Probability Definitions and Interpretations

Consider, for example, tossing a coin twice. The sample space can be described as 

If probability j is assigned to each element of S and A is tlie event of at least one head, then 

The description of tlie sample space is not unique. The sample space S in tlie case of tossing a coin twice could be described in tenns of the number of heads obtained. Then 

How probabilities, are assigned to tlie elements of the sample space depends on tlie desired interpretation of the probability of an event. Thus P(A) 

As anoUier example, consider a single valve tliat can stick in an open (O) or closed (C) position. The sample space can be described as follows  

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The aforementioned interpretation of (x,f) 2 as a probability density is possible because positive definite and (b) when integrated over all space, i.e., J i//(x,t) 2d3x, is time independent. (By a suitable normalization of (0 > this integral can always be made equal to 1.)... 

Elucidation of the non-local holistic nature of quantum theory, first discerned by Einstein [3] and interpreted as a defect of the theory, is probably the most important feature of Bohm s interpretation. Two other major innovations that flow from the Bohm interpretation are a definition of particle trajectories directed by a pilot wave and the physical picture of a stationary state. 

The picture to this point then is the collision of hard spheres given in Figure 2.7, with energies as defined in equation (2-14) and probabilities as given in equation (2-15). How do we turn this into a reaction rate expression that is compatable with all the work done previously to get to equation (2-12) This has been done over the years, following the terminology of the theorists, in terms of yet a new quantity that is generally called the reactive cross-section. For now, let us write down the definition and worry about the physical interpretation later. Thus, we define a reactive cross-section as... 

The frameworks can provide useful guidance, but their scientific basis deserves further work. If a concept is introduced in a framework it should be properly defined. The frameworks are for instance unclear about definitions of probabilities and uncertainties. Referring to a probability is not sufficient as probabilities can be interpreted in different ways. And depending on the chosen interpretation, we are led in different directions for assessing risk. 

Have clear definitions of terms like probability , uncertainty and reliability so that the results are intuitive and easy to interpret. 

Figure 8.11 Graphical representations of the definition and implications of the EPA definition of an MDL. (a). Assumed normal frequency distribution of measured concentrations of MDL test samples spiked at one to five times the expected MDL concentration, showing the standard deviation s. (b) Assumed standard deviation as a function of analyte concentration, with a region of constant standard deviation at low concentrations, (c) The frequency distribution of the low concentration spike measurements is assumed to be the same as that for replicate blank measurements (analyte not present), (d) The MDL is set at a concentration to provide a false positive rate of no more than 1% (t = Student s t value at the 99 % confidence level), (e) Probability of a false negative when a sample contains the analyte at the EPA MDL concentration. Reproduced with permission from New Reporting Procedures Based on Long-Term Method Detection Levels and Some Considerations for Interpretations of Water-Quality Data Provided by the US Geological Survey NationalWater Quality Laboratory (1999), Open-File Report 99-193.

Frequency with the dimensions of per unit time, ranges from zero to infinity and means the number of occurrences per time interval. Probability is dimensionless, ranges from zero to one, and has several definitions. The confusion between frequency and probability arises from the need to determine the probability that a given system will fail in a year. Such a calculation of probability explicitly considers the time interval and, hence, is frequency. However, considerable care must be used to ensure that calculations are dimensionally correct as well as obeying the appropriate algebra. Three interpretations of the meaning of probability are ... 

I.I.I. Comment. The anhaptoglobinemia frequently found at birth has been discussed by Fine et al. (FI), who conclude on the basis of immunological studies that Hp probably exists in cord blood, but has not got its definitive structure. Their experimental observations and suggestions are interesting, but owing to lack of quantitative data their interpretation is debatable. 

At this point we again stress the sequence of definitions leading to Eq. (4.2.16). First, the correlation function is defined as a measure of the extent of the dependence between the two events in Eq. (4.2.12) [or, equivalently, in Eq. (4.2.13)]. The probabilities used in the definition of g a, b) were read from the GPF of the system, e.g., (4.2.1). This side of g a, b) allows us to investigate the molecular content of the correlation function, which is the central issue of this book. The other side of g a, b) follows from the recognition that the limiting value of g(a, b), denoted by g a, b), connects the binding constants ah and kg A. This side of g a, b) allows us to extract information on the cooperativity of the system from the experimental data. In other words, these relationships may be used to calculate the correlation fimction from experimental data, on the one hand, and to interpret these correlation functions in terms of molecular properties, on the other. 

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