What Is Probability

Here arc two statements of probabilily. In 1990, the probability that a person in the United States was a scientist or an engineer was 1/2S0. That is, there were about a million scientists and engineers out of a total of about 250 million people. In 1992, the probability that a child under 13 years old in the United States ate a fast-food hamburger on any given day w-as 1/30 1]. 

Let s generalize. Suppose that the possible outcomes or events fall into categories, B, or C. Event and outcome are generic terms. An event might be the flipping of a coin, resulting in heads or tails. Alternatively it might be one of the possible conformations of a molecule. Suppose that outcome A occurs 20% of the time, B 50% of the time, and C 30% of the time. Then the probability of A is 0.20, the probability of B is 0.50, and the probability of C is 0.30. 

1 he definition of probability is If N is the total number of possible outcomes, and n, of the outcomes fall into category A, then p, the probability of outcome A, is 

Probabilities are quantities in the range from zero to one. If only one outcome is possible, the process is deterministic—the outcome has a probability of one. An outcome that never occurs has a probability of zero. 

Probabilities can be computed for different combinations of events. Consider one roll of a six-sided die, for example (die, unfortunately, is the singular of dice). The probability that a 4 appears face up is 1/6 because there are N = 6 possible outcomes and only 114 = 1 of them is a 4. But suppose you roll a six-sided die three times. You may ask for the probability that you w ill observ e the sequence of two 3 s followed by one 4. Or ou may ask for the probabilit of rolling two 2 s and one 6 in any order. The rules of probability and combinatorics provide the machinery for calculating such probabilities. Here we define the relationships among events that we need to formulate the rules. 

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The values of d and n are given in Table 3 typical values for can be found in Table 4. The exponent of 0.5 on the Schmidt number (l-L /PiLj) supports the penetration theory. Further examples of empirical correlations provide partial experimental confirmation of equation 78 (3,64—68). The correlation reflecting what is probably the most comprehensive experimental basis, the Monsanto Model, also falls in this category (68,69). It is based on 545 observations from 13 different sources and may be summarized as... 

Nickel and other transition metals function as solvent-catalysts for the transformation of carbon species into the diamond aHotrope. At temperatures high enough to melt the metal or metal—carbon mixture and at pressures high enough for diamond to be stable, diamond forms by what is probably an electronic mechanism (see Carbon, diamond-synthetic). 

Though the literature is replete with methods of measuring the moisture content, truly accurate as well as practical methods are virtually nonexistent in the food field. The situation is well illustrated in what is probably the best compendium on this subject, the Official and Tentative Methods of Analysis of the Association of Official Agricultural Chemists (2). It becomes apparent from an examination of this volume that the stress is laid not so much on accuracy as on reproducibility and practicability of a method. Though these last two factors are for the most part the only ones of importance in the control of processing procedures and in standardization of products of commerce, the factor of accuracy is, nevertheless, of extreme importance to the research worker who endeavors to establish broad quantitative generalizations for the conditions that govern the stability of foods. 

For what is probably the earliest microscopic calculations of thermodynamic cycles in proteins see Ref. 12, that reported a PDLD study of the pKtt s of some groups in lysozyme. The use of FEP approaches for studies of proteins is more recent and early studies of catalysis and binding were reported in Refs. 11, 12, and 13 of Chapter 4. 

It must again be emphasized that the numbers in the quantitative discussions of the preceding paragraphs provide nothing more than a cmdc hint as to the scale of the problems. But they are consistent with what is probably obvious without numbers the risks from the open-ended problems are far greater than those from the confined problems. For the former, the potential worldwide toll is probably on the scale of millions of lives per century while for the latter it probably is on the scale of a thousand lives per century. 

Sakurai et al. have provided what is probably the most important mechanistic finding in the area of intermolecular additions of silenes in recent years, namely a detailed proposal for the mechanism of alcohol addition to the silicon-carbon double bond.68 A cyclic silene 116 was synthesized in the presence of various amounts of methanol and other alcohols, and varying proportions of methanol adducts 117 and 118 were obtained. It was concluded that the methanolysis involved two steps, the first being the association of the oxygen lone pairs with the sp 2-hybridized silicon atom of the silene. The second step, proton transfer, could occur in two ways. If the proton was transferred from the complexed methanol molecule (path a) its delivery would result in syn addition. However, if a second molecule of methanol participated (path b), it would deliver its proton... 

The fast Fourier transform can be carried out by rearranging the various terms in the summations involved in the discrete Fourier transform. It is, in effect, a special book-keeping scheme that results in a very important simplification of the numerical evaluation of a Fburier transform. It was introduced into the scientific community in the mid-sixties and has resulted in what is probably one of the few significant advances in numerical methods of analysis since the invention of the digital computer. 

The biological activity of a compound can often be affected dramatically by the presence of even a single fluorine substituent that is placed in a particular position within the molecule. There are diverse reasons for this, which have been discussed briefly in the preface and introduction of this book. A few illustrative examples of bioactive compounds containing a single fluorine substituent are given in Fig. 3.1. These include what is probably the first example of enhanced bioactivity due to fluorine substitution, that of the corticosteroid 3-1 below wherein Fried discovered, in 1954, that the enhanced acidity of the fluorohydrin enhanced the activity of the compound.1 Also pictured are the antibacterial (3-fluoro amino acid, FA (3-2), which acts as a suicide substrate enzyme inactivator, and the well-known anti-anthrax drug, CIPRO (3-3). 

In conclusion, we are at a very preliminary step on what is probably a long but promising path. The modulation of estrogen action seems a powerful mechanism in the control of CVD risk. Additional advances in the knowledge... 

There have been several attempts over the years to classify adverse drug reactions, primarily based on the perception of what can and what cannot be predicted given knowledge of the pharmacology and toxicology of the drug. For example, Edwards and Aronson [13] proposed what is probably the most extensive classification system ... 

The water data plotted in Figures 3-16 showed that most of the pesticides are sorbed on soil particles. This confirmed what is probably the most beneficial effect of soil in the containers it guaranteed almost complete containment even when the liquid contents were lost. 

The theory of representations of symmetric groups is intimately connected with the idea of partitions of integers. Rutherford[7] gives what is probably the most accessible treatment of these matters. A partition of an integer is a set of smaller... 

In turn, Robert Lebel, another very close acquaintance of Duchamp, has rendered what is probably the best overall summation of the total significance of the endlessly enigmatic artist s meta-optical, and therefore for him similarly pseudoscientific, obsessions. As described by Lebel,... 

Less than half a century after Balboa had stood silent on a peak in Darien, facing the unknown ocean, a famous Italian scholar and poet, Julius Caesar Scaliger, or della Scala, recorded the presence there of an unknown noble metal. In 1557 he made what is probably the first definite allusion to platinum. Girolamo Cardano (1501—1576), in his well-known work On Subtlety, had defined a metal as a substance which... 

There exists a much simpler example. Let x and y be independent positive real number. This means that vector (x, y) is uniformly distributed in the first quadrant. What is probability that x y Following the definition of probability based on the density of sets, we take the correspondent angle and find immediately that this probability is Vi. This meets our intuition well. But let us take the first number x and look for possible values of y. The result for given x the... 

This section would not be complete without a reference to what is probably the best-known form and the most common usage of break-seals. 

Bromination of 1-phenylpyrazoles resembles nitration in that it occurs initially in the benzene ring, and only then in the heterocyclic nucleus to give what is probably 4,5-dibromo-l-(p-bromophenyl)pyrazole [61JCS2769 66AHC(6)391]. 3-Methyl-1-phenylpyrazole was brominated at both C-4 and in the para-position (61JCS2769), and there are other examples (83JHC277). 

A quinazolone moiety also provides the nucleus for a highly simplified leukotriene antagonist (compare this compound with verlukast (29-6), discussed earlier in this chapter). Condensation of the anthranilate ester (85-1) with formamide leads to the formation of the quinazolone (85-2). Reaction of the salt from the reaction of this product with a strong base with ethyl 3-bromoacrylate leads to vinylation on nitrogen by what is probably an addition-elimination sequence the product is largely the E isomer (85-3). Saponification then affords tiacrilast (85-4) [95]. 

The top right hand cell in Figure 3.1, leadership through innovation , represents what is probably most usually regsu -ded as leadership the proactive, creative initiation of change. 

Recently, a report appeared of what is probably a conclusive optimization of the system [102]. After isolating a (25 kDa) lipase-like enzyme from crude PPL, applying intricate immobilization methods, and carefully checking experimental conditions, the authors report E-values in the kinetic resolution of glycidyl butyrate that are still no higher than 10. However, illustrating the importance of organic solvents on enzyme enantioselectivity, when dioxane (10% v/v) is added as a cosolvent, E > 100 is found ... 

Thus, the susceptibility is the result of accumulation of the drug in the target organ to reach concentrations not achieved in other tissues. This is then followed by what is probably a combination of events such as formation of a reactive intermediate, possibly a free radical, stimulation of lipid peroxidation and depletion of GSH, and then peroxidative damage to cell membranes and mitochondria. Whether metabolic activation by cytochromes P-450, or chemical rearrangement, or reductive activation, or all the three are involved is not currently clear. 

Alkyl amines add photochemically to olefins a condensation occurs between the a-carbon atom of the amine and the terminal carbon atom of 1-olefins by what is probably a free-radical, chain reaction.291 Internally illumination of solutions, containing piperidine and octene-1, with a mercury discharge tube led to the formation of 2-n-oetylpiperidine (CXLV). 

From the principle of microscopic reversibility it may be inferred that what is probable in absorption should also be probable in emission. A high emission probability for (it, it ) state is predicted. The rate constant for emission is observed to be large for dipole allowed it it transitions. As a consequence, other deactivating processes cannot compete with the radiative process, and a high fluorescence efficiency for such a system is usually observed. 

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