Probability of occurrence

If there is insufficient data to describe a continuous probability distribution for a variable (as with the area of a field in an earlier example), we may be able to make a subjective estimate of high, medium and low values. If those are chosen using the p85, p50, pi 5 cumulative probabilities described in Section 6.2.2, then the implication is that the three values are equally likely, and therefore each has a probability of occurrence of 1/3. Note that the low and high values are not the minimum and maximum values. 

Note that the low value of the combination Is not the absolute minimum (which would be 4, and is still a possible outcome), just as the high value is not the maximum. The three values (which are calculated by taking the mean of the three lowest values In the matrix etc.) represent equally likely outcomes of the product A B, each with a probability of occurrence of 1/3. 

The normal distribution of measurements (or the normal law of error) is the fundamental starting point for analysis of data. When a large number of measurements are made, the individual measurements are not all identical and equal to the accepted value /x, which is the mean of an infinite population or universe of data, but are scattered about /x, owing to random error. If the magnitude of any single measurement is the abscissa and the relative frequencies (i.e., the probability) of occurrence of different-sized measurements are the ordinate, the smooth curve drawn through the points (Fig. 2.10) is the normal or Gaussian distribution curve (also the error curve or probability curve). The term error curve arises when one considers the distribution of errors (x — /x) about the true value. 

The second type of error occurs when the null hypothesis is retained even though it is false and should be rejected. This is known as a type 2 error, and its probability of occurrence is [3. Unfortunately, in most cases [3 cannot be easily calculated or estimated. 

Likelihood A measure of the expected frequency with which an event occurs. This may be expressed as a frequency (e.g., events per year), a probability of occurrence during a time interval (e.g., annual probability), or a conditional probability (e.g., probability of occurrence, given that a precursor event has occurred). 

Failure in the context here means that product performance does not meet requirements and is related back in the design FMEA to some component/character-istic being out of specified limits - a fault. The probability of occurrence of failure (O) caused by a fault can be expressed as ... 

While 30 ppm may be acceptable as a maximum probability of occurrence for a failure of low severity, it is not acceptable as severity increases. An example table of FMEA Severity Ratings was shown in Figure 2.20. In the definite return to manufacturer (a warranty return) or violation of statutory requirement region (S = 5 or S = 6), the designer would seek ways to enhance the process capability or else utilize some inspection or test process. Reducing d will reduce occurrence, as indicated by equation 2.11, but inspection or test is of limited efficiency. 

In some cases where the ASME Code woidd not require pressure relief protection, the 1.5 Times Design Pressure Rule is apphcable. This rule is stated as follows Equipment may be considered to be adequately protected against overpressure from certain low-probability situations if the pressure does not exceed 1.5 times design pressure. This criterion has been selected since it generally does not exceed yield stress, and most Ukety would not occur more frequently than a hydrostatic test. Thus, it will protect against the possibility of a catastrophic failure. This rule is applied in special situations which have a low probability of occurrence but which cannot be completely ruled out. 

A number of vendors offer software based hazard assessment tools that help determine the magnitude of the hazards involved. With this software, calculations can be made to reflect the hazard for various failures. Some risk ranking software combines hazard assessment with probabilities of occurrence so that the relative risk levels can be assessed. 

The technique is particularly useful in evaluating the effect of alternative actions on reducing the probability of occurrence of the undesired event. 

Analyses are types of calculations but may be comparative studies, predictions, and estimations. Examples are stress analysis, reliability analysis, hazard analysis. Analyses are often performed to detect whether the design has any inherent modes of failure and to predict the probability of occurrence. The analyses assist in design improvement and the prevention of failure, hazard, deterioration, and other adverse conditions. Analyses may need to be conducted as the end-use conditions may not be reproducible in the factory. Assumptions may need to be made about the interfaces, the environment, the actions of users, etc. and analysis of such conditions assists in determining characteristics as well as verifying the inherent characteristics. (See also in Part 2 Chapter 14 under Detecting design weaknesses.)... 

Start by doing a risk assessment and identify those things on which continuity of business depends power, water, labor, materials, components, services, etc. Determine what could cause a termination of supply and estimate the probability of occurrence. For those with a relatively high probability (1 in 100) find ways to reduce the probability. For those with lower probability (1 in 10000) determine the action needed to minimize the effect. The FMEA technique works for this as well as for products and processes. 

A failure modes and effects analysis is a systematic analytical technique for identifying potential failures in a design or a process, assessing the probability of occurrence and likely effect, and determining the measures needed to eliminate, contain, or control the effects. Action taken on the basis of an FMEA will improve safety, performance, reliability, maintainability and reduce costs. The outputs are essential to balanced and effective quality plans for both development and production as it will help focus the controls upon those products, processes, and characteristics that are at risk. It is not the intention here to give a full appreciation of the FMEA technique and readers are advised to consult other texts. 

Fire and explosion are much more serious events than pollution. For one thing, fire and explosion can create catastrophes that will lead to poi ludon anyway, but for another thing, they can injure people. We clearl> want to have more levels of safety (that is, a lower probability of occurrence) in the chain leading to fire or explosion than is necessary in the chain leading to pollution. That is, whatever the acceptable risk lor <.)ii pollution, a lower risk is required for fire or explosion. 

A Safety Case is a narrative that literally makes the case that an adequate level of safety has been reached for an installation. It requires looking at all potential hazards which could lead to a loss of the installation, a loss of life, or a major pollution event. A risk analysis is performed on each hazard evaluating the probability of the event occurring and describing the magnitude of the consequences. A discussion is then given of the measure undertaken to lower the probability of occurrence or to mitigate the consequences and a case is made that the risk for the installation meets the ALARP safety criteria. 

This report documents the development of data on the severity as well as the frequency of accidents involving truck, rail, and air transport. Volume 1 includes a summary giving the probability of occurrence of accidents as a function of accident severity. Subsequent Volumes give supporting data, calculations and analysis. 

If the probability of the accident and tlie severity of its consequences are low, then die risk is usually deemed acceptable and the plant should be allowed to operate. If die probability of occurrence is too high or die damage to the surroundings is too great, dien the risk is usually unacceptable and die system needs to be modified to minimize these defects. 

Societal Risk - This represents a measure of the risk to a group of people, including tlie risk of incidents potentially affecting more tlian one person. Individual risk (see above) is generally not significantly affected by the number of people involved in an incident. The risk to a person at a particular location depends on tlie probability of occurrence of the luizardous event, and on the probability of an adverse imptict at that location should the event occur. 

Likelihood estimation, sometimes called frequency estimation, cluuactcrizes the probability of occurrence for each potential incident considered in tlie iiiuilysis. The major tools used for likelihood estimation are listed below. ... 

Figure 18.4.1 for a chemical release. Qualitative definitions regarding tlie probability of occurrence are as follows ... 

Low Tlie probability of occurrence is considered unlikely during the... 

This is approximately equivalent to 3x10 occasions/yr, i.e., tlie probability of occurrence should not exceed 3x10 (yr). If tlie worker is killed every 10 time the incident occurs, then tlie target liazard rate is ... 

Cause-consequence analysis serx es to characterize tlie physical effects resulting from a specific incident and the impact of these physical effects on people, the environment, and property. Some consequence models or equations used to estimate tlie potential for damage or injury are as follows Source Models, Dispersion Models, Fire Explosion Models, and Effect Models. Likelihood estimation (frequency estimation), cliaractcrizcs the probability of occurrence for each potential incident considered in tlie analysis. The major tools used for likelihood estimation are as follows Historical Data, Failure sequence modeling techniques, and Expert Judgment. 

Descriptive statistics quantify central tendency and variance of data sets. The probability of occurrence of a value in a given population can be described in terms of the Gaussian distribution. 

P 1 probability of occurrence of ruitile-like slabs in crystal structure of yls- Mn02 II. I... 

Pr probability of occurrence of ramsdellite-like building blocks inll.l yts-y[n02 ... 

By identifying the probability of occurrence of a crystal with the flux corresponding to its thickness as explained above, the average thickness of an ensemble of crystals is given by ... 

Networks obtained by anionic end-linking processes are not necessarily free of defects 106). There are always some dangling chains — which do not contribute to the elasticity of the network — and the formation of loops and of double connections cannot be excluded either. The probability of occurrence, of such defects decreases as the concentration of the reaction medium increases. Conversely, when the concentration is very high the network may contain entrapped entanglements which act as additional crosslinks. It remains that, upon reaction, the linear precursor chains (which are characterized independently) become elastically effective network chains, even though their number may be slightly lower than expected because of the defects. 

Otherwise, one is left with the possibility to accept that an individual OOS result constitutes failure, and must be avoided. This can be done by reducing the probability of occurrence of one OOS result out of N measurements to some predefined level, e.g., p < 0.05. The acceptable level of risk is management s decision, because they will have to face the press (Wall Street, FDA, etc.) if scandal errupts, and approve the budget overruns incurred by increased testing and wasted batches. 

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