1. 1 NA387 Lecture 6: Bayes’ Theorem, Independence Devore, Sections: 2.4 – 2.5

2. Topics Conditional Probability –Multiplication Rule –Law of Total Probability –Bayes’ Theorem Independence of Events

3. Conditional Probability The intersection of two events may be re-written from the above using the “multiplication rule”. Multiplication rule is useful for determining probability of an event that depends on other events: Conditional Probability Relationships:

4. Multiplication Rule P(A B) = P(B) * P(A|B) Or P(A)* P(B|A) Examples: Car Crash Injuries: –Event A – Getting injured in a crash (injuries: hospital visits or fatalities) –Event B – US resident involved in a car crash; P(B) = 0.01; P(A | B) = 0.30; -- Of those US residents involved in crashes, 30% get injured –P (A B) = What is the probability that a US resident will be in a car crash and get injured? P (getting injured and involved in a car crash) = ? U U

5. Conditional Probability and Tree Diagrams Suppose you produce three brands of TVs. –Event A: sell a TV (brands: A 1, A 2, A 3 ) –Event B: repair a sold TV –Suppose Selling Mix: A 1 = 50%, A 2 = 30% and A 3 = 20% –Likelihood to Repair Given Model A 1 = 25% –Likelihood to Repair Given Model A 2 = 20% –Likelihood to Repair Given Model A 3 = 10% Draw a tree diagram and determine the probability that any unit will be repaired…

6. Conditional Probability and Venn Diagrams Event B – repair sold TV; B’ – no repair Event A – sell TV Brand (A 1, A 2, A 3 ) B B’ With multiple events forming an exhaustive set, S, a general expression for total probability can be developed.

7. Law of Total Probability Given a collection of k mutually exclusive and exhaustive events (A 1,.. A k ), for any event B Law of Total Probability: –P(B) = P(B|A 1 )P(A 1 ) + … + P(B|A k )P(A k )

8. TV Example – Revisited Using the law of total probability, compute the Probability that a TV will be repaired P(B). P(A 1 ) = 50%P(B|A 1 ) = 25% P(A 2 ) = 30%P(B|A 2 ) = 20% P(A 3 ) = 20%P(B|A 3 ) = 10% Find P(B) = P(repair) = Find P(B’) = P(Not repair) =

9. Bayes’ Theorem Using the law of total probability, we may develop a useful application of the multiplication rule: –given a set of prior probabilities P(Ai) (where P(Ai) > 0) and the conditional probabilities P(B | Ai), we may compute posterior probabilities provided P(B) > 0 –Convert: P(B | Ai) to P(Aj | B) Bayes’ Theorem i – prior j - posterior

10. Prior and Posterior Probability To appreciate Bayes’ Theorem, we must understand prior and posterior probability. Prior Probability – initial probability -- e.g., P(Ai) –Often based on background data Posterior Probability – updated probability arising from new information (i.e., condition -- P(Aj | B)). –Often based on empirical evidence (i.e., sample observation) TV Example: –Prior Probability – Prob A 1 is sold ~ P(A 1 ) = 50% –Posterior Probability Example – What is the Prob (A 1 ) if you know that the TV was repaired? P(A 1 | B) = ?

11. Prior and Posterior Probabilities So, why is Bayes’ Theorem so useful? Consider the following TV results: Computing posterior probabilities allows us to update our decisions with new empirical information.

12. Rare Disease Problem (Devore Example: 2.30) Event A1: Have Disease Event A2: Do Not Have Disease Event B: Positive Test Result for Disease What % of tests yield positive results P(B)? What % of positive test results actually will indicate the person has the disease?

13. Sensitivity Analysis What happens if % of false positives for the test reduces? What happens to the test results if the disease is not so rare?

14. Independent Events If and only if (iff) A and B are independent If you have a set of mutually independent events (A 1.. A n ), then

15. Independence – Process Control Application Suppose you take samples of size 5 from a population to test for process stability. You assume your process has a normal distribution such that the mean = 1000 mm. –So, Event A i : sample i has a mean > 1000 –What is the Prob (A i )? What is the probability that three samples in a row will have a mean greater than 1000 (assume samples are independent)? If you get seven samples in a row, would you conclude the underlying mean of the population is still 1000, or would you conclude the mean has changed?

16. Independence and Reliability: Series Reliability  Reliability – probability that a system does not fail during a time interval (0 to t, or t1 to t2)  Series Reliability (one unit fails, system fails) R1R1 R2R2 i = unit in series

17. Independence and Reliability - Parallel  Parallel System Reliability (if either unit survives, system survives) R1R1 R2R2

18. Series and Parallel Problem: Twin Engine Example: –If both engines are independent and if P(Engine Survives) = 0.99, –What is the probability that both will survive? –What is the probability that at least one will survive? –What is the purpose of adding redundancy to a system (i.e., multiple components performing the same function)? –Other Example: alarm clock & wake-up service

19. 19 Systems consisting of independent components Suppose you have a complex systems of independent components. To operate, component 3 must survive and either component 1 or 2. Determine the overall system reliability (Rs) of this system if R 1 =R 2 =.95 and R 3 =.80. R1R1 R2R2 R3R3 Rs?