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

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. See exercise 67

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?

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

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)!)

Empirical Probabilities Observation How 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 Probability Evidence/experience based Expert opinion

Working With Probabilities If it were 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

Contingency Table

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

Complemntarity P(Towboat) = P(Towboat’) = 1 – =.9914

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 = 37125/37288 =

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 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)

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(Casualty|Deep)=29/22 07= P(Casualty)= 433/37288=0.0116

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

Probability --Language of Variability & Uncertainty Addresses likelihood of chance events Allows us to bound what we don’t know * Know nothing * Know little * Some theory

Probability—Language of Random Variables Constant Variables Some things vary predictably Some things vary unpredictably Random variables It can be something known but not known by us

Types of Random Variables Discrete Given any interval on a number line only some of the values in that interval are possible Continuous Given any interval on a number line any value in that interval is possible

Discrete Variables Barges in a tow Houses in floodplain People at a meeting Results of a diagnostic test Casualties per year Relocations and acquisitions

Continuous Variables Average number of barges per tow Weight of an adult striped bass Sensitivity or specificity of a diagnostic test Transit time Expected annual damages Duration of a storm Shoreline eroded Sediment loads

Effectively One or the Other Effectively discrete Weight of grain exported (tons) Levee length (yards) Effectively continuous Dollar amounts

Populations & Samples Population All of the things we are interested in Numerical characteristics called parameters They are constants Sample Part of a population Many kinds of sample, many ways to take one Numerical characteristics called statistics (sample statistics) They are variables Population Sample We’d really like our samples to be representative of the population from which they are taken.

Numerical Characteristics Minimum Maximum Fifth Second largest Mean Mode Standard deviation Range Variance 27 th percentile Interquartile range And so on

Take Away Points Probability is human construct, number [0,1] Estimates are analytical, frequency, subjective There are laws that govern calculations It is language of variability and uncertainty Learning about populations is important function of probability