Computing Probability

This chapter starts out by showing you how to do the computations needed for solving probability problems. (If you need a more thorough review of the relationships between fractions, decimals, and percents, and how to change from one to another, refer to Chapter 5.) In this chapter, you use the percentages to make the predictions. And you see what the odds are that you ll just love this topic. 

The probability of an event is the likelihood that it ll happen. The most common way to express a probability is with a percent, such as 60 percent probability of rain or 70 percent likelihood that he ll hit the ball. Probability is also expressed in terms of fractions — in fact, a fraction supplies one of the nicest ways of defining how you get the probability of something happening. 

When the probability of something happening is 95 percent, you can be pretty sure that the event will happen — 95 out of 100 times it does. A probability of 15 percent is pretty low. That sounds like the chance that I ll make a free throw in basketball. 

Following is a standard version of a probability formula. This formula allows for you to count up how many different ways something can be done and then determine the probability that just a few of those things will happen. 

The probability, P, that an event, e, will happen is found with  

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For computing probabilities, the q in the names of the functions has to be replaced by p, for computing densities it has to be replaced by d, and for generating random numbers it has to be replaced by r (using the appropriate function parameters) see the help files in R for further information. 

Explosive Stimulus Transfer. Under a program to develop a technique to compute probability of detonation transfer between a donor and an acceptor, experiments were performed to measure the effects of seven variables. These included two donor parameters fragment slack energy and five acceptor parameters confinement, closure thickness, explosive material, explosive particle size and closure material... 

High available energy is only one reason for the new chemistry that is made possible in the hot and compressed cluster. The simulations show that equally important is that collisions in the cluster necessarily occur with rather low impact parameters. (Two molecules moving relative to one another with a high impact parameter will collide with other molecules before they can collide with each other.) The same is true for nonadiabatic transitions. The computed probability of crossing to the upper electronic state decreases rapidly with increasing impact parameter. It is because the cluster favors low impact parameter collisions that the yield of reactive collisions is high. 

The structure-based thermodynamic method combines the derived binding free energy model with the formalism which computes probabilities of individual amino acids being folded in native-like conformations and thereby allows to determine structural stability of different protein regions [48-53]. In a single site thermodynamic mutation approach, the cooperativity of in-... 

The idea of the dynamic Monte Carlo method is not to compute probabilities Pa(t) explicitly, but to start with some particular configuration, representative for the initial state of the experiment one wants to simulate, and then generate a sequence of other configurations with the correct probability. The integral formulation directly gives us a useful algorithm to do this. 

Probability assignment The next step is to compute probabilities for possible mediated schemas that we have generated. As a basis for the probability assignment, we first define when a mediated schema is consistent with a source schema. The probability of a mediated schema in M will be the proportion of the number of sources with which it is consistent. 

The individual series here deviate from the general mean more than the magnitude of its probable error would lead us to suppose. The constant errors, in consequence, must be greater than the probable errors. In such a case as this, the computed probable error 0 0085 has no real meaning, and we can only conclude that the atomic weight of barium is, at its best, not known more accurately than to five units in the second decimal place.1... 

Golden and Peiser use the Eyring-Polanyi potential, whereas Bauer and Wu use an even more simphfied potential. Furthermore, Bauer and Wu treat the linear case and do not consider rotation. A major difference is that whereas Golden and Peiser bypass the activated complex and compute transitions between initial and final states, Bauer and Wu have taken the activated complex made up of Hj and Br as an actual state and have computed probabilities from the initial state to this activated state . Then they have merely chosen a transmission coefficient of for the transition from the activated state to the final state. Undoubtedly, if the interaction potential is well known, the introduction of the activated state is an unnecessary complexity, but Bauer and Wu may well be justified in their claim that the interaction potential is sufficiently mysterious at present to warrant the crutch. 

From a computational point of view, it should be stressed that the computational tool of Francisco et al. [35] results in obtaining the electron number probability distribution functions of an -electron molecule through an exhaustive partitioning of the real space into arbitrary regions. From the computed probabilities, several magnitudes relevant to chemical bonding theory are obtained, such as average electronic populations and locahzation/delocalization indices. 

Therefore caution must be taken when considering which kind of wave-function (normalized or note) to be used to compute probability density on which interval as well they have to be compatible to assure the correct Bom normalization condition at any time. 

For both perspectives we conclude that risk cannot be adequately described and evaluated simply by reference to the summarising probabilities and expected values. In the classical case we have to take into account the uncertainties in the estimates, and in the Bayesian perspective we have to acknowledge that the computed probabilities are subjective probabilities conditional on a specific background information (Aven 2008a,b). 

Second step Probability computation In order to compute probabilities, we consider that we have a system that receives segments ofbits. According to whether the segment received is error free or not, the system transitions from state to state. Thus, we can build a Markov chain. 

Shanthikumar, J.G. Sumita, VJAPPUED PROBABILITY AND STOCHASTIC PROCESSES Liu, B. Esogbue, A.O./ DECISION CRITERIA AND OPTIMAL INVENTORY PROCESSES Gal, T., Stewart, T.J., Hanne, T. / MULTICRITERIA DECISIONMAKING Advances in MCDM Models, Algorithms, Theory, and Applications Fox, B.L. / STRATEGIES FOR QUASI-MONTE CARLO Hall, R.W. / HANDBOOK OF TRANSPORTATION SCIENCE Grassman, W.K. / COMPUTATIONAL PROBABILITY... 

There is another aspect that makes measurements of transition probabilities very attractive with regard to a more detailed knowledge of molecular structure. Transition probabilities derived from computed wave functions of upper and lower states are much more sensitive to approximation errors in these functions than are the energies of these states. Experimentally determined transition probabilities are therefore well suited to test the validity of calculated approximate wave functions. A comparison with computed probabilities allows theoretical models of electronic charge distributions in excited molecular states to be improved [2.19,2.20]. 

The choice between these two approaches depends on many factors, such as the risk consequences of the failure, the availability of statistical information, the feasibility of computing probabilities with sufficient accuracy, the computational cost, etc. 

In the other side, in modem age of quantum mechanics, Wigner and Seitz stated in 1955 in front of the quantum challenges of solid state of the matter and of its combinations and transformations if someone would have a huge computer, probable was able to solve the Schrbdinger problem for each (crystal/solid/) metal and thus obtained interesting physical sizes... which most likely will be in concordance with the experimental measures, but nothing vast and new could not be obtained in such a procedure. Instead, there is preferable a lively picture of the behavior of wave functions, as basis for the description of the essence of the factors... and... 

Bayesian approach This approach uses the likelihood and prior distributions to constmct the posterior distribution, and would compute probability that activation exceeds some threshold directly from the posterior distribution. This approach can determine activation/no activation and can compare models of the data as well as gives the probability distribution of the activation given the data. 

The reference water level in an enclosed body of water which is not subject to human control should be taken as the mean value of all data on the water level for a certain time period. Surge and seiche effects cause changes in the transient water level only and do not significantly change the mean water level. The reference water level upon which the computed probable maximum storm surge or probable maximum seiche is superimposed should be seleaed so that the probability of its being exceeded over the lifetime of the plant is sufficiently low. ... 

For sites on rivers or estuaries that flow into large bodies of water, the computed probable maximum storm surge will require empirical or mathematical routing upstream of the point of interest. Sites on large enclosed bodies of water should be analysed for surges by the use of one or two dimensional surge models. 

Equation (3.25) holds for equal intervals but also for arbitrary (unequal) intervals, if the coefficients are computed accordingly. For unequal intervals, the coefficients must be computed (probably precomputed in a given program), as they cannot be tabulated, and this is best done using the Fornberg algorithm [10], to be described in the later Chap. 7. It is implemented in the routine GOFORN, also described in Appendix E. 

The ESS is right on schedule for the computer power that we would expect based on the ffends of the last 30 years. The architecture is vector parallel, so it s a combination of the kind of vectorization used 20 years ago on Cray supercomputers, with the parallelization that has become common on distributed machines over the last decade. It s no surprise that this architecture leads to a fast computer. Probably the main reason that it s newsworthy is that it was built in Japan rather than the United States. Politicians may be concerned about that, but as scientists 1 think we should just stay our course and do what we feel is right for science and engineering. 

Insertion of component failure rates Failure rates X are stored in each component model that is enhanced with failures. Constant failure rates (exponentially distributed lifetimes) are assumed per default. Since the stress level of a component is known in the simulation, its failure rate can be adapted accordingly. Failure rates are used to compute probability of system operation Rsyff) or failure from the detected minimal path sets. 

Notice that reliability problems do not impose bounds on the random variables, which is not possible to model in the PC framework, where P c must be a bounded box. Thus, to guarantee the safety of the computed probability enclosures when DczQ, a small correction term must be added to such enclosure. This is done by computing [P](I>), an enclosure for the probabiUty of event V = F((jr,P, )) and [P](I X>) = 1 - [P](X>), an enclosure for the neglected probability. Then the term [0] W [P](Qx ) is added to the enclosure computed by Algorithm 2. 

The same probabilistic model checking techniques are used for computing the probability of the top-level event in fault trees. They can be extended to computing probabilities for dynamic fault trees [6]. Akin to checking the correctness of FDIR measures, we use the same probabilistic techniques to evaluate FDIR performance. For example, in addition to checking whether a fault is detected or not, we compute the probability of detection in case of fault recovery, we compute the probability that the system will recover from a fault. 

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