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The Erik Jonsson School of Engineering and Computer Science
Chapter 1pp. 1-48
William J. Pervin
The University of Texas at Dallas
Richardson, Texas 75083
The Erik Jonsson School of Engineering and Computer Science
Chapter 1
Chapter 1
Experiments, Models, and Probabilities
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
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)
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[iAi] = ΣiP[Ai], Ai mutually exclusive
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[Ac] = 1 – P[A]
P[AB] = P[A] + P[B] – P[AB]
If AB then P[A] ≤ P[B]
The Erik Jonsson School of Engineering and Computer Science
Chapter 1
For any event A and event space {Bi}i,
P[A] = Σi P[ABi]
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[AB]/P[B]
Note: P[A B] = P[A]P[B|A] = P[B]P[A|B]
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[iAi|B] = ΣiP[Ai|B],
Ai mutually exclusive
The Erik Jonsson School of Engineering and Computer Science
Chapter 1
Law of Total Probability:
For an event space {Bi}i with P[Bi] > 0,i,
P[A] = Σi P[A|Bi]P[Bi]
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(AB) = P[A|B]P[B]
The Erik Jonsson School of Engineering and Computer Science
Chapter 1
1.6 Independence:
Two: (A&B): P[AB] = P[A]P[B]
Three or more: (Ai): Every set of n-1 are independent and P[iAi] = ∏iP[Ai]
The Erik Jonsson School of Engineering and Computer Science
Chapter 1
1.7 Sequential Experiments and Tree Diagrams
1.8 Counting Methods:
The Erik Jonsson School of Engineering and Computer Science
Chapter 1
1.9 Independent Trials:
The probability of n0 failures and n1 successes in n = n0+n1 independent trials is
P[Sn0,n1] = C(n,n1)(1-p)n0pn1
= C(n,n0)(1-p)n0pn1
The Erik Jonsson School of Engineering and Computer Science
Chapter 1
1.10 Reliability Problems
1.11 MATLAB
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