Simple Event Formula

The formula for calculating simple event probabilities is as follows (Hays 1988,27)  

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Suppose a safety manager collected data from accident reports involving back injuries. Over a three-year period, 300 reports were filed, of which 18 involved back injuries. Using the simple event formula, the probability of randomly selecting a report for which a back injury was recorded from the 300 reports filed is determined by ... 

What are the probability formulas for a joint event, compound event, and simple event probability ... 

In addition, Perrin tended to interest himself in individual atomic events rather than in the molecule as a complex entity. The radiation theory of chemical activation attempted a mathematical representation in simple and general formulas, based on axiomatic principles. Perrin s work in radiation bears more resemblance to his reformulation of thermodynamics in the early 1900s than to his experimental work on x-rays and cathode rays in the 1890s or on colloids and Brownian motion in the early 1900s. What binds all his work together is interest in single events or the individual corpuscle. 106... 

We will consider the MWD in two simple cases. The first is when chain transfer is sufficiently rapid to ensure that all other chain-stopping events can be ignored. In such a situation, whereas the compartmentalized nature of the reaction may affect the rate of initiation of new chains, it will not affect the lifetime distributions of the chains once they are formed. The MWD may then be found from the bulk formulas, provided only that the average number of free radicals per particle, is known. Such an approach has been used by Friis et al. (1974) to calculate the MWD evolved in a vinyl acetate emukion polymerization. These authors included in addition the mechanisms of terminal bond polymerization and of transfer to polymer (both of which cause broadening). The formulas required for the in corporation of these mechanisms could be taken from bulk theory. 

The main difference between the cases discussed in this section on selectivity and the previously discussed case of a simple polystep reaction resides in the fact that here the single component which generates the species that becomes the intermediate in the polyfunctional composite can itself generate a distinct product species with appreciable yield. Since the coupling between the Y- and the X-system occurs in any event through mass-transport of intermediates between X-sites and T-sites, the diffusion criteria already discussed must apply or the kinetic schemes which accomplish interception or selectivity control will not be physically and effectively accomplished. The criterion, of formula (15) should be satisfied. ... 

Suppose there are 20 nonoverlapping subgroups. A simple application of the aforementioned formula with a = 0.025 for K = 20 gives a false-positive probability of approximately 40%. If we consider 50 adverse event (AE) terms that, for simplicity, are assumed to be nearly independent, then there would be 1000 comparisons resulting from possible combinations of 20 subgroup and 50 AE terms. Similar application of the formula would lead to a falsepositive probability of nearly one. A more strict criterion of a = 0.005 does not improve the situation by much. 

As an example, in evaluating the safe behavior of 1000 men and women. Table 12.1 shows the percent of the total that either did or did not have occupational injuries. The contributing events (men having injuries and women having injuries) are not mutually exclusive since either or both may have suffered occupational injuries and, subsequently, affected the primary or top event (safe behavior). Hgure 12.7 shows the fault tree for this extremely simple example. By applying the modified formula for non-mutually exclusive events, the probability of an injury event I involving a man M can be calculated as follows ... 

We first must establish that the data are homoscedastic in order to apply the simple (unweighted) linear regression formulae. In this case simple examination of the data shows that there is no detectable trend of the absolute values Yj j — Yj j with increasing Xj, as we would expect since the range of values of Yj would increase with Xj if heteroscedasticity were present, compare Figure 8.9. (This preliminary conclusion of homoscedacity is confirmed after the event by examining the residuals calculated on the basis of this assumption (columns 7 and 8) that demonstrate no trend of rjj with Xj, the implications of these residuals for the goodness of fit are discussed later.)... 

Probability distribution functions were first developed to measure the possibility of the occurrence of a specific event. Depending on the number of possible events, they can be discrete functions, when the number of possible events is discrete, or they can be classified as continuous functions. In the present contribution, only continuous distribution functions are studied. Probability distribution functions are defined for a reduced number of parameters and have simple formulae for calculating mean, mode, variance, etc. They have two main forms the probability density function (PDF) and the cumulative distribution function (CDF). The former is the most commonly used form of the probability distribution functions a very well known example of this type of function is the classical Gaussian bell (Evans et al, 1993). CDFs increase monotonically and generally describe the same behavior that is observed with distillation curves. Due to their simplicity, probability distribution functions are easily included in computer programs for modeling, optimization, and control purposes. 

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