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Golay, Michael, 22.38 Probability And Its Applications To Reliability, Quality Control, And Risk Assessment, Fall 2005. (Massachusetts Institute of Technology: MIT OpenCourseWare), (Accessed 07 Jul, 2010). License: Creative Commons BY-NC-SA

Probability And Its Applications To Reliability, Quality Control, And Risk Assessment

Fall 2005

Fault tree for external tank and solid rocket booster.
A fault tree from the NASA Accident Analysis Team in Report of the Presidential Commission on the Space Shuttle Challenger Accident. (Courtesy of NASA.)

Course Highlights

This course features a complete set of homework assignments and extensive lecture notes.

Course Description

This course covers interpretations of the concept of probability. Topics include basic probability rules; random variables and distribution functions; functions of random variables; and applications to quality control and the reliability assessment of mechanical/electrical components, as well as simple structures and redundant systems. The course also considers elements of statistics; Bayesian methods in engineering; methods for reliability and risk assessment of complex systems (event-tree and fault-tree analysis, common-cause failures, human reliability models); uncertainty propagation in complex systems (Monte Carlo methods, Latin Hypercube Sampling); and an introduction to Markov models. Examples and applications are drawn from nuclear and other industries, waste repositories, and mechanical systems.


Material Orientation

Probabilistic risk and reliability analyses provide methods for integrating the contribution of all contributors to ultimate system performance. They also permit the effects of uncertainties to be reflected directly in the understanding of system outcomes. In order to do this correctly, a good understanding of probabilistic concepts and methods is essential. This subject covers all of these topics.


 Ang, A. H-S., and W. H. Tang. Probability Concepts in Engineering Planning and Design, vol. 1, Basic Principles. New York, NY: John Wiley & Sons, 1975. ISBN: 9780471032007.


Homework is assigned approximately weekly, due two class sessions after the date of assignment.


Homework 20%
Exam 1 20%
Exam 2 20%
Final Exam 40%


I. The Logic of Certainty
1-2 I.1 Events and Boolean Operations

I.2 Event Sequence Identification (Failure Modes and Effects Analysis; Hazard and Operability Analysis; Fault Tree Analysis; Event Tree Analysis)

I.3 Coherent Structure Functions

I.4 Minimal Cut (Path) Sets
II. Probability
3-4 II.1 Definitions and Interpretations (Axiomatic; Subjectivistic; Frequentistic)

II.2 Basic Rules

II.3 Theorem of Total Probability

II.4 Bayes' Theorem
Problem set 1 due
III. Random Variables and Distribution Functions
5-6 III.1 Discrete and Continuous Random Variables

III.2 Cumulative Distribution Functions

III.3 Probability Mass and Density Functions

III.4 Moments

III.5 Failure Models and Reliability

III.6 Failure Rates
IV. Useful Probability Distributions
7-8 IV.1 Bernoulli Trials and the Binomial Distribution

IV.2 The Poisson Distribution

IV.3 The Exponential Distribution

IV.4 The Normal and Lognormal Distributions

IV.5 The Concept of Correlation
Problem set 2 due
V. Multivariate Distributions
9-10 V.1 Joint and Conditional Distribution Functions

V.2 Moments

V.3 The Multivariate Normal and Lognormal Distributions
Problem set 3 due
  Exam 1  
VI. Functions of Random Variables
11-12 VI.1 Single Random Variable

VI.2 Multiple Random Variables

VI.3 Moments of Functions of Random Variables

VI.4 Approximate Evaluation of the Mean and Variance of a Function

VI.5 Analytical Results for the Normal and Lognormal Distributions
Problem set 4 due
VII. Statistical Methods
13-14 VII.1 Student's t-distribution

VII.2 Chi-Squared Distribution

VII.3 Hypothesis Testing
Problem set 5 due
VIII. Elements of Statistics
15 VIII.1 Random Samples

VIII.2 Method of Moments

VIII.3 Method of Maximum Likelihood

VIII.4 Probability Plotting
IX. Applications to Reliability
16 IX.1 Simple Logical Configurations (Series; Parallel; Standby Redundancy)

IX.2 Complex Systems

IX.3 Stress-Strength Interference Theory

IX.4 Modeling of Loads and Strength

IX.5 Reliability-Based Design

IX.6 Elementary Markov Models
Problem set 6 due
X. Bayesian Statistics
17 X.1 Bayes' Theorem and Inference

X.2 Conjugate Families of Distributions

X.3 Comparison with Frequentist Statistics

X.4 Elicitation and Utilization of Expert Opinions
  Exam 2  
XI. Monte Carlo Simulation
18 XI.1 The Concept of Simulation

XI.2 Generation of Random Numbers

XI.3 Generation of Jointly Distributed Random Numbers

XI.4 Latin Hypercube Sampling

XI.5 Examples from Risk and Reliability Assessment
Problem set 7 due
XII. Probabilistic Risk Assessment of Complex Systems
19-23 XII.1 Risk Curves and Accident Scenario Identification

XII.2 Event-Tree and Fault-Tree Analysis

XII.3 Unavailability Theory of Repairable and Periodically Tested Systems

XII.4 Dependent (Common-Cause) Failures

XII.5 Human Reliability Models

XII.6 Component Importance

XII.7 Examples from Risk Assessments for Nuclear Reactors, Chemical Process Systems, and Waste Repositories
Problem set 8 due

Problem set 9 due

Problem set 10 due
  Final Exam   Tell A Friend