1. Copyright © 2010 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill/Irwin Chapter 4 Probability

2. 4-2 Chapter Outline 4.1 The Concept of Probability 4.2 Sample Spaces and Events 4.3 Some Elementary Probability Rules 4.4 Conditional Probability and Independence 4.5 Bayes’ Theorem (Optional)

3. 4-3 4.1 The Concept of Probability An experiment is any process of observation with an uncertain outcome The possible outcomes for an experiment are called the experimental outcomes Probability is a measure of the chance that an experimental outcome will occur when an experiment is carried out

4. 4-4 Probability If E is an experimental outcome, then P(E) denotes the probability that E will occur and: Conditions 1. 0  P(E)  1 such that: If E can never occur, then P(E) = 0 If E is certain to occur, then P(E) = 1 2. The probabilities of all the experimental outcomes must sum to 1

5. 4-5 Assigning Probabilities to Experimental Outcomes Classical Method For equally likely outcomes Long-run relative frequency In the long run Subjective Assessment based on experience, expertise or intuition

6. 4-6 4.2 Sample Spaces and Events Sample Space: The set of all possible experimental outcomes Sample Space Outcomes: The experimental outcomes in the sample space Event: A set of sample space outcomes

7. 4-7 Events If A is an event, then 0 ≤ P(A) ≤ 1 1.If an event never occurs, then the probability of this event equals 0 2.If an event is certain to occur, then the provability of this event equals 1

8. 4-8 Example 4.3 Figure 4.2

9. 4-9 4.3 Some Elementary Probability Rules 1.Complement 2.Union 3.Intersection 4.Addition 5.Conditional probability

10. 4-10 Complement Figure 4.4

11. 4-11 Union and Intersection 1.The intersection of A and B are elementary events that belong to both A and B Written as A ∩ B 2.The union of A and B are elementary events that belong to either A or B or both Written as A  B

12. 4-12 Union and Intersection Diagram Figure 4.5

13. 4-13 Mutually Exclusive Figure 4.6

14. 4-14 The Addition Rule If A and B are mutually exclusive, then the probability that A or B will occur is P(A  B) = P(A) + P(B) If A and B are not mutually exclusive: P(A  B) = P(A) + P(B) – P(A ∩ B) where P(A ∩ B) is the joint probability of A and B both occurring together

15. 4-15 4.4 Conditional Probability and Independence The probability of an event A, given that the event B has occurred, is called the conditional probability of A given B Denoted as P(A|B) Further, P(A|B) = P(A ∩ B) / P(B) P(B) ≠ 0

16. 4-16 Interpretation Restrict sample space to just event B The conditional probability P(A|B) is the chance of event A occurring in this new sample space In other words, if B occurred, then what is the chance of A occurring

17. 4-17 General Multiplication Rule

18. 4-18 Independence of Events Two events A and B are said to be independent if and only if: P(A|B) = P(A) This is equivalently to P(B|A) = P(B)

19. 4-19 The Multiplication Rule The joint probability that A and B (the intersection of A and B) will occur is P(A ∩ B) = P(A) P(B|A) = P(B) P(A|B) If A and B are independent, then the probability that A and B will occur is: P(A ∩ B) = P(A) P(B) = P(B) P(A)

20. 4-20 4.5 Bayes’ Theorem (Optional) S 1, S 2, …, S k represents k mutually exclusive possible states of nature, one of which must be true P(S 1 ), P(S 2 ), …, P(S k ) represents the prior probabilities of the k possible states of nature If E is a particular outcome of an experiment designed to determine which is the true state of nature, then the posterior (or revised) probability of a state S i, given the experimental outcome E, is calculated using the formula on the next slide

21. 4-21 Bayes’ Theorem Continued

22. 4-22 Example 4.18 Oil drilling on a particular site P(S 1 = none) =.7 P(S 2 = some) =.2 P(S 3 = much) =.1 Can perform a seismic experiment P(high|none) =.04 P(high|some) =.02 P(high|much) =.96

23. 4-23 Example 4.18 Continued