1. LECTURE#2 Refreshing Concepts on Probability
Event Definition
Classical Vs. Experimental Definition of Probability
Independence/Dependence of Events
Mutually Exclusiveness of Events
BAYE’S Theorem in terms of Probabilities
Mr. BAYE’S Life-His Contribution
Likelihood and Probability – the difference?
Random Variables and PDFs
Trivia
Ideas for Class Projects
CEE6430: Probabilistic Methods in Hydrosciences

2. CEE6430: Probabilistic Methods in Hydrosciences
Event
(Random) Event - an outcome of a random experiment
Sample Space – all possible outcomes.
Example 1: Tossing a coin 10 times – an experiment. (tossing is
random as we cannot predict outcome ‘definitively’)
Sample space ?– Head and Tail
Getting Head 3 times ? – a random event.
Example 2: Predicting Avg. Rainrate for tomorrow (assume you are in
Key West Florida, not Arizona!)
0 mm/hr <Rainrate< infinite mm/hr (Statistical Sample Space)
0 mm/hr < Rainrate< 1500 mm/hr (Realistic/Hydrologic Sample Space)
A Random Event- Predicting an average Rainrate between 10 and 20
mm/hr.
CEE6430: Probabilistic Methods in Hydrosciences

3. CEE6430: Probabilistic Methods in Hydrosciences
Probability
Refer to Lecture Note #2 (First Page)
Probability of an Event: (Intuitively) the Ratio of the times the Event occurred to number of times N, the experiment was attempted (N goes to infinity).
Probability can sometimes be deduced logically (e.g. for a ‘fair coin’, chances of getting a Head is 50%)
Logical deduction often doesn’t work in Hydroscience
Example: Say we want to predict the probability that the river stage at Caney Fork river will exceed 15 ft during a thunderstorm. Is the problem as straight forward as tossing a coin?
CEE6430: Probabilistic Methods in Hydrosciences

4. Independence/Dependence
Mutually Independent Events – Occurrence of events bears no relation to another.
Mutually Exclusive Events – Occurrence of one event precludes the other. A binary concept.
In Nature, there are many M.E events – Rain/No-rain; Drought/Flood; Day/Night etc…
The Probability Laws for Independence and mutually-exclusiveness – look at Lecture Note#2 (page 2).
CEE6430: Probabilistic Methods in Hydrosciences

5. Conditional Probability
Conditional Probability – In engineering, many problems are formulated based on the assumption that an event has occurred. E.g. For a dam that has not failed in 50 years, what is its probability not to fail the next 25 years? We need Conditional Probability to answer this question.
Bayes’ Theorem
Derivation of Bayes’ Theorem
CEE6430: Probabilistic Methods in Hydrosciences

6. Bayes’ Theorem – Quizz#1
Relax - not due today! (Due Next Class)
A city has a uniformly gridded water distribution system (20 km wide X10 km long). Pressures and flowrates are uniform everywhere. Assume Cartesian coordinate system the bottom left corner as origin. Show on your system, the following events (loss in region):
A= (water loss in region bounded by 0<x<6km, 0<y<3km)
B=(water loss in region bounded by 4<x<10km, 2<y<6km)
Assume probability of loss proportional to affected area.
What is the ‘prior’ probability of a loss in region A?
What is the ‘prior’ probability of a loss in region B?
If there is a loss in region B, what is the probability that it is also in
region A?
CEE6430: Probabilistic Methods in Hydrosciences

7. How indebted are we to Mr. Bayes?
Mr. Bayes (actually his theorem) has facilitated combining different kinds of information and updating knowledge based on newly acquired information. It can be used to incorporate additional observations to improve apriori estimates of probability of events.
Mr. Bayes’ Theorem got acceptance/publication after his death.
In essence, Bayes's Theorem is a simple mathematical formula used for calculating conditional probabilities. Now used in almost all fields of science.
Looks like John Wayne?
CEE6430: Probabilistic Methods in Hydrosciences

8. CEE6430: Probabilistic Methods in Hydrosciences
Recap
Solving Trivia#2 (Lecture Note#2, page 4)
Certainty Vs. Uncertainty
What’s the difference Likelihood Vs. Probability?
Optional reading assignment – Chapters 1 and 2 of ‘Introduction to the Theory of Statistics’ (Mood).
CEE6430: Probabilistic Methods in Hydrosciences

9. Random Variables and PDFs (Lecture Note#2 Pages 4-7)
Cumulative Probability Distribution Function (CDF)
Probability Density Function (PDF)
Joint Probability Distribution Function
Distribution and Density?
Conditional CDF, PDF
Bayes’ Theorem in Continuous form.
CEE6430: Probabilistic Methods in Hydrosciences

10. CEE6430: Probabilistic Methods in Hydrosciences
Class Project
Discussion of Ideas.
CEE6430: Probabilistic Methods in Hydrosciences