Probability Review Water Resource Risk Analysis Davis, CA 2009

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.

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. Information changes probabilities.

Definition Probability => Chance something will or will not happen. A state of belief A historical frequency The math is more settled than the perspective

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?

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

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

One of these four endpoints must occur. Endpoints define the sample space.

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

Where Do We Get Probabilities Classical/analytical probabilities Empirical/frequentist probabilities Subjective probabilities

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

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

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

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

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

Levee Condition Contingency Table

Marginal Probabilities Marginal Probability => Probability of a single event P(A) P(private) = 100/300 = 0.333

Complementarity P(Private) = P(Private’) = 1 – =.667

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 = 160/300 = 0.533

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

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

Dependence & Independence Here 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.

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)=80/1 00=0.8 P(Inadequate)= 140/300=0.4667

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 BirthProbability 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 Again reevaluate your probabilities and answer all three questions. John Tyler was inaugurated as President in Answer the same three questions. March 29, 1790

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

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 It is trivially easy with the table, 80/140 But what if there was no table?

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

Calculation P(P|I) = P(P)P(I|P)/P(I) (100/300 * 80/100)/ (140/300) = 80/140

Bayes Helps Us Answer Useful Questions 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? % 80% P(P)=33.33% P(I)=46.67% But suppose we had more pointed Q’s?

You Need to Know the Laws So you can construct rational models

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 branches conditional on what happened before

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

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 was not door 3. That meant there is a 2/3 chance it is door 1 and if you want to maximize your chance of winning you should switch.

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

Charles Yoe, Ph.D. Questions?