Chapter 18 Section 8.5 Fault Trees Analysis (FTA) Don t get caught out on a limb of your fault tree.
|
|
- Valerie Banks
- 6 years ago
- Views:
Transcription
1 Chapter 18 Section 8.5 Fault Trees Analysis (FTA) Don t get caught out on a limb of your fault tree. C. Ebeling, Intro to Reliability & Maintainability Engineering, 2 nd ed. Waveland Press, Inc. Copyright 2010
2 Characteristics Graphical design technique Alternative to reliability block diagrams Broader in scope Perspective on faults rather than reliability Model events rather than components Faults include failures Focus on a catastrophic event (top event) Top-down deductive analysis FTA 2
3 The Four Steps to a FTA (1) Define the system, its boundaries, and the top event, (2) Construct the fault tree representing symbolically the system and its relevant events, (3) Perform a qualitative evaluation by identifying those combinations of events which will cause the top event, (4) Perform a quantitative evaluation by assigning failure probabilities or unavailabilities to the basic events and computing the probability of the top event. FTA 3
4 Fault Tree Symbols gate - a logic gate where an output event occurs only when all the input events have occurred. gate - a logic gate where an output event occurs if at least one of the input events have occurred. Resultant event - a fault event resulting from the logical combination of other fault events and usually an output to a logic gate. Basic event - an elementary event representing a basic fault or component failure. Incomplete event - an event that has not been fully developed because of lack of knowledge or its unimportance. FTA 4
5 General Structure of a Fault Tree Top Event System Failure Resultant Events / gates Basic Events FTA 5
6 Example of / Gates Tank Ruptures Overpressure Overpressure Wall Fatigue Failure Excessive Temperature Relief Valve Fails (a) (b) FTA 6
7 Example of / Gates Tank Ruptures Overpressure Wall Fatigue Failure Excessive Temperature Relief Valve Fails (c) FTA 7
8 Alarm System Example meter observer switch alarm backup power power source automatic sensor FTA 8
9 Alarm System Fault Tree T- Alarm Failure A-Power Failure B-Sensor Failure C-Alarm Failure D Secondary alarm failure E-Primary power fails F-Backup fails G-Manual Alarm Failure H-Auto Sensor Fails I-Human Error J-Switch fails K-Meter Fails FTA 9
10 Boolean representation of Top Event T- Alarm Failure A-Power Failure B-Sensor Failure C-Alarm Failure D-Secondary alarm failure E-Primary power fails F-Backup fails G-Manual Alarm Failure H-Auto Sensor Fails T = A B C D = ( E F ) ( G H ) C D I-Human Error J-Switch fails K-Meter Fails = ( E F ) [ (I J K ) H ] C D FTA 10
11 Example A Top Event Event B Event C Event D T = A B = A ( C D ) = A [ (E F ) ( E A )] = A [ E (F A)] = A E since A (F A) = A E F E A FTA 11
12 Example Equivalent Fault Tree T = A U E Top Event Event A Event E FTA 12
13 Minimal Cut Sets A cut set is a collection of basic events which will cause the top event. A minimal cut set is one with no unnecessary events. That is, all the events within the cut set must occur to cause the top event M1 Top Event M2 MK E1 E2 En T = M 1 M 2... M k where M i = E 1 E 2... E ni and E i are basic events. FTA 13
14 Example T- Alarm Failure A-Power Failure E-Primary power fails F-Backup fails I-Human Error B-Sensor Failure G-Manual Alarm Failure J-Switch fails H-Auto Sensor Fails C-Alarm Failure K-Meter Fails D-Secondary alarm failure A B C E,F G,H C E,F I,H J,H K,H C D D D T = (E F) (I H) (J H) (K H) C D FTA 14
15 Example Cut Sets A A A A A B C, D E, D E, E E F, D E, A F, E F, A A Top Event Event B Event C Event D E F E A since E E = E, E A A, F E E, and F A A. Therefore T = A E FTA 15
16 Quantitative Analysis If cut sets are mutually exclusive: If P(M) < 10-3 P(T) = P( M 1 U M 2 U U M k ) = P(M 1 ) + P(M 2 ) P(M k ) If not: P(T) = P(A E) = P(A) + P(E) - P(A E). P(T) = P(M 1 ) +P(M 2 ) - P(M 1 ) P(M 2 ) + etc. P(M i ) = P(E 1 E 2... E ni ) = P(E 1 ) P(E 2 )... P(E n1 ) Then P(M 1 ) P(M 2 ) < 10-6 if independent FTA 16
17 Example P(T) = P{ ( E F ) [ (I J K ) H ] C D } P( E F) + P [ (I J K ) H ] + P(C) +P(D) P(E) P(F) + [ P(I) + P(J) +P(K) ] P(H) + P(C) + P(D) If each basic event has a probability of.01, then P(T) (.01) 2 + ( ) (.01) =.0204 FTA 17
18 One Last Example Top Event T = A B C = (D E) (H I) (L M N) D A B C E H I L M N = [D (F G) [H (J k)] [L (P Q) N] = [D (F G) [H (J k)] [L (R S Q) N] F G J K Q P R S P(T) P(D) [P(F) + P(G)] + P(H) + P(J)P(K) + P(L) [ P(R) + P(S) + P(Q)] P(N) FTA 19
19 The Cut Sets Top Event #1 #2 #3 #4 A B C A D,E D,F D,F D,G D F E G H I L M N J K Q P B C H I L,M,N H J,K L,P,N L,Q,N H J,K L,R,N L,S,N L,Q,N R S FTA 20
12 - The Tie Set Method
12 - The Tie Set Method Definitions: A tie set V is a set of components whose success results in system success, i.e. the presence of all components in any tie set connects the input to the output in the
More informationRisk Analysis of Highly-integrated Systems
Risk Analysis of Highly-integrated Systems RA II: Methods (FTA, ETA) Fault Tree Analysis (FTA) Problem description It is not possible to analyse complicated, highly-reliable or novel systems as black box
More informationFailures in Process Industries
Fault Tree Analysis Failures in Process Industries Single Component Failure Data for failure rates are compiled by industry Single component or single action Multiple Component Failure Failures resulting
More informationOn Qualitative Analysis of Fault Trees Using Structurally Persistent Nets
On Qualitative Analysis of Fault Trees Using Structurally Persistent Nets Ricardo J. Rodríguez rj.rodriguez@unileon.es Research Institute of Applied Sciences in Cybersecurity University of León, Spain
More informationNuclear reliability: system reliabilty
Nuclear reliability: system reliabilty Dr. Richard E. Turner (ret26@cam.ac.uk) December 3, 203 Goal of these two lectures failures are inevitable: need methods for characterising and quantifying them LAST
More informationPROBABILISTIC AND POSSIBILISTIC FAULT TREE ANALYSIS
PROBABILISTIC AD POSSIBILISTIC FAULT TREE AALYSIS M. Ragheb 12/28/2017 ITRODUCTIO In the design of nuclear power plants, it is important to analyze the probable and possible mechanisms of failure. Fault
More informationOverview of Control System Design
Overview of Control System Design General Requirements 1. Safety. It is imperative that industrial plants operate safely so as to promote the well-being of people and equipment within the plant and in
More information200 2 10 17 5 5 10 18 12 1 10 19 960 1 10 21 Deductive Logic Probability Theory π X X X X = X X = x f z x Physics f Sensor z x f f z P(X = x) X x X x P(X = x) = 1 /6 P(X = x) P(x) F(x) F(x) =
More informationModule No. # 03 Lecture No. # 11 Probabilistic risk analysis
Health, Safety and Environmental Management in Petroleum and offshore Engineering Prof. Dr. Srinivasan Chandrasekaran Department of Ocean Engineering Indian Institute of Technology, Madras Module No. #
More informationCausal & Frequency Analysis
Causal & Frequency Analysis Arshad Ahmad arshad@utm.my Fishbone Diagram 2 The Cause and Effect (CE) Diagram (Ishikawa Fishbone) Created in 1943 by Professor Kaoru Ishikawa of Tokyo University Used to investigate
More informationQuantitative Reliability Analysis
Quantitative Reliability Analysis Moosung Jae May 4, 2015 System Reliability Analysis System reliability analysis is conducted in terms of probabilities The probabilities of events can be modelled as logical
More informationEE 445 / 850: Final Examination
EE 445 / 850: Final Examination Date and Time: 3 Dec 0, PM Room: HLTH B6 Exam Duration: 3 hours One formula sheet permitted. - Covers chapters - 5 problems each carrying 0 marks - Must show all calculations
More informationNumber, Number Sense, and Operations Data Analysis and Probability
Algebra 1 Unit 1 Numbers 3 weeks Number, Number Sense, and Operations Data Analysis and Probability NC Apply properties of operations and the real number system, and justify when they hold for a set of
More informationRISK-INFORMED OPERATIONAL DECISION MANAGEMENT (RIODM): RISK, EVENT TREES AND FAULT TREES
22.38 PROBABILITY AND ITS APPLICATIONS TO RELIABILITY, QUALITY CONTROL AND RISK ASSESSMENT Fall 2005, Lecture 1 RISK-INFORMED OPERATIONAL DECISION MANAGEMENT (RIODM): RISK, EVENT TREES AND FAULT TREES
More informationASTRA 3.0: LOGICAL AND PROBABILISTIC ANALYSIS METHODS
ASTRA 3.: LOGICAL AND PROBABILISTIC ANALYSIS METHODS Description of the main phases and algorithms of the fault tree analysis procedure implemented in ASTRA 3. Sergio Contini and Vaidas Matuzas EUR 2452
More informationThe following number (percentage) of students scored in the following ranges:
The following number (percentage) of students scored in the following ranges: 100: 6 (18%) 95-99: 11 (33%) 90-94: 5 (15%) [22 (67%) scored 90 or better] 85-89: 3 (9%) 80-84: 3 (9%) [28 (85%) scored 80
More informationEngineering Risk Benefit Analysis
Engineering Risk enefit nalysis.55, 2.943, 3.577, 6.938, 0.86, 3.62, 6.862, 22.82 ESD.72J, ESD.72 RPR. The Logic of ertainty George E. postolakis Massachusetts Institute of Technology Spring 2007 RPR.
More informationReliability of Technical Systems
Reliability of Technical Systems Main Topics 1. Short Introduction, Reliability Parameters: Failure Rate, Failure Probability, etc. 2. Some Important Reliability Distributions 3. Component Reliability
More informationConditional Probability. CS231 Dianna Xu
Conditional Probability CS231 Dianna Xu 1 Boy or Girl? A couple has two children, one of them is a girl. What is the probability that the other one is also a girl? Assuming 50/50 chances of conceiving
More informationChapter 5 Reliability of Systems
Chapter 5 Reliability of Systems Hey! Can you tell us how to analyze complex systems? Serial Configuration Parallel Configuration Combined Series-Parallel C. Ebeling, Intro to Reliability & Maintainability
More informationChapter 6. a. Open Circuit. Only if both resistors fail open-circuit, i.e. they are in parallel.
Chapter 6 1. a. Section 6.1. b. Section 6.3, see also Section 6.2. c. Predictions based on most published sources of reliability data tend to underestimate the reliability that is achievable, given that
More informationAssessing system reliability through binary decision diagrams using bayesian techniques.
Loughborough University Institutional Repository Assessing system reliability through binary decision diagrams using bayesian techniques. This item was submitted to Loughborough University's Institutional
More informationA Gentle Introduction to Gradient Boosting. Cheng Li College of Computer and Information Science Northeastern University
A Gentle Introduction to Gradient Boosting Cheng Li chengli@ccs.neu.edu College of Computer and Information Science Northeastern University Gradient Boosting a powerful machine learning algorithm it can
More informationAxioms of Probability
Sample Space (denoted by S) The set of all possible outcomes of a random experiment is called the Sample Space of the experiment, and is denoted by S. Example 1.10 If the experiment consists of tossing
More informationChapter 12. Spurious Operation and Spurious Trips
Chapter 12. Spurious Operation and Spurious Trips Mary Ann Lundteigen Marvin Rausand RAMS Group Department of Mechanical and Industrial Engineering NTNU (Version 0.1) Lundteigen& Rausand Chapter 12.Spurious
More information2011 Pearson Education, Inc
Statistics for Business and Economics Chapter 3 Probability Contents 1. Events, Sample Spaces, and Probability 2. Unions and Intersections 3. Complementary Events 4. The Additive Rule and Mutually Exclusive
More informationWYOMING COMMUNITY DEVELOPMENT AUTHORITY DISCLOSURE REPORT FOR THE 1994 INDENTURE SINGLE FAMILY HOUSING REVENUE BOND SERIES
WYOMING COMMUNITY DEVELOPMENT AUTHORITY DISCLOSURE REPORT FOR THE 1994 INDENTURE SINGLE FAMILY HOUSING REVENUE BOND SERIES 1994-1&2 THROUGH 2014-1, 2, 3, 4 & 5 AS OF SEPTEMBER 30, 2014 INDENTURE 007 IND.
More informationAn-Najah National University Faculty of Engineering Industrial Engineering Department. Course : Quantitative Methods (65211)
An-Najah National University Faculty of Engineering Industrial Engineering Department Course : Quantitative Methods (65211) Instructor: Eng. Tamer Haddad 2 nd Semester 2009/2010 Chapter 2 Probability 2-1
More informationChapter 3 : Conditional Probability and Independence
STAT/MATH 394 A - PROBABILITY I UW Autumn Quarter 2016 Néhémy Lim Chapter 3 : Conditional Probability and Independence 1 Conditional Probabilities How should we modify the probability of an event when
More informationLecture 5 Probability
Lecture 5 Probability Dr. V.G. Snell Nuclear Reactor Safety Course McMaster University vgs 1 Probability Basic Ideas P(A)/probability of event A 'lim n64 ( x n ) (1) (Axiom #1) 0 # P(A) #1 (1) (Axiom #2):
More informationCOMP538: Introduction to Bayesian Networks
COMP538: Introduction to ayesian Networks Lecture 4: Inference in ayesian Networks: The VE lgorithm Nevin L. Zhang lzhang@cse.ust.hk Department of Computer Science and Engineering Hong Kong University
More informationWill Monroe July 5, with materials by Mehran Sahami and Chris Piech. image: Therightclicks. Independence
Will Monroe July 5, 2017 with materials by Mehran Sahami and Chris Piech image: Therightclicks Independence Announcements: Problem Set 1 due! Solutions to be posted next Wednesday (no submissions allowed
More informationSafety analysis and standards Analyse de sécurité et normes Sicherheitsanalyse und Normen
Industrial Automation Automation Industrielle Industrielle Automation 9.6 Safety analysis and standards Analyse de sécurité et normes Sicherheitsanalyse und Normen Prof Dr. Hubert Kirrmann & Dr. B. Eschermann
More informationCS626 Data Analysis and Simulation
CS626 Data Analysis and Simulation Instructor: Peter Kemper R 104A, phone 221-3462, email:kemper@cs.wm.edu Today: Probability Primer Quick Reference: Sheldon Ross: Introduction to Probability Models 9th
More informationBayesian networks. Instructor: Vincent Conitzer
Bayesian networks Instructor: Vincent Conitzer Rain and sprinklers example raining (X) sprinklers (Y) P(X=1) =.3 P(Y=1) =.4 grass wet (Z) Each node has a conditional probability table (CPT) P(Z=1 X=0,
More informationReliability Analysis of Hydraulic Steering System with DICLFL Considering Shutdown Correlation Based on GO Methodology
2015 ICRSE&PHM-Beijing Reliability Analysis of Hydraulic Steering System with DICLFL Considering Shutdown Correlation Based on GO Methodology YI Xiaojian, SHI Jian, MU Huina, DONG Haiping, GUO Shaowei
More informationAnalysis methods for fault trees that contain secondary failures
Loughborough University Institutional Repository Analysis methods for fault trees that contain secondary failures This item was submitted to Loughborough University's Institutional Repository by the/an
More informationChapter 8. Calculation of PFD using FTA
Chapter 8. Calculation of PFD using FTA Mary Ann Lundteigen Marvin Rausand RAMS Group Department of Mechanical and Industrial Engineering NTNU (Version 0.1) Lundteigen& Rausand Chapter 8.Calculation of
More informationLecture 4. Selected material from: Ch. 6 Probability
Lecture 4 Selected material from: Ch. 6 Probability Example: Music preferences F M Suppose you want to know what types of CD s males and females are more likely to buy. The CD s are classified as Classical,
More informationDIN 2445 Part-2 SEAMLESS STEEL TUBES FOR DYNAMIC LOADS SUPPLEMENT BASIS FOR CALUCULATION OF STRAIGHT TUBES
DIN Part- SEAMLESS STEEL TUBES FOR DYNAMIC LOADS SUPPLEMENT BASIS FOR CALUCULATION OF STRAIGHT TUBES. General information During the operation of hydraulic installations, when the control valve is operated
More informationChapter 3. P{E has 3, N S has 8} P{N S has 8}
Instant download and all chapters SOLUTIONS MANUAL A First Course in Probability 9th Edition Sheldon Ross https://testbankdata.com/download/solutions-manual-a-first-course-in-probability-9th-edition-sheldon-ross
More informationSafety and Reliability of Embedded Systems. (Sicherheit und Zuverlässigkeit eingebetteter Systeme) Fault Tree Analysis Obscurities and Open Issues
(Sicherheit und Zuverlässigkeit eingebetteter Systeme) Fault Tree Analysis Obscurities and Open Issues Content What are Events? Examples for Problematic Event Semantics Inhibit, Enabler / Conditioning
More informationCommon Cause Failure (CCF)
Common Cause Failure (CCF) 건국대학교컴퓨터공학과 UC Lab. 정혁준 & 박경식 amitajung@naver.com, kyeongsik@konkuk.ac.kr Contents Common Cause Failure (CCF) Types of CCF Examples Reducing CCF Common Cause Failure (CCF) Definition
More informationCommon Cause Failures: Extended Alpha Factor method and its Implementation
Common Cause Failures: Extended Alpha Factor method and its Implementation Alexandra Sitdikova Reactor Engineering Division, Jožef Stefan Institute Jamova 39, SI-1000 Ljubljana, Slovenia Institute of Physics
More informationIntroduction to Probability 2017/18 Supplementary Problems
Introduction to Probability 2017/18 Supplementary Problems Problem 1: Let A and B denote two events with P(A B) 0. Show that P(A) 0 and P(B) 0. A A B implies P(A) P(A B) 0, hence P(A) 0. Similarly B A
More informationChapter 9 Part II Maintainability
Chapter 9 Part II Maintainability 9.4 System Repair Time 9.5 Reliability Under Preventive Maintenance 9.6 State-Dependent Systems with Repair C. Ebeling, Intro to Reliability & Maintainability Chapter
More informationPLEASE MARK YOUR ANSWERS WITH AN X, not a circle! 1. (a) (b) (c) (d) (e) 2. (a) (b) (c) (d) (e) (a) (b) (c) (d) (e) 4. (a) (b) (c) (d) (e)...
Math 020, Exam II October, 206 The Honor Code is in effect for this examination. All work is to be your own. You may use a calculator. The exam lasts for hour 5 minutes. Be sure that your name is on every
More informationLaw of Total Probability and Bayes Rule
MATH 382 Law of Total Probability and Bayes Rule Dr Neal, WKU Law of Total Probability: Suppose events A 1, A 2,, A n form a partition of Ω That is, the events are mutually disjoint and their union is
More informationThe enumeration of all possible outcomes of an experiment is called the sample space, denoted S. E.g.: S={head, tail}
Random Experiment In random experiments, the result is unpredictable, unknown prior to its conduct, and can be one of several choices. Examples: The Experiment of tossing a coin (head, tail) The Experiment
More informationDigital Secondary Control Architecture for Aircraft Application
Digital Secondary Control Architecture for Aircraft Application PhD researchers: Henri C. Belan Cristiano C. Locateli Advisors (SAAB): Birgitta Lantto (LiU): Petter Krus (UFSC): Victor J. De Negri Introduction
More informationReliability of Safety-Critical Systems Chapter 8. Probability of Failure on Demand using fault trees
Reliability of Safety-Critical Systems Chapter 8. Probability of Failure on Demand using fault trees Mary Ann Lundteigen and Marvin Rausand mary.a.lundteigen@ntnu.no &marvin.rausand@ntnu.no RAMS Group
More informationQuantification of Temporal Fault Trees Based on Fuzzy Set Theory
Quantification of Temporal Fault Trees Based on Fuzzy Set Theory Sohag Kabir, Ernest Edifor, Martin Walker, Neil Gordon Department of Computer Science, University of Hull, Hull, UK {s.kabir@2012.,e.e.edifor@2007.,martin.walker@,n.a.gordon
More informationPart I: Propositional Calculus
Logic Part I: Propositional Calculus Statements Undefined Terms True, T, #t, 1 False, F, #f, 0 Statement, Proposition Statement/Proposition -- Informal Definition Statement = anything that can meaningfully
More informationPlease do NOT write in this box. Multiple Choice Total
Name: Instructor: ANSWERS Bullwinkle Math 1010, Exam I. October 14, 014 The Honor Code is in effect for this examination. All work is to be your own. Please turn off all cellphones and electronic devices.
More informationSTAT:5100 (22S:193) Statistical Inference I
STAT:5100 (22S:193) Statistical Inference I Week 3 Luke Tierney University of Iowa Fall 2015 Luke Tierney (U Iowa) STAT:5100 (22S:193) Statistical Inference I Fall 2015 1 Recap Matching problem Generalized
More informationThe Failure-tree Analysis Based on Imprecise Probability and its Application on Tunnel Project
463 A publication of CHEMICAL ENGINEERING TRANSACTIONS VOL. 59, 2017 Guest Editors: Zhuo Yang, Junjie Ba, Jing Pan Copyright 2017, AIDIC Servizi S.r.l. ISBN 978-88-95608-49-5; ISSN 2283-9216 The Italian
More informationReliability of Technical Systems
Reliability of Technical Systems Main Topics. Short Introduction, Reliability Parameters: Failure Rate, Failure Probability, etc. 2. Some Important Reliability Distributions 3. Component Reliability 4.
More informationFault-Tolerant Computer System Design ECE 60872/CS 590. Topic 2: Discrete Distributions
Fault-Tolerant Computer System Design ECE 60872/CS 590 Topic 2: Discrete Distributions Saurabh Bagchi ECE/CS Purdue University Outline Basic probability Conditional probability Independence of events Series-parallel
More informationCSE 311: Foundations of Computing. Lecture 14: Induction
CSE 311: Foundations of Computing Lecture 14: Induction Mathematical Induction Method for proving statements about all natural numbers A new logical inference rule! It only applies over the natural numbers
More informationFinite Automata Theory and Formal Languages TMV027/DIT321 LP4 2017
Finite Automata Theory and Formal Languages TMV027/DIT321 LP4 2017 Lecture 4 Ana Bove March 24th 2017 Structural induction; Concepts of automata theory. Overview of today s lecture: Recap: Formal Proofs
More informationChapter 5. System Reliability and Reliability Prediction.
Chapter 5. System Reliability and Reliability Prediction. Problems & Solutions. Problem 1. Estimate the individual part failure rate given a base failure rate of 0.0333 failure/hour, a quality factor of
More informationOverview of Control System Design
Overview of Control System Design Introduction Degrees of Freedom for Process Control Selection of Controlled, Manipulated, and Measured Variables Process Safety and Process Control 1 General Requirements
More informationChapter. Probability
Chapter 3 Probability Section 3.1 Basic Concepts of Probability Section 3.1 Objectives Identify the sample space of a probability experiment Identify simple events Use the Fundamental Counting Principle
More informationENGI 4421 Introduction to Probability; Sets & Venn Diagrams Page α 2 θ 1 u 3. wear coat. θ 2 = warm u 2 = sweaty! θ 1 = cold u 3 = brrr!
ENGI 4421 Introduction to Probability; Sets & Venn Diagrams Page 2-01 Probability Decision trees u 1 u 2 α 2 θ 1 u 3 θ 2 u 4 Example 2.01 θ 1 = cold u 1 = snug! α 1 wear coat θ 2 = warm u 2 = sweaty! θ
More informationReliability of sequential systems using the causeconsequence diagram method
Loughborough University Institutional Repository Reliability of sequential systems using the causeconsequence diagram method This item was submitted to Loughborough University's Institutional Repository
More informationENGI 3423 Introduction to Probability; Sets & Venn Diagrams Page 3-01
ENGI 3423 Introduction to Probability; Sets & Venn Diagrams Page 3-01 Probability Decision trees θ 1 u 1 α 1 θ 2 u 2 Decision α 2 θ 1 u 3 Actions Chance nodes States of nature θ 2 u 4 Consequences; utility
More informationIndependence 1 2 P(H) = 1 4. On the other hand = P(F ) =
Independence Previously we considered the following experiment: A card is drawn at random from a standard deck of cards. Let H be the event that a heart is drawn, let R be the event that a red card is
More informationMath Exam 1 Review. NOTE: For reviews of the other sections on Exam 1, refer to the first page of WIR #1 and #2.
Math 166 Fall 2008 c Heather Ramsey Page 1 Math 166 - Exam 1 Review NOTE: For reviews of the other sections on Exam 1, refer to the first page of WIR #1 and #2. Section 1.5 - Rules for Probability Elementary
More informationApplication of the Cause-Consequence Diagram Method to Static Systems
Application of the ause-onsequence Diagram Method to Static Systems L.M.Ridley and J.D.Andrews Department of Mathematical Sciences Loughborough University Loughborough Leicestershire LE11 3TU Keywords:
More informationSystem Reliability Analysis. CS6323 Networks and Systems
System Reliability Analysis CS6323 Networks and Systems Topics Combinatorial Models for reliability Topology-based (structured) methods for Series Systems Parallel Systems Reliability analysis for arbitrary
More informationof an algorithm for automated cause-consequence diagram construction.
Loughborough University Institutional Repository Development of an algorithm for automated cause-consequence diagram construction. This item was submitted to Loughborough University's Institutional Repository
More informationThe Applications of Inductive Method in the Construction of Fault Trees MENG Qinghe 1,a, SUN Qin 2,b
The Applications of Inductive Method in the Construction of Fault Trees MENG Qinghe 1,a, SUN Qin 2,b 1 School of Aeronautics, Northwestern Polytechnical University, Xi an 710072, China 2 School of Aeronautics,
More informationBoosting. 1 Boosting. 2 AdaBoost. 2.1 Intuition
Boosting 1 Boosting Boosting refers to a general and effective method of producing accurate classifier by combining moderately inaccurate classifiers, which are called weak learners. In the lecture, we
More informationIntroduction to Probabilistic Reasoning. Image credit: NASA. Assignment
Introduction to Probabilistic Reasoning Brian C. Williams 16.410/16.413 November 17 th, 2010 11/17/10 copyright Brian Williams, 2005-10 1 Brian C. Williams, copyright 2000-09 Image credit: NASA. Assignment
More informationCprE 281: Digital Logic
CprE 28: Digital Logic Instructor: Alexander Stoytchev http://www.ece.iastate.edu/~alexs/classes/ Code Converters CprE 28: Digital Logic Iowa State University, Ames, IA Copyright Alexander Stoytchev HW
More informationMarkov Reliability and Availability Analysis. Markov Processes
Markov Reliability and Availability Analysis Firma convenzione Politecnico Part II: Continuous di Milano e Time Veneranda Discrete Fabbrica State del Duomo di Milano Markov Processes Aula Magna Rettorato
More informationConditional Statement: Statements in if-then form are called.
Monday 9/21 2.2 and 2.4 Wednesday 9/23 2.5 and 2.6 Conditional and Algebraic Proofs Algebraic Properties and Geometric Proofs Unit 2 Angles and Proofs Packet pages 1-3 Textbook Pg 85 (14, 17, 20, 25, 27,
More informationP B A. conditional probabilities A B and unconditional probabilities are neither 0 nor 1, this note demonstrates two consequences when
1 When P P A 1. Introduction Many students encountering probability theory for the first time have difficulty distinguishing conditional probabilities from joint or unconditional probabilities and they
More informationSixth Edition. Chapter 2 Probability. Copyright 2014 John Wiley & Sons, Inc. All rights reserved. Probability
Applied Statistics and Probability for Engineers Sixth Edition Douglas C. Montgomery George C. Runger Chapter 2 Probability 2 Probability CHAPTER OUTLINE 2-1 Sample Spaces and Events 2-1.1 Random Experiments
More informationIntroduction to Probability. Experiments. Sample Space. Event. Basic Requirements for Assigning Probabilities. Experiments
Introduction to Probability Experiments These are processes that generate welldefined outcomes Experiments Counting Rules Combinations Permutations Assigning Probabilities Experiment Experimental Outcomes
More informationConditional Probability P( )
Conditional Probability P( ) 1 conditional probability and the chain rule General defn: where P(F) > 0 Implies: P(EF) = P(E F) P(F) ( the chain rule ) General definition of Chain Rule: 2 Best of 3 tournament
More informationEngineering Risk Benefit Analysis
Engineering Risk Benefit Analysis 1.155, 2.943, 3.577, 6.938, 10.816, 13.621, 16.862, 22.82, ESD.72, ESD.721 RPRA 3. Probability Distributions in RPRA George E. Apostolakis Massachusetts Institute of Technology
More informationMutually Exclusive Events
172 CHAPTER 3 PROBABILITY TOPICS c. QS, 7D, 6D, KS Mutually Exclusive Events A and B are mutually exclusive events if they cannot occur at the same time. This means that A and B do not share any outcomes
More informationLecture 10: 09//25/03 A.R. Neureuther Version Date 09/14/03 EECS 42 Introduction to Digital Electronics Andrew R. Neureuther
EECS 42 Intro. Digital Electronics Fall 23 Lecture : 9//25/3.R. Neureuther Version Date 9/4/3 EECS 42 Introduction to Digital Electronics ndrew R. Neureuther Lecture # Prof. King: asic Digital locks 2
More informationPHM Engineering Perspectives, Challenges and Crossing the Valley of Death. 30 September, 2009 San Diego, CA
PHM Engineering Perspectives, Challenges and Crossing the Valley of Death 30 September, 2009 San Diego, CA The views, opinions, and/or findings contained in this article/presentation are those of the author/presenter
More informationCOMPSCI 276 Fall 2007
Exact Inference lgorithms for Probabilistic Reasoning; OMPSI 276 Fall 2007 1 elief Updating Smoking lung ancer ronchitis X-ray Dyspnoea P lung cancer=yes smoking=no, dyspnoea=yes =? 2 Probabilistic Inference
More informationA Probability Primer. A random walk down a probabilistic path leading to some stochastic thoughts on chance events and uncertain outcomes.
A Probability Primer A random walk down a probabilistic path leading to some stochastic thoughts on chance events and uncertain outcomes. Are you holding all the cards?? Random Events A random event, E,
More informationCSSTP. Given CSSTP. Statements Reasons. Given CSSTP. Mult. Prop. = Div. Prop. = Sym. Prop. = or 1 Mult. Prop. = Div. Prop. =
: If the triangles are similar (~), then all of the sides must be congruent proportional (create equal scale fractions). Example: A~ F Before you start your proof, it is important to plan! Setup the three
More informationIntroduction and basic definitions
Chapter 1 Introduction and basic definitions 1.1 Sample space, events, elementary probability Exercise 1.1 Prove that P( ) = 0. Solution of Exercise 1.1 : Events S (where S is the sample space) and are
More informationFinite Automata Theory and Formal Languages TMV027/DIT321 LP Recap: Logic, Sets, Relations, Functions
Finite Automata Theory and Formal Languages TMV027/DIT321 LP4 2017 Formal proofs; Simple/strong induction; Mutual induction; Inductively defined sets; Recursively defined functions. Lecture 3 Ana Bove
More informationReliability Analysis of Electronic Systems using Markov Models
Reliability Analysis of Electronic Systems using Markov Models István Matijevics Polytechnical Engineering College, Subotica, Serbia and Montenegro, matistvan@yahoo.com Zoltán Jeges Polytechnical Engineering
More informationExam 1 - Math Solutions
Exam 1 - Math 3200 - Solutions Spring 2013 1. Without actually expanding, find the coefficient of x y 2 z 3 in the expansion of (2x y z) 6. (A) 120 (B) 60 (C) 30 (D) 20 (E) 10 (F) 10 (G) 20 (H) 30 (I)
More informationProbability COMP 245 STATISTICS. Dr N A Heard. 1 Sample Spaces and Events Sample Spaces Events Combinations of Events...
Probability COMP 245 STATISTICS Dr N A Heard Contents Sample Spaces and Events. Sample Spaces........................................2 Events........................................... 2.3 Combinations
More informationIntro to Probability Day 4 (Compound events & their probabilities)
Intro to Probability Day 4 (Compound events & their probabilities) Compound Events Let A, and B be two event. Then we can define 3 new events as follows: 1) A or B (also A B ) is the list of all outcomes
More informationCogs 14B: Introduction to Statistical Analysis
Cogs 14B: Introduction to Statistical Analysis Statistical Tools: Description vs. Prediction/Inference Description Averages Variability Correlation Prediction (Inference) Regression Confidence intervals/
More informationr. Matthias Bretschneider amburg - Dept. Safety Fehleranalyse mit Hilfe von Model Checkern
r. Matthias Bretschneider amburg - Dept. Safety Fehleranalyse mit Hilfe von Model Checkern otivation: Design of safe embedded systems X y Sensor(s) Controller Actuator Design Phase Study the effect of
More information= (, ) V λ (1) λ λ ( + + ) P = [ ( ), (1)] ( ) ( ) = ( ) ( ) ( 0 ) ( 0 ) = ( 0 ) ( 0 ) 0 ( 0 ) ( ( 0 )) ( ( 0 )) = ( ( 0 )) ( ( 0 )) ( + ( 0 )) ( + ( 0 )) = ( + ( 0 )) ( ( 0 )) P V V V V V P V P V V V
More informationCprE 281: Digital Logic
CprE 28: Digital Logic Instructor: Alexander Stoytchev http://www.ece.iastate.edu/~alexs/classes/ Decoders and Encoders CprE 28: Digital Logic Iowa State University, Ames, IA Copyright Alexander Stoytchev
More informationSTAT 516: Basic Probability and its Applications
Lecture 3: Conditional Probability and Independence Prof. Michael September 29, 2015 Motivating Example Experiment ξ consists of rolling a fair die twice; A = { the first roll is 6 } amd B = { the sum
More informationQuantitative Methods for Decision Making
January 14, 2012 Lecture 3 Probability Theory Definition Mutually exclusive events: Two events A and B are mutually exclusive if A B = φ Definition Special Addition Rule: Let A and B be two mutually exclusive
More information