1. The Erik Jonsson School of Engineering and Computer Science Chapter 1 pp. 1-48 William J. Pervin The University of Texas at Dallas Richardson, Texas 75083

2. The Erik Jonsson School of Engineering and Computer Science Chapter 1 Experiments, Models, and Probabilities

3. The Erik Jonsson School of Engineering and Computer Science Chapter 1 1.1 Set Theory Set, elements, , subset(  ), union(  ), intersection(  ), complement( c ), difference(-), disjoint, mutually exclusive, collectively exhaustive. DeMorgan’s Laws

4. The Erik Jonsson School of Engineering and Computer Science Chapter 1 1.2 Applying Set Theory to Probability Experiment (procedure and observation) Models Outcome; Sample Space; Event; Event Space (NOTE: Definition)

5. The Erik Jonsson School of Engineering and Computer Science Chapter 1 1.3 Probability Axioms A probability measure P[.] is a function that maps events in the sample space S to numbers such that 1.  A, P[A] ≥ 0 2. P[S] = 1 3. P[  i A i ] = Σ i P[A i ], A i mutually exclusive

6. The Erik Jonsson School of Engineering and Computer Science Chapter 1 1.4 Some Consequences of the Axioms For any A,B: P[Ø] = 0 P[A c ] = 1 – P[A] P[A  B] = P[A] + P[B] – P[A  B] If A  B then P[A] ≤ P[B]

7. The Erik Jonsson School of Engineering and Computer Science Chapter 1 For any event A and event space {B i } i, P[A] = Σ i P[A  B i ]

8. The Erik Jonsson School of Engineering and Computer Science Chapter 1 1.5 Conditional Probability The conditional probability of the event A given the occurrence of the event B is P[A|B] = P[A  B]/P[B] Note: P[A  B] = P[A]P[B|A] = P[B]P[A|B]

9. The Erik Jonsson School of Engineering and Computer Science Chapter 1 Conditional Probability Axioms: 1. P[A|B] ≥ 0 2. P[B|B] = 1 3. P[  i A i |B] = Σ i P[A i |B], A i mutually exclusive

10. The Erik Jonsson School of Engineering and Computer Science Chapter 1 Law of Total Probability: For an event space {B i } i with P[B i ] > 0,  i, P[A] = Σ i P[A|B i ]P[B i ]

11. The Erik Jonsson School of Engineering and Computer Science Chapter 1 Bayes’ Theorem: P[B|A] = P[A|B]P[B]/P[A] Proof: P[B|A]P[A] = P(A  B) = P[A|B]P[B]

12. The Erik Jonsson School of Engineering and Computer Science Chapter 1 1.6 Independence: Two: (A&B): P[A  B] = P[A]P[B] Three or more: (A i ): Every set of n-1 are independent and P[  i A i ] = ∏ i P[A i ]

13. The Erik Jonsson School of Engineering and Computer Science Chapter 1 1.7 Sequential Experiments and Tree Diagrams 1.8 Counting Methods:

14. The Erik Jonsson School of Engineering and Computer Science Chapter 1 1.9 Independent Trials: The probability of n 0 failures and n 1 successes in n = n 0 +n 1 independent trials is P[S n 0,n 1 ] = C(n,n 1 )(1-p) n 0 p n 1 = C(n,n 0 )(1-p) n 0 p n 1

15. The Erik Jonsson School of Engineering and Computer Science Chapter 1 1.10 Reliability Problems 1.11 MATLAB