Relative probability

Equally foreign is the property that when 9 1 the relative probability... 

Ihese relative probabilities can be easily determined by simply counting the number c imes during the simulation that the relevant value of lambda reaches unity. In the case ( the para-substituted benzamidines it was possible after only a relatively short simulatio (110 po) to observe that the p-chloro and p-methyl derivatives were significantly weaki than the p-amino and the parent compound (Figure 11.18). In this particular case, all foe... 

The statistical problem. The relative probability associated with the placement of the two coils such that d = °° is unity The solution is so dilute that we can place the polymer molecules anywhere. It is at smaller d s that the placement probability drops off because of a generally unfavorable AG associated with the overlap. We assume that the decrease in probability is described by the Boltzmann factor and write... 

That means that the transition rate is equal to the relative probability of being an activated reactant state times the average forward flux... 

The average flux across the transition state dividing surface will be proportional to the relative probability of being found at the transition state,... 

J 9 Assess the relative probability of finding an electron at a given distance from the nucleus of an atom (Example 1.9). 

The relative probabilities of Reactions 24, 25, and 26 were, respectively, 1.00, 0.25, and 0.12 at a hydrogen pressure of about 1 atmosphere (9). These numbers could be derived either by analyzing the stable alkanes formed in the unimolecular decompositions (Reactions 24-26) or from the products of the hydride transfer reactions between C5Hi2 and the alkyl ions. Elimination of H2 from protonated pentane may also occur, but it is difficult (although not impossible) to establish this reaction through neutral product analysis. 

If in the ideal case the noise on the measurements is zero, such that N = 0, then the measurements are explained exactly and the basis coefficients are assigned by their relative probability, which is determined by C. 

Here the information on the relative probability of the basis coefficients contained in C is lost. 

The possibility of experimental investigation is, of course, restricted to systems with particular ligand partition, where the number of isomers is not too big. The aim of such investigations is to obtain information about the relative probabilities of the five mechanisms which have been defined, till now, on symmetry grounds only. We think that a comparison of the experimental spectra of relaxation times with the various possible theoretical ones (see Table 3) should furnish answers to that question, provided the assumptions of the model are a good description of the physical system. 

The probability of finding the center of gravity of polymer molecule Z in a volume element dV far removed from any other polymer molecule naturally will be proportional to the size of the element. (This volume element should not be confused with the dV represented in Fig. 114 and employed above in the investigation of segment interactions in the vicinity of the pair kyL) Presuming pi — Ki to be positive, the probability that molecule Z is found near another molecule, such as fc, will be diminished to an extent depending on AFa for the pair. If we consider volume elements of the same size, one at a finite distance a from molecule k and the other far away from any polymer molecule, the relative probability that the center of molecule I will occur in the former compared with the latter is... 

Not all wave functions can be normalized. In such cases the quantity (x, t)p may be regarded as the relative probability density, so that the ratio... 

The term exp[iax] in equations (2.47) indicates travel in the positive x-direction, while exp[—iax] refers to travel in the opposite direction. The coefficient A is, then, the amplitude of the incident wave, B is the amplitude of the reflected wave, and F is the amplitude of the transmitted wave. In region III, the particle moves in the positive x-direction, so that G is zero. The relative probability of tunneling is given by the transmission coefficient T... 

These four relations are sufficient to determine any four of the constants A, B, C, D, F in terms of the fifth. If the particle were confined to a finite region of space, then its wave function could be normalized, thereby determining the fifth and final constant. However, in this example, the position of the particle may range from -oo to oo. Accordingly, the wave function cannot be normalized, the remaining constant cannot be evaluated, and only relative probabilities such as the transmission and reflection coefficients can be determined. 

The transmission coefficient T in equation (2.58) is the relative probability that a particle impinging on the potential barrier tunnels through the barrier. The reflection coefficient R in equation (2.59) is the relative probability that the particle bounces off the barrier and moves in the negative v-direction. Since the particle must do one or the other of these two possibilities, the sum of T and R should equal unity... 

As these functions cannot be normalized, it is sufficient here to pose A[2 = 1 and calculate the relative probability densities in each succeeding step. Then, R = 1512 represents the reflection coefficient and T = F 2 the transmission coefficient Assuming that the particle cannot remain trapped within the hairier, the relation... 

Figure 9.4. Relative probability of being involved in a road traffic accident with rising levels of alcohol. The shaded area indicates typical legal drink-driving limits in Westernised countries (80mg/100ml in the UK and some states of the USA, whereas lOOmg/lOOml is still the most prevalent legal level elsewhere in the USA).

The algorithm also incorporates tacticity control for vinyl chains. The random number generator is used to choose between d- and l-versions of the transformation matrix. A single parameter controls the relative probability of d- and l-residues. The poly(p-fluorostyrene) results presented here are for atactic (stereochemically irregular) chains. 

Similarly, when CH4 containing a little n-C5D12 is irradiated the immediate product of the proton-transfer reaction, C5D12H+ will dissociate in several different ways. It is believed that (1) the relative probability of dissociative proton transfer reaction increases with exothermicity (2) at high pressures, the different modes of dissociation on proton transfer from H3+ are about equally competitive and (3) there is little or no H atom reshuffling on proton transfer (Ausloos and Lias, 1967). The last-mentioned item may be established by isotope studies. 

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