Probability Rules

There are rules that apply to probabilities that allow safety managers to manipulate and interpret the results (Hays 1988, 22-25). For example, suppose a safety manager wanted to find the probability that there is not a chemical spill this month. Something not happening is not an event however, the safety manager can use the following rules and concepts to determine this probability anyway. 

1) The probability of an event must be between 0 and 1.00. A probability of zero means an event will not happen. A probability of 1 means an event is certain to happen. (This rule is called the rule of probability range.) 

2) For every event A, there is also a complementary event A, read as not A.  

3) The sum of the probabilities of events A and A is always 1.00. This is called the rule of complementary probability. 

In the example above, if A is a spill, then its complementary event, A s there is no spill. According to the rule of complementary probability, P(A) + P( A) = 1. (This is simply another way of saying that there will either be a spill or there will not.) Therefore, if the safety manager want to find the probability of there not being a spill, he or she could calculate P(A) = 1 -P(A).Fromabove,P(A) =. 12,thereforeP(A) = 1 -. 12 =. 88, or an 88% probability that there will not be a spill this month. 

---

In this application the reader will examine the influence of the interrelation between the two liquids on the extent and rate of the demixing process. In Example 5.1, the two liquids have a modest affinity for each other, characterized by rules describing a relatively low breaking probability rule between them. These setup data are found in Parameter setup 5.1. 

The pentacyclic component at m/e 412 resembles the spectrum of oleanane (structure shown) or gammacerane. It is also similar to that of lupane (13, 15) but lacks the M-43 (369) fragment indicative of any isopropyl sidechain. Therefore, ring E is probably 6-member-ed, not 5-membered as in lupane or the hopanes. Although the mass spectrum is a slightly better match to gammacerane, the short GC retention time probably rules out gammacerane but not oleanane (13). 

Nucleophilic reactions of the spin-paired tris(o-phenanthroline) iron(II) ion are bimolecular 70-72). The tris complex is close to the spin-free complex in energy since dithiocyanatobis(o-phenanthroline) iron(II) exists in a spin-free = spin-paired equilibrium 53). The corresponding tris(o-phenanthroline)nickel(II) ion is unaffected by the same nucleophile, which probably rules out Sat2 attack on the organic ring as the predominant factor. 

A chemically pure compound, such as 99.99 % vanillin, is in fact a complex mixture of isotopomers, themselves a combination of isotopes of the elements concerned (C, H and 0), and distributed according to their isotopic abundances. Probability rules predict 884736 isotopomers of vanillin molecule, C8H803. At the natural abundance of the stable isotopes of C, H and O (Table 1) the light isotopomer of vanillin, 12C81H81603, has an occurrence probability of 0.90685, whereas the heavy isotopomer, 13C82HgI803, has only a 6.4478 E-55 chance of occurring in nature. In other words, we would have to produce a mass of vanillin much larger than the mass of the Sun to be able to observe its heavy isotopomer ... 

The same probability rules that apply to elementary events also apply to composite events. Moreover, pa, Pb, and pc are each products of elementary event probabilities because the first and second rolls of the die are independent ... 

The same probability rules apply to the composite events that apply to elementary events. For example, Equation (1.21) for the normalization of discrete distributions requires that the probabilities must sum to one ... 

Kafri R, Lancet D (2004) Probability rule for chiral recognition. Chirality 16 369-378 Mesecar AD, Koshland DE Jr (2(XX)) A new model for protein stereospedficity. Nature 403 614-615... 

The kinetic study on tertiary-amine-catalyzed ester formation from benzoyl chloride (BC) and phenol in dichloromethane at 0°C showed the following (1) the lack of phenoxide in the UV-visible spectral study, which probably rules out specific base catalysis, (2) the presence of benzoylammonium salt (AAS) observed in low-temperature NMR experiments, (3) the presence of small or negligible deuterium isotope effect (i.e., kjj/ko 1), which rules out the general base catalysis, and (4) the rates of the reactions of BC with quinuclidine and triethyl amine (the most basic amines in pure aqueous solvent) differed by at least 3 orders of magnitude. These observations suggest the occurrence of nucleophilic catalysis for phenyl benzoate formation in the reaction of BC with phenol under the presence of tertiary amines. However, butyl benzoate formation... 

---