2. Set theory and Bayes’ Rule Sets and set operations Axioms of probability Probabilities of sets Bayes’ rule Chapter 2 of Haldar and Mahadevan’s Probability, Reliability and Statistical Methods in Engineering Design, John Wiley, 2000. Slides based in part on lecture by Prof. Joo-Ho Choi of Korea Aerospace University

3. Emergency car depot Three car sets –Sample space has 8 mutually exclusive outcomes. –Figure defines 4 other mutually exclusive events. –Random variable X is the number of good cars.

4. Set notation An event with no sample point is an impossible event or a null set –Give an example for car space? The complement of an event E is denoted as –What is the complement of X=2 ? Union of events denoted, intersection The answer to the previous question may be written as Events E 1 and E 2 both happening is written as E 1 E 2 or

5. Venn diagrams Intersection Other relationships

6. Axioms of Probability 1.P( E ) ≥ 0: probability of an event E is non-negative. 2.Probability of certain event P(S)=1. 3.Probability of two mutually exclusive events Corollaries: How do you prove them ?

7. Design office quiz P( E1∩E2 ) = ? P( E1 U E2 ) = ? –Are E1, E2 mutually exclusive ? P( E3∩E2 ) = ? P( E3 U E2 ) = ? –Are E3, E2 mutually exclusive ? E1: 80, E2: ≥90., E3=80,100

8. Conditional probability P(E 1 |E 2 ) denotes the probability that E 1 will happen given the fact that E 2 did. When joint occurrence is possible For statistically independent events Bayes’ rule

9. California example Are F,E mutually exclusive? Calculate P( E U F ) Recall that

10. Bridge damage questions Draw Venn diagram –P( F U S ) ? (0.01 is probability of S alone) –P( F U S ) when F, S are independent ? –Which probability is higher ?, i.e., more conservative ?

11. Bridge damage answer Divide space into F and its complement

12. Bayes’ rule example During an Ebola epidemic in Africa, it was estimated that a passenger arriving from Africa has 10 -5 chance of being infected. We have a test that is 99% accurate for detecting Ebola (1% of the time a positive result is false and 1% of the time a negative result is false). Calculate the probability that a person testing positive is infected.

13. Calculate with Bayes’ rule 1.For the design office example, what is the probability of a drawing taking 120 hours if it was not completed after 80? 2.For the California damage example, what is the probability that an earthquake occurred based on finding damage? 3. For the Ebola example, what is the probability of being infected when testing negative? Source: Smithsonian Institution Number: 2004-57325