1. ENM 207 Lecture 7

2. Example: There are three columns entitled “Art” (A),“Books” (B) and “Cinema” (C) in a new magazine. Reading habits of a randomly selected reader with respect to choose columns are:

3.  What is the probability of reading A given that they are reading B or C columns?

5. Example: Mendenhall 3.63 solution There are five suppliers and a company will choose at least 2 suppliers, therefore, we must find the number of ways to select 2, 3, 4 and 5 suppliers from the 5. At least 2 means that, a company can choose 2 or 3 or 4 or 5 suppliers from 5. The total number of obtions are

6. Example:3.65 Define the following events : (A) Part is supplied by company A (B) Part is supplied by company B (C) Part is defective From the problem, P(A)=0.8, P(B)=0.2, P(C\A)= 0.05, P(C\B)= 0.03 We know the given part is defective and we want to find The probability it come from company A And the probability it come from company B P(A\C)=?, P(B\C)=?

9. Total Probability And Bayes Theorem Partitions Total Probability And Bayes Theorem A partition of the sample space may be defined like this.

10. Total Probability Law

11. Bayes Theorem Another important result of the total probability law is known as “Bayes Theorem”:

12. Ex: 2.35 page 56 (Montgomery)

14. Example: Three machines A,B,C produce respectively 50%, 30%, 20% of the total number of items of a factory. The percentages of defective output of these machines are 3%, 4%, 5%,respectively. If an item is selected at random, find the probability that the item is defective. Let X be the event that an item is defective. Which theorem or which rule is used to solve this problem? Total probability law.

15. Another way to solve this problem: Tree Diagram

16. Example: Consider the factory in the preceeding example. Suppose an item is selected at random and is found to be defective. Find the probability that the item was produced by machine A; What is the P(A\X)? Which theorem or which law can be used to solve this problem?

17. Ex: 4.11 p 62/schaums outline series In a certain college, 25% of the students failed math, 15% of the students failed chemistry, 10% of the students failed both math and chemistry. A student is selected at random. a) If he failed chemistry what is the probability that he failed math? b) If he failed math, what is the probability that he failed chemistry? c) What is the probability that he failed math or chemistry ?

18. SYSTEM RELIABILITY System- electronic, mechanical or a combination of both –are composed of components. A component of a system is represented by a capital letter. Two system each composed of tree components A; B; C are shown below. Systems according to their components connections can be classified in two groups. Such as series and parallel.

19. Definition: If the system fails when any of the components fails, it is called a series system. If the system fails only when all of its components fail, it is called a parallel system.

20. This system is composed of five components A, B, C, D and E as shown above. Components D and E are from a two-component parallel system. This subsystem is connected in series with A, B and C.

23. Summary of system reliability

25. Example: 17.22 : First subcircuits involves comp. A,B,C in parallel.