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Probability Review
Water Resource Risk AnalysisDavis, CA
2009
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Learning Objectives
• At 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.
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Probability Is Not Intuitive
Pick a door.What is the probability you picked the winning door?What is the probability you did not?
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Suppose you picked door #2
Should you switch doors or stay with your original choice if your goal is to win the game?
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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. Information changes probabilities.
http://math.ucsd.edu/~crypto/Monty/monty.html
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Definition
Probability => Chance something will or will not happen.
A state of beliefA historical frequencyThe math is more settled than the perspective
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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?
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Probability
• Human construct to understand chance events and uncertainty
• A number between 0 and 1
• 0 is impossible
• 1 is certain
• 0.5 is the most uncertain of all
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Probability
• One of our identified possibilities has to occur or we have not identified all the possibilities
• Something has to happen
• The sum of the probability of all our possibilities equals one
• Probability of all branches from a node =1
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80.0% 24.0%
30.0% Cracking
20.0% 6.0%
Liquifiable Soil
60.0% 42.0%
70.0% Cracking
0
40.0% 28.0%
Earthquake Model
Yes
Yes
No
No
Yes
No
One of these four endpoints must occur.Endpoints define the sample space.
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Expressing Probability
• Decimal = 0.6
• Percentage = 60%
• Fraction = 6/10 = 3/5
• Odds = 3:2 (x:y based on x/(x + y))
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Where Do We Get Probabilities
• Classical/analytical probabilities
• Empirical/frequentist probabilities
• Subjective probabilities
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Analytical Probabilities
• Equally likely events (1/n)
– Chance of a 1 on a die = 1/6
– Chance of head on coin toss = ½
• Combinatorics
– Factorial rule of counting
– Permutations (n!/(n - r)!)
– Combinations (n!/(r!(n - r)!)
• Probability of a 7
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Empirical Probabilities• Observation-how many times the event of interest
happens out of the number of times it could have happened
• P(light red)
• Useful when process of
interest is repeated many
times under same
circumstances
• Relative frequency is
approximation of true probability
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Subjective Probability
• Evidence/experience based
• Expert opinion• Useful when we deal
with uncertainty of events that will occur once or that have not yet occurred
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Repeatable and Unique
• Frequency of flooding
• House has basement
• Pump motor lasts two years
• Grounding
• County manager won’t reassign personnel
• >30% rock in channel bottom
• Structure damage in earthquake <6.2
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Working With Probabilities
• If it was that simple anyone could do it
• It ain’t 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
• Most of us do not naturally assess probabilities well
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Levee ConditionContingency Table
Inadequate Maintenance
Adequate Maintenance Total
Private 80 20 100Locally Constructed 50 50 100Federal Construction 10 90 100Total 140 160 300
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Marginal Probabilities
• Marginal Probability => Probability of a single event P(A)
• P(private) = 100/300 = 0.333
Inadequate Maintenance
Adequate Maintenance Total
Private 80 20 100Locally Constructed 50 50 100Federal Construction 10 90 100Total 140 160 300
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Complementarity
• P(Private) = 0.333• P(Private’) = 1 –
0.333 = .667Inadequate Maintenance
Adequate Maintenance Total
Private 80 20 100Locally Constructed 50 50 100Federal Construction 10 90 100Total 140 160 300
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General Rule of Addition
• For two events A & B• P(A or B) = P(A) + P(B) -
P(A and B)
• P(Private or Inadequate)=P(P)+P(I)-P(P and I)
• 100/300 + 140/300 -80/300 = 160/300 = 0.533
Inadequate Maintenance
Adequate Maintenance Total
Private 80 20 100Locally Constructed 50 50 100Federal Construction 10 90 100Total 140 160 300
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Addition Rules
• For mutually exclusive events P(A and B) is zero
• P(A and B) is a joint probability
• P(Private and Local) = 0
• For events not mutually exclusive P(A and B) can be non-zero and positive
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Multiplication Rules of Probability
• Independent 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)
• Engineering involves a lot of dependence
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Dependence & IndependenceHere is a “picture” of our table.
Notice how inadequate and adequate probabilities vary. They depend on the ownership. Thus, ownership changes the probability.
If maintenance condition was independent of ownership all probabilities would be the same.
20.0% 6.6667%
20
33.3333% Chance
100
80.0% 26.6667%
300 80
Chance
50.0% 16.6667%
50
33.3333% Chance
100
50.0% 16.6667%
50
90.0% 30.0%
90
33.3333% Chance
100
10.0% 3.3333%
10
Bayes Theorem Example1
Private
Adequate Maintenance
Inadequate Maintenance
Locally constructed
Adequate Maintenance
Inadequate Maintenance
Federal construction
Adequate Maintenance
Inadequate Maintenance
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Conditional Probabilities
• Information can change probabilities
• P(A|B) is not same as P(A) if A and B are dependent
• P(A|B) = P(A and B)/P(B)• P(Inadequate|
Private)=80/100=0.8• P(Inadequate)=
140/300=0.4667
Inadequate Maintenance
Adequate Maintenance Total
Private 80 20 100Locally Constructed 50 50 100Federal Construction 10 90 100Total 140 160 300
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Information Changes Probabilities
• John Tyler’s birth year• Which of the four statements
do you believe is most likely? • Which of the statements do
you believe is least likely? • Give probabilities to the four
events that are consistent with the answers you made above.
Year of Birth Probability
no later than 1750
between 1751 and 1775
between 1776 and 1800
after 1800
John Tyler was the tenth president of the United States. Use this information to reevaluate the probabilities you made above. Before you assign probabilities, answer the first two questions stated above
George Washington, the first President of the United States, was born in 1732. Again reevaluate your probabilities and answer all three questions. John Tyler was inaugurated as President in 1841. Answer the same three questions.
March 29, 1790
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Important Point
• We often lack data and rely on subjective probabilities
• Subjectivists, maintain rational belief is governed by the laws of probability and lean heavily on conditional probabilities in their theories of evidence and their models of empirical learning. Bayes' Theorem is a simple mathematical formula used for calculating conditional probabilities
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A Question
• Suppose a levee is inspected and is found to be inadequately maintained
• What is the probability it is a private levee?– This flips the previous
example
Inadequate Maintenance
Adequate Maintenance Total
Private 80 20 100Locally Constructed 50 50 100Federal Construction 10 90 100Total 140 160 300
It is trivially easy with the table, 80/140
But what if there was no table?
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Bayes Theorem for Calculating Conditional Probabilities
• P(A|B) = P(A)P(B|A)/P(B)
• Translated: P(P|I) = P(P)P(I|P)/P(I)
• In words, the probability a levee is private given it is inadequate equals the probability it is private times the probability it is inadequate given it is private all divided by the probability it is inadequate
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Calculation
• P(P|I) = P(P)P(I|P)/P(I)
• (100/300 * 80/100)/ (140/300) =
• 80/140
Inadequate Maintenance
Adequate Maintenance Total
Private 80 20 100Locally Constructed 50 50 100Federal Construction 10 90 100Total 140 160 300
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Bayes Helps Us Answer Useful Questions
20.0% 6.6667%
20
33.3333% Chance
100
80.0% 26.6667%
300 80
Chance
50.0% 16.6667%
50
33.3333% Chance
100
50.0% 16.6667%
50
90.0% 30.0%
90
33.3333% Chance
100
10.0% 3.3333%
10
Bayes Theorem Example1
Private
Adequate Maintenance
Inadequate Maintenance
Locally constructed
Adequate Maintenance
Inadequate Maintenance
Federal construction
Adequate Maintenance
Inadequate Maintenance
57.1429% 26.6667%
80
35.7143% 16.6667%
50
46.6667% Private
140
7.1429% 3.3333%
300 10
Inadequate Maintenance
12.5% 6.6667%
20
31.25% 16.6667%
50
53.3333% Private
160
56.25% 30.0%
90
Bayes Theorem Example2
Yes
Locally constructed
Federal construction
Private
No
Private
Locally constructed
Federal construction
1. We have an inadequate levee, what’s the probability it’s private? 2. We have a private levee, what’s the probability it is inadequate?
57.1429% 80%
P(P)=33.33% P(I)=46.67%But suppose we had more pointed Q’s?
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You Need to Know the Laws
• So you can construct rational models
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Marginal=>P(contains oil)
Additive=>This times this times
this time this equals this
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Conditional probability=>P(D>CD|Oil)
Conditional probability=>P(D>CD| No Oil)
Probabilities on branchesconditional on whathappened before
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Conclusions
• Risk assessors must understand probability to do good assessments
• Risk managers must understand probability to make good decisions
• Risk communicators must understand probability to communicate effectively with those who do not
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It’s True
Your original choice had a 1/3 chance of winning and there was a 2/3 chance it was the doors you did not pick. I gave you some information I told you it wasnot door 3. That meant there is a 2/3 chance it is door 1 and if you want to maximizeyour chance of winning you should switch.
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Take Away Points
• Probability is human construct, number [0,1]
• Estimates are analytical, frequency, subjective
• There are laws that govern probability calculations but philosophies differ
• It is language of variability and uncertainty
• You need to have people who know probability to do risk analysis