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 & UncertaintyAddresses likelihood of chance eventsAllows us to bound what we dont knowKnow nothingKnow littleSome theory

ProbabilityLanguage of Random VariablesConstantVariablesSome things vary predictablySome things vary unpredictablyRandom variablesIt can be something known but not known by us

Types of Random VariablesDiscreteGiven any interval on a number line only some of the values in that interval are possibleContinuousGiven 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 discreteWeight of grain exported (tons)Levee length (yards)Effectively continuousDollar amounts

Populations & SamplesPopulationAll of the things we are interested inNumerical characteristics called parametersThey are constantsSamplePart of a populationMany kinds of sample, many ways to take oneNumerical characteristics called statistics (sample statistics)They are variablesPopulationSampleWed 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

*A constant is a numerical characteristic that does not change. There are many important constants in life numerical and otherwise. Pi, Avogadros number, inches in a foot, pounds in a ton, and so on.

Ask: Tell me something that is constant. Your name, number of eyes, number of children.

Is there anything relative about a constant? Time? Place?

What varies predictably? Crowds at stadiums--if actual number is not important we sure know there will be many people at a game. Maybe who will be home for dinner at your house.

Our level of resolution is important in determining what is predictable. Big or little may be predictable, but the exact number is not.

*Go to Excel exercise.

Ask questions and enter data from students.*From a small number to a huge number

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