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John Arbuthnot's celebrated but flawed paper in the Philosophical Transactions of 1711–12 is a philosophically and historically plausible target of Hume's probability theory. Arbuthnot argues for providential design rather than chance as a cause of the annual birth ratio, and the paper was championed as a successful extension of the new calculations of the value of wagers in games of chance to wagers about natural and social phenomena. Arbuthnot replaces the earlier anti-Epicurean notion of chance with the equiprobability assumption of Huygens's mathematics of games of chance, and misrepresents the birth ratio data to rule out chance in favour of design. The probability sections of Hume's Treatise taken together correct the equiprobability assumption and its extension to other kinds of phenomena in the estimation of wagers or expectations about particular events. Hume's probability theory demonstrates the flaw in this version of the design argument.

The distribution of the chi-square goodness-of-fit statistic is studied in the equiprobable case. Tables of exact critical values are given for a = .1, .05, .01, .005; k = 2(1)4, N = 26(1)50; k = 5, N = 26(1)40; k = 6(1)10, N = 26(1)30, where a is the desired significance level, k is the number of cells and N is the sample size. Methods of fitting the true distribution are compared. If k> 3, it is found that a simple additive adjustment to the asymptotic chi-square fit leads to high accuracy even for N between 10 and 20. For k = 2, the Yates corrected chi-square statistic is very accurately fitted by the usual chi-square distribution.

This paper presents a very simple approximation, having “graph” accuracy, for calculating bivariate normal probability contours. Such contours are useful in simultaneous inference and other applications. This method is then adapted to approximate roughly bivariate and multivariate normal union probabilities.

Complementary color coding is used in conjunction with nontraditional legend arrangements to create two-variable maps. The logic of a complementary color-coding scheme is simpler and easier for the map user to comprehend than the spectrally encoded scheme used to produce most of the two-variable maps published to date. In addition to the complementary color-coding scheme, an unclassed legend based on a line of best fit and a legend arrangement based on the binormal distribution are used to represent more accurately the structure of the bivariate scatter diagram. These legends avoid many of the statistical distortions introduced by the rectangular legend currently being used to produce two-variable maps. Although the unclassed two-variable map presents a true graphic correlation, the large number of color tones contained in the map confounds the map user. The two-variable map based on the binormal distribution combines the rectangular legend and an equiprobability ellipse to represent the legend d...

Boltzmann's hypothesis of the equiprobability of the various ways of combining microslates (giving rise to the different configurations of a system) is erroneous. This renders untenable the ergodic hypothesis according to which the configurations that prime the order in an isolated system are reached randomly and points to the need to postulate the existence of a hidden variable in order to explain the system's shift away from its state of maximum entropy.

The strength of a significance test is defined as a weighted average of the power especially if the weights are a prior density. The concept is a compromise between Neyman-Pearsonian and Bayesian methods and is applied to tests for multinomial equiprobability and for no association in contingency tables. The differences of the power functions of two criteria, X 2 and G, are illustrated for the trinomial, where G arose earlier from a partially Bayesian approach. Other methods for apprehending power functions of several variables are discussed. For 2 × 2 contingency tables with equal row or column totals the power and strength functions of several statistics are equal.

In sampling from general populations, the skewness and kurtosis statistics are subject to the constraint b 2 > 1 + b 1. A bivariate product-density model for the distribution of √b 1 and b 2 is studied, consisting of a Johnson Su approximation to the marginal density of √b 1, and a gamma density for the conditional distribution of b 2. Equiprobability contours are given for sampling from normal and nonnormal populations. In the normal case, an eight-parameter model is completely specified.

The parity distribution of nuclear excited states is investigated in the frame of a statistical microscopic Nilsson-BCS formalism. Calculations performed for medium-mass nuclei (51≤A≤66) show that the parity equiprobability holds at excitation energies greater than 15 MeV, in agreement with the experimental information.

SummaryThe evenness of behavioural observations of a group of honeybees is indicated by the easily calculable D1 measure (divergence from equiprobability) of information theory. A low D1 measure indicates that the members of a cohort of bees used for collecting behavioural data have been selected without bias. Values of D1 ranging from approximately 2% to 7% of H1max (first-order diversity) are reported for 13 colonies of bees observed using scan sampling in experiments made in the past several years. The interpretation and importance of these results are discussed.

For the m-way contingency table, we discuss both the direct estimation of the multiplicative interactions among the m variables, and the indirect testing of hypotheses pertaining to these interactions. We consider, among other things, hierarchical hypotheses pertaining to the interactions among the m variables, including hypotheses that can be expressed in terms of one or more of the following kinds of concepts: (a) the usual concepts of independence and equiprobability; (b) concepts describing conditional properties (e.g., conditional independence) pertaining to a subset of the m variables, given the level of some of the remaining variables; (c) concepts related to the usual logit-analysis or to a generalized form of logit-analysis; and (d) concepts related to a more general log-linear model. Methods of partitioning these hypotheses are introduced which provide, among other things, insight into the relationship between tests applied to the m-way table and tests applied to marginal tables formed ...

A step-by-step method is presented herein for partitioning a certain kind of hypothesis H about the m-way contingency table into (a) a series of hypotheses about marginal tables formed from the m-way table by ignoring one or more of table's m dimensions; and (b) a hypothesis about independence, conditional independence, or conditional equiprobability in the m-way table. This step-by-step method facilitates both the testing of H and the calculation of , the estimated expected frequencies in the m-way table under H. The method introduced herein for calculating is easier to apply than the usual iterative-scaling method in many cases.