Binomial Probabilities

For np > 5 and n( 1 - p) > 5, an approximation of binomial probabilities is given by the standard normal distribution where z is a standard normal deviate and... 

The probability function that has been displayed is a very special case of the more general case, which is called the binomial probability distribution. 

Table 12 illustrates the computation procedure in the case of m = 5 the plant may be envisaged, as in Section V.l, to consist of m cell banks, the quantity j denoting the number of banks switched back into operation. In the specific case of X = p (equi -probability of switching into either direction), Eq. (47) reduces to the binomial probability distribution of selecting j elements out of m identical elements with a single-event probability of xh. 

For further information see Reference 18.] The event might be the presence of any particular attribute in a sample, such as the detection of a pesticide. Only two levels of the attribute are possible, present or not present. If many attributes contribute to the result of an observation, the binomial probability distribution approaida.es a limiting curve whose equation is given by y = (1/ /211) exp[-(2 jx) As... 

A compound is flagged as either active or inactive on the basis of the activity threshold determined as in Section 6.2.2. The probability that the observed number of actives x occurs in a cluster of size n given a background hit rate p (expressed as a fraction) is determined by the binomial probability mass function [38], shown as follows ... 

A binomial probability model is to be based on the following index function model ... 

Mathematical methods for the calculation of theoretical relative abundances within the isotopic duster, for comparison with experiment, usually rely on expansion of the polynomial expression based on an extension of the binomial probability distribution [12, 13]. Indeed, for an element with x isotopes and relative abundances /1,/2, ., /, and for... 

To check that the method can be used for isobaric data a set of perfect data are generated and random errors added to x, y, T, and tt in turn and all together to see what effect they have on our standard procedure. For large samples we expect 68% of the sample values to lie within one standard deviation of the perfect value of the selected variable. In the case of small samples, e.g., twelve data, error bounds are calculated using binomial probabilities for each of the above variables so that, with probability of 0.95, we expect 41-95% of the sample observations to lie within one standard deviation of the perfect value of the selected variable (the normal distribution is assumed). Twelve is a common number of data points with salt-saturated solutions and this shows the desirability of taking more experimental observations. 

The one-tailed probability obtained using the normal approximation to the binomial probability is 0.0091, and the logarithm of the inverse of this probability value is used as the activity weight for the considered substructure, i.e. 

To establish the statistical significance of the likeliness function of each th SEV, the binomial probability py is calculated as ... 

We should note the absence of dose standardization and probably of randomization because Lind s two seawater patients were noted to have tendons in the ham rigid , unlike the others. However, the result had been crudely replicated by using n = 2 in each group. If we accept that the hypothesis was that the citrus-treated patients alone would improve (Lind was certainly skeptical of the anecdotal support for the other five alternative treatments), then, using a binomial probability distribution, the result has p = 0.0075. But statistics had hardly been invented, and Lind had no need of them to interpret the clinical significance of this brilliant clinical trial. 

Example 2.17 Binomial Probability fiom Tooth Cavities. Suppose that the incidence of tooth cavities in a population of childien is known to be 10%. A toothpaste manufacturer... 

This is reasonably close to the binomial probability of 0.392 found in Example 2.17. The R code for this example is as follows ... 

Table 6.3 illustrates the prediction accuracy obtained by ensemble majority voting. When p = 0, the standard binomial probability in Eq. (6.1) is used for n<25 and the normal approximation is used for a larger n. The beta-binomial model is used when the correlation is positive, and the extended beta-binomial model is used when the correlation is negative. The table illustrates that negatively correlated classifiers improve the prediction accuracy more rapidly than the independent classifiers. 

The two functions specify the deviations of the discount functions from the implied forward functions. To satisfy arbitrage-free conditions, they define an implied binomial probability tt that is independent of time T, while the initial discoxmt function P(T) is given by ... 

Generally, to calculate the probability of a certain number of successes in a series of trials, where the probability of success is constant, and where the trials are independent, it is appropriate to use the binomial probability mass function ... 

A total of 12 simultaneous choice trials were run each having a duration of five minutes. Males showed a significant preference for females compared to she-males in all 12 simultaneous choice tests (binomial probability, p <. 005). Similar results were obtained when males were presented with females and she-males in a sequential choice paradigm. 

In situations where there are only two possible outcomes for an event—such as a male/female or yes/no situation—the distribution is said to be binomial Observing a safety valve a number of times and determining the probability that it will be open or closed a specific number of times meets the binomial probability since there are only two possible outcomes, open or closed. The formula presented below can be used to determine the binomial probability (Hays 1988) ... 

A safety inspector observes a safety valve 5 times and finds that it is open 4 of the 5 times. Past observations have indicated that there is a 50% chance at any given time that the valve will be open. What is the probability the inspector s observation That is, what is the probability of observing this valve 5 times and finding it open 4 times In the binomial probability formula, n is equal to the number of observations—in this case, the number of times the valve is observed so n = 5. The number of desired... 

General Industrial Safety Application of the Binomial Probability... 

Safety managers can use binomial probabilities in a variety of situations to determine the probability of a particular outcome. In this application, a safety manager has collected safety inspection data over a period of time for a particular power press. A summary of the inspection data is as follows ... 

Binomial Probability a probability in which only two possible outcomes for an event are possible—such as a heads/tails or yes/no situation the distribution of these probabilities is considered to be binomial. 

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