Simple Probabilities

In a simple euewf probability, the researcher compares the number of times an event occurred compared to the total number of possible outcomes. 

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Note that in practice the abstract species A and B may themselves already be molecules or supermolecules formed from prior condensations, but simple probability arguments make condensation reactions simultaneously involving more than two species impossible under most sets of experimental conditions. The rate law associated with eq. 15.11 is... 

Now if the boundary forces are thought of as drawn at random from a mixture of a proportion 9 of w(x, t), 9 of w x, t), etc., simple probability reasoning shows that, with regard to this body of knowledge, the actual... 

The statistical treatment of random stepwise crosslinking reactions (e.g. polycondensation) neglecting ring formation originates from Stock-mayer and Flory and is explained in Flory s book (55) on a number of examples. Using simple probability statistics, it is possible to calculate the molecular size distribution in the sol and in the gel, fractions of sol and gel, the crosslinking density and the fraction of free functionalities in... 

What is the probability that a given fragment among the total produced will be found in one of the recombinant phage in the library From simple probability theory the number of clones that must be isolated and screened is given by Eq. 26-2.180 Here N is the number of clones needed, p is the probability of... 

Equation 5.16 may be used to determine the simple probability (Ojj) of finding a molecule of type J in a cavity of type i. This value may be obtained by dividing the number of molecules of k in cavity i by the total number of cavities of type i,... 

The genomist asserts that this experiment has shown beyond reasonable doubt that chemistry is not a random process The reader, chemist or not, can verify the genomists assertion by counting the atoms in the mixture, (assuming that they are spheres without features) and do a simple probability calculation. The result shows that amino acids and purines and pyrimidines should have occurred at such a low concentration that they would not be detectable with our technology. Carbon, nitro-... 

A similar analysis holds for molecules that have more than two chirality centers Each chirality center may have either the R or the 5 configuration. The number of possible stereoisomers can be calculated by using simple probability theory. A molecule with a number of chirality centers equal to n has a maximum of 2" stereoisomers ... 

In this model, where no preference of Si-Al over Al-Al pairing is assumed, the NMR spectrum is determined by the simple probabilities of uncorrelated occupancies of the four nearest neighbors. Therefore, the peak intensities for the completely random model are independent of the details of the zeolite framework topology. They are... 

In connection with the stability of aqueous solutions of phenothiazine drugs, it has been suggested that simple probability theory could be used to determine the composition of the ionic environment around a drug molecule, and to predict thereby the stability of various salts of the same drug. ... 

L.G.Gibilaro, H.W.KropholIer and D.Spikis, Solution of a Mixing Model due to van de Vusse by a Simple Probability Method, Chemical Engineering Science, 22, 517-523(1967). 

Problem 7.2 Use simple probability concepts to justify the following statement a value of unity for the product rir2 signifies that the likelihood that an Mj unit in the copolymer chain follows an Mi unit is the same as the likelihood that it follows an M2 unit. 

Further details about the molecular weight distribution of the polymer chains can be obtained by simple probability arguments. If the probability of adding another monomer unit to a chain is p, the probability of a chain length P (number distribution) with random addition is... 

Understanding the connection between entropy and spontaneity will allow us to answer such questions. We will begin to explore this connection by considering a very simple process, the expansion of an ideal gas into a vacuum, as represented in Fig. 10.3. Why is this process spontaneous What causes the gas to expand to a uniform state The driving force can be explained in terms of simple probability. Because there are more ways of having the gas evenly spread throughout the container than there are ways for it to be in any other possible state, the gas spontaneously attains the uniform distribution. 

One difficulty with this type of model is that it is difficult to provide a probability value for an absolute repair time. For example, the Mean Downtime (MDT) for P-lOlB is 3 hours this is the time it takes to repair it when it fails. It may be that this time is an absolute minimum (say due to the time it takes to isolate the pump, drain it down, repair the failure, and bring the pump back into service). Hence a simple probability value cannot truly represent this downtime. 

Simple probability of perforation as a function of strike velocity. 

Another model introduced by Fritsch-Faules and Faulkner suggests that or De should first have an exponential rise with increasing c and then flatten at high concentrations. The exponential rise occurs because d becomes smaller as the concentration increases, which promotes intersite electron transfer. As the minimum center-to-center separation is approached, when each redox center has a nearest neighbor that is practically in contact, k or asymptotically approaches its theoretical maximum value. A similar result has been obtained by a microscopic model which describes electron (or hole) diffusion in a rigid three-dimensional network. This concept is based on simple probability distribution arguments and on a random walk [36]. 

Self-assembly defies simple probability calculations. You have to drop your calculator and run the experiment. 

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