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

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1.2 Applying Set Theory to Probability

Experiment (procedure and observation)

Models

Outcome; Sample Space;

Event; Event Space (NOTE: Definition)

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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

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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]

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For any event A and event space {Bi}i,

P[A] = Σi P[ABi]

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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]

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Conditional Probability Axioms:

1. P[A|B] ≥ 0

2. P[B|B] = 1

3. P[iAi|B] = ΣiP[Ai|B],

Ai mutually exclusive

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Law of Total Probability:

For an event space {Bi}i with P[Bi] > 0,i,

P[A] = Σi P[A|Bi]P[Bi]

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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]

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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]

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1.7 Sequential Experiments and Tree Diagrams

1.8 Counting Methods:

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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

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1.10 Reliability Problems

1.11 MATLAB