1. “ Building Strong “ Delivering Integrated, Sustainable, Water Resources Solutions Probability Review Charles Yoe, Ph.D. Institute for Water Resources 2010

2. “ Building Strong “ Delivering Integrated, Sustainable, Water Resources Solutions 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.

3. “ Building Strong “ Delivering Integrated, Sustainable, Water Resources Solutions Probability Is Not Intuitive Pick a door. What is the probability you picked the winning door? What is the probability you did not?

4. “ Building Strong “ Delivering Integrated, Sustainable, Water Resources Solutions Suppose you picked door #2 Should you switch doors or stay with your original choice if your goal is to win the game?

5. “ Building Strong “ Delivering Integrated, Sustainable, Water Resources Solutions 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

6. “ Building Strong “ Delivering Integrated, Sustainable, Water Resources Solutions Definition Probability => Chance something will or will not happen. A historical frequency-good for things that recur A state of belief-good for things that don’t recur The math is more settled than the perspective

7. “ Building Strong “ Delivering Integrated, Sustainable, Water Resources Solutions 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?

8. “ Building Strong “ Delivering Integrated, Sustainable, Water Resources Solutions 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

9. “ Building Strong “ Delivering Integrated, Sustainable, Water Resources Solutions 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

10. “ Building Strong “ Delivering Integrated, Sustainable, Water Resources Solutions Sample Space One of these four endpoints must occur. Endpoints define the sample space.

11. “ Building Strong “ Delivering Integrated, Sustainable, Water Resources Solutions Expressing Probability Decimal = 0.6 Percentage = 60% Fraction = 6/10 = 3/5 Odds = 3:2 (x:y based on x/(x + y))

12. “ Building Strong “ Delivering Integrated, Sustainable, Water Resources Solutions Where Do We Get Probabilities Classical/analytical probabilities Empirical/frequentist probabilities Subjective probabilities

13. “ Building Strong “ Delivering Integrated, Sustainable, Water Resources Solutions 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

14. “ Building Strong “ Delivering Integrated, Sustainable, Water Resources Solutions 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

15. “ Building Strong “ Delivering Integrated, Sustainable, Water Resources Solutions Subjective Probability Evidence/experienc e based Expert opinion Useful when we deal with uncertainty of events that will occur once or that have not yet occurred

16. “ Building Strong “ Delivering Integrated, Sustainable, Water Resources Solutions Different Kinds of Events Unique County manager won’t reassign personnel >30% rock in channel bottom Structure damage in earthquake <6.2 Repeatable Frequency of flooding House has basement Pump motor lasts two years Grounding

17. “ Building Strong “ Delivering Integrated, Sustainable, Water Resources Solutions 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

18. “ Building Strong “ Delivering Integrated, Sustainable, Water Resources Solutions Levee Condition Contingency Table

19. “ Building Strong “ Delivering Integrated, Sustainable, Water Resources Solutions Marginal Probabilities Marginal Probability => Probability of a single event P(A) P(private) = 100/300 = 0.333

20. “ Building Strong “ Delivering Integrated, Sustainable, Water Resources Solutions Complementarity P(Private) = 0.333 P(Private’) = 1 – 0.333 =.667

21. “ Building Strong “ Delivering Integrated, Sustainable, Water Resources Solutions 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

22. “ Building Strong “ Delivering Integrated, Sustainable, Water Resources Solutions 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

23. “ Building Strong “ Delivering Integrated, Sustainable, Water Resources Solutions 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!

24. “ Building Strong “ Delivering Integrated, Sustainable, Water Resources Solutions Dependence & Independence A “picture” of our table. Notice how inadequate and adequate probabilities vary. They depend on ownership. Thus, ownership changes the probability. If maintenance condition was independent of ownership all probabilities would be the same.

25. “ Building Strong “ Delivering Integrated, Sustainable, Water Resources Solutions 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

26. “ Building Strong “ Delivering Integrated, Sustainable, Water Resources Solutions 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 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

27. “ Building Strong “ Delivering Integrated, Sustainable, Water Resources Solutions Important Point We often lack data and rely on subjective probabilities Subjectivists say rational belief is governed by the laws of probability so they 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

28. “ Building Strong “ Delivering Integrated, Sustainable, Water Resources Solutions 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?

29. “ Building Strong “ Delivering Integrated, Sustainable, Water Resources Solutions 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

30. “ Building Strong “ Delivering Integrated, Sustainable, Water Resources Solutions Calculation P(P|I) = P(P)P(I|P)/P(I) (100/300 * 80/100)/ (140/300) = 80/140

31. “ Building Strong “ Delivering Integrated, Sustainable, Water Resources Solutions 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? 57.1429% 80% P(P)=33.33% P(I)=46.67% But suppose we had more pointed Q’s?

32. “ Building Strong “ Delivering Integrated, Sustainable, Water Resources Solutions You Need to Know the Laws So you can construct rational models

33. “ Building Strong “ Delivering Integrated, Sustainable, Water Resources Solutions Marginal=> P(contains oil) Additive=> This times this times this time this equals this

34. “ Building Strong “ Delivering Integrated, Sustainable, Water Resources Solutions Conditional probability=> P(D>CD|Oil) Conditional probability=> P(D>CD| No Oil) Probabilities on branches conditional on what happened before

35. “ Building Strong “ Delivering Integrated, Sustainable, Water Resources Solutions 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

36. “ Building Strong “ Delivering Integrated, Sustainable, Water Resources Solutions 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.

37. “ Building Strong “ Delivering Integrated, Sustainable, Water Resources Solutions 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

38. “ Building Strong “ Delivering Integrated, Sustainable, Water Resources Solutions Charles Yoe, Ph.D. cyoe1@verizon.net Questions?