RASM Chapter 6b

Information about RASM Chapter 6b

Published on July 21, 2014

Author: pthomp

Source: authorstream.com

Content

PowerPoint Presentation: Union Union is the event of X or Y occurring = X U Y P(XUY ) is the probability that X or Y occurs Shown as shaded area of Venn diagram So P (X  Y) = P (X) + P (Y) - P (X  Y ) The last term avoids counting the overlap twice PowerPoint Presentation: Union For example: Fraction of cars failing test because: A – steering is faulty B – brakes inadequate Random sample of 1000 cars A – 600 have faulty steering B – 500 have faulty brakes Obviously total # of cars is not 600+500 – some will have both Assuming steering and brakes are independent: P(A∩B ) = 0.6 x 0.5 = 0.3 P(AUB ) = P(A) + P(B) - P(A ∩ B ) = 0.6 + 0.5 - 0.6 x 0.5 = 0.8 PowerPoint Presentation: Or gate Probability of the union of events = OR gate P(C) = P(A) + P(B) - P(A∩B ) PowerPoint Presentation: Mutually exclusive events If A occurs but B cannot then A and B are mutually exclusive e.g. you cannot have both heads and tails the intersection cannot happen P(‘heads’ ∩ ’tails’) = 0 For mutually exclusive events the union is P (X  Y) = P (X) + P (Y ) The Venn diagram does not overlap This should be specified as an XOR gate to indicate it is mutually exclusive and to ensure the right mathematical formula is used PowerPoint Presentation: Extension to many events The results can be generalised to 3 or more events Union is P(XUYUZ) Intersection is P(X∩Y ∩ Z ) PowerPoint Presentation: Extension to many events For independent events intersection is easy: P (X  Y  Z) = P (X)  P (Y)  P (Z) Harder if not independent and it may be wise to design systems and fault trees to avoid dependency Union calculations are harder: Need to avoid counting overlapping areas more than once P (X  Y  Z) = P (X ) + P (Y) + P (Z) - P (X) P (Y) - P (Y) P (Z) - P (Z) P (X) + P (X) P (Y) P (Z) Can sort of simplify by splitting calculations into a piecewise approach, i.e. add one event at a time. See example in text. PowerPoint Presentation: Parallel Systems To improve reliability parallel systems are often used e.g. a dual braking system on a car If one fails then the other can take over Often called redundancy Shown graphically in a reliability block diagram In nuclear industry four fold redundancy may even be used PowerPoint Presentation: Parallel Systems Example: Probability of failure in a 12 hour storm of Pump A = P(A) = 0.1 Pump B = P(B) = 0.2 If the failures of these pumps are independent and there is no common cause failure: P(both fail) = P(A∩B) = 0.1 x 0.2 = 0.02 This is considerably more reliable than either pump separately The reliability is 0.98. With a third Pump C = P(C) = 0.2 P(all fail) = P(A∩B∩C) = 0.1 x 0.2 x 0.2 = 0.004 This improves the probability of failure to 0.4% and the reliability to 0.996 in a 12 hour storm PowerPoint Presentation: Series systems Series systems tend to have very low reliability Shown as a reliability block diagram: For example in the previous case had the pumps been connected in series then the probability of failure would be a union calculation: P(both fail) = P(A) + P(B) - P(A∩B) = 0.1 + 0.2 – 0.1 x 0.2 = 0.28 There would therefore be nearly a 30% chance of failure if the pumps had been connected in series! Series systems are best avoided otherwise accidents are very likely This also illustrates that a union calculation or OR gate leads to much higher failure rates. Where possible design systems with AND gates at the top. PowerPoint Presentation: Failure Rates/Statistical Uncertainty Reliability often estimated from factory/laboratory based tests or from field experience of failures in service. Key failure rates are: PowerPoint Presentation: Confidence Intervals If we have, e.g., an emergency generating system that has started in all of 20 (= N ) routine tests so far then we can say that we are 95% confident that the maximum probability of it not starting next time will be: Upper confidence limit ( p+ ) = 1 – (1 – 0.95) 1/N where N = number of tests (No need to remember!) Based on the Binomial distribution – assumes each trial to be independent of the others and that all trials occur under identical circumstances So p+ = 1 - 0.86 = 0.14 . Or: 0 < probability not starting < 0.14 95 % confident the probability of starting next time > (1–0.14) = 0.86 Can also use others e.g. 99% If N is bigger interval decreases (improves) PowerPoint Presentation: Time to Failure/ Continuous data Time to failure not binary = continuous instead Uncertainty / confidence intervals determined differently e.g. Computer controlled by a fan Time to failure of similar fans (years ): 4.9, 1.8, 2.1, 2.1. Average or mean time to failure (MTTF) is = Sample mean , where ^ shows estimate of true population mean μ How good is estimate? How can uncertainty be measured? Commonly use sample standard deviation ( s.d. or s ) PowerPoint Presentation: Uncertainty If only small samples mean and s.d. may vary significantly MTTF may be misleading if don’t know sample size and s.d. Two key principles are larger sample = more accurate the estimate of mean spread of data are important - usually expressed as standard deviation E.g. “ As the mean time to failure of the fans is 4.9 years we have decided to replace them every two years to be on the safe side.” BUT if only 4 samples and s.d. is 4.39 you should think twice! PowerPoint Presentation: Uncertainty in Normal Distribution If know the data fits a normal classic bell shaped distribution: 95% confidence interval is given by μ – 2 σ < time to failure < μ + 2 σ , e.g. device with MTTF ( μ ) of 1000hours with s.d. ( σ ) of 60 hours then 880 hours < time to failure < 1120 hours 2.5 % of the failures will happen in less than 880 hours and 2.5 % will fail after more than 1120 hours 95% of the normal distribution lies within +/- two σ of the mean In practice have to estimate μ and σ from a sample There are other distributions! PowerPoint Presentation: Data: Quality and Uncertainty Other factors/sources of uncertainty (than statistical): Accuracy and Quality Age Applicability Bias of the Assessor PowerPoint Presentation: Summary Methods that can be used for quantifying risk assessment. Estimating probability data - theory , empirical data, subjective assessments - can be combined using probabilistic modelling. Probabilistic modelling was explained using sets. The three axioms of probability, Independent events, Intersection and union AND and OR gates, Mutually exclusive events Extension to many events Series and parallel systems. Need to consider confidence intervals/uncertainty in estimates . PowerPoint Presentation: Discussion Questions Now we recommend: Read through these sections in the notes Tackle the discussion the questions at the end Work with your fellow students on the discussion boards PowerPoint Presentation: Any Questions ? If so – post them in the discussion boards

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