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Chapter 4

Chapter 4. Probability. 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). 4.1 The Concept of Probability.

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Chapter 4

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  1. Chapter 4 Probability

  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. 4.1 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. Probability If E is an experimental outcome, then P(E) denotes the probability that E will occur and: Conditions • 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 • The probabilities of all the experimental outcomes must sum to 1

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

  8. Example 4.3 Figure 4.2

  9. 4.3 Some Elementary Probability Rules • Complement • Union • Intersection • Addition • Conditional probability

  10. Complement Figure 4.4

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

  12. Union and Intersection Diagram Figure 4.5

  13. Mutually Exclusive Figure 4.6

  14. The Addition Rule • If A and B are mutually exclusive, then the probability that A or B will occur isP(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. 4.4 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. 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. General Multiplication Rule

  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 toP(B|A) = P(B)

  19. The Multiplication Rule • The joint probability that A and B (the intersection of A and B) will occur isP(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. 4.5 Bayes’ Theorem (Optional) • S1, S2, …, Sk represents k mutually exclusive possible states of nature, one of which must be true • P(S1), P(S2), …, P(Sk) 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 Si, given the experimental outcome E, is calculated using the formula on the next slide

  21. Bayes’ Theorem Continued

  22. Example 4.18 • Oil drilling on a particular site • P(S1 = none) = .7 • P(S2 = some) = .2 • P(S3 = much) = .1 • Can perform a seismic experiment • P(high|none) = .04 • P(high|some) = .02 • P(high|much) = .96

  23. Example 4.18 Continued

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