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The embedded Markov chain approach is widely used in queuing theory, in particular in M/G/1 and GI/M/c queues. In these cases, one has to relate the embedded equilibrium probablities to the corresponding random-time probabilities. The classical method to do this is based on Markov renewal theory, a rather complex approach, especially if the population is finite or if there is balking. In this article we present a much simpler method to derive the random-time probabilities from the embedded Markov chain probabilities. The method is based on conditional probability. Our approach might also be applicable in such situations.

We study the uncertain dichotomous choice model. In this model, a group of expert decision makers is required to select one of two alternatives. The applications of this model are relevant to a wide variety of areas. A decision rule translates the individual opinions of the members into a group decision, and is optimal if it maximizes the probability of the group making a correct choice. In this paper, we assume the correctness probabilities of the experts to be independent random variables selected from some given distribution. Moreover, the ranking of the members in the group is (at least partly) known. Thus, one can follow rules based on this ranking. The extremes are the expert rule and the majority rule. The probabilities of the two extreme rules being optimal were compared in a series of early papers, for a variety of distributions. In most cases, the asymptotic behaviours of the probabilities of the two extreme rules followed the same patterns. Do these patterns hold in general? If not, what are the ranges of possible asymptotic behaviours of the probabilities of the two extreme rules being optimal? In this paper, we provide satisfactory answers to these questions.

Our aim in this paper is to clarify the notion of independence for imprecise probabilities. Suppose that two marginal experiments are each described by an imprecise probability model, i.e., by a convex set of probability distributions or an equivalent model such as upper and lower probabilities or previsions. Then there are several ways to define independence of the two experiments and to construct an imprecise probability model for the joint experiment. We survey and compare six definitions of independence. To clarify the meaning of the definitions and the relationships between them, we give simple examples which involve drawing balls from urns. For each concept of independence, we give a mathematical definition, an intuitive or behavioural interpretation, assumptions under which the definition is justified, and an example of an urn model to which the definition is applicable. Each of the independence concepts we study appears to be useful in some kinds of application. The concepts of strong independence and epistemic independence appear to be the most frequently applicable.

We shall study continuous-time Markov chains on the nonnegative integers which are both irreducible and transient, and which exhibit discernible stationarity before drift to infinity “sets in”. We will show how this ‘quasi’ stationary behaviour can be modelled using a limiting conditional distribution: specifically, the limiting state probabilities conditional on not having left 0 for the last time. By way of adualchain, obtained by killing the original process on last exit from 0, we invoke the theory of quasistationarity forabsorbingMarkov chains. We prove that the conditioned state probabilities of the original chain are equal to the state probabilities of its dual conditioned on non-absorption, thus allowing to establish the simultaneous existence and then equivalence, of their limiting conditional distributions. Although a limiting conditional distribution for the dual chain is always quasistationary distribution in the usual sense, a similar statement is not possible for the original chain.

Motivated by problems that arise in computing degrees of belief, we consider the problem of computing asymptotic conditional probabilities for first-order sentences. Given first-order sentences φ and θ, we consider the structures with domain {1, …, N} that satisfy θ, and compute the fraction of them in which φ is true. We then consider what happens to this fraction as N gets large. This extends the work on 0-1 laws that considers the limiting probability of first-order sentences, by considering asymptotic conditional probabilities. As shown by Liogon'kiĭ [24], if there is a non-unary predicate symbol in the vocabulary, asymptotic conditional probabilities do not always exist. We extend this result to show that asymptotic conditional probabilities do not always exist for any reasonable notion of limit. Liogon'kiĭ also showed that the problem of deciding whether the limit exists is undecidable. We analyze the complexity of three problems with respect to this limit: deciding whether it is well-defined, whether it exists, and whether it lies in some nontrivial interval. Matching upper and lower bounds are given for all three problems, showing them to be highly undecidable.

Numerous new properties of stochastic conditional independence are introduced. They are aimed, together with two surprisingly trivial examples, at a further reduction of the problem of probabilistic representability for four-element sets, i.e. of the problem which conditional independences within a system of four random variables can occur simultaneously. Proofs are based on fundamental properties of conditional independence and, in the discrete case, on the use of I-divergence and algebraic manipulations with marginal probabilities. A duality question is answered in the negative.

We discuss two general issues concerning diverging sets of Bayesian (conditional) probabilities—divergence of “posteriors”—that can result with increasing evidence. Consider a set of probabilities typically, but not always, based on a set of Bayesian “priors.” Incorporating sets of probabilities, rather than relying on a single probability, is a useful way to provide a rigorous mathematical framework for studying sensitivity and robustness in Classical and Bayesian inference. See: Berger (1984, 1985, 1990); Lavine (1991); Huber and Strassen (1973); Walley (1991); and Wasserman and Kadane (1990). Also, sets of probabilities arise in group decision problems. See: Levi (1982); and Seidenfeld, Kadane, and Schervish (1989). Third, sets of probabilities are one consequence of weakening traditional axioms for uncertainty. See: Good (1952); Smith (1961); Kyburg (1961); Levi (1974); Fishburn (1986); Seidenfeld, Schervish, and Kadane (1990); and Walley (1991).

In a recent paper [4] it was shown that, for an absorbing Markov chain where absorption is not guaranteed, the state probabilities at time t conditional on non-absorption by t generally depend on t. Conditions were derived under which there can be no initial distribution such that the conditional state probabilities are stationary. The purpose of this note is to show that these conditions can be relaxed completely: we prove, once and for all, that there are no circumstances under which a quasistationary distribution can admit a stationary conditional interpretation.

The use of negative probabilities is discussed for certain problems in which a stochastic process approach is indicated. An extension of probability theory to include signed (negative and positive) probabilities is outlined and both philosophical and axiomatic examinations of negative probabilities are presented. Finally, a class of applications illustrates the use and implications of signed probability theory.

The conditional probability of h given e is commonly claimed to be equal to the probability that h would have if e were learned. Here I contend that this general claim about conditional probabilities is false. I present a counter-example that involves probabilities of probabilities, a second that involves probabilities of possible future actions, and a third that involves probabilities of indicative conditionals. In addition, I briefly defend these counter-examples against charges that the probabilities they involve are illegitimate.

The aim of the paper is to draw a connection between a semantical theory of conditional statements and the theory of conditional probability. First, the probability calculus is interpreted as a semantics for truth functional logic. Absolute probabilities are treated as degrees of rational belief. Conditional probabilities are explicitly defined in terms of absolute probabilities in the familiar way. Second, the probability calculus is extended in order to provide an interpretation for counter-factual probabilities—conditional probabilities where the condition has zero probability. Third, conditional propositions are introduced as propositions whose absolute probability is equal to the conditional probability of the consequent on the antecedent. An axiom system for this conditional connective is recovered from the probabilistic definition. Finally, the primary semantics for this axiom system, presented elsewhere, is related to the probabilistic interpretation.

The theory of Probabilities has been characterized by Laplace, one of those who have contributed most largely to its advance,—as “good sense reduced to a system of calculation;” and such, no doubt, it is. But it must be especially noticed that there is hardly any subject to which thought can be applied, which calls for so continuous an application of that excellent quality, or in which it i s easier to make mistakes from simple want of circumspection. And, moreover, that its reduction to calculation is attended with difficulties of a very peculiar nature, such as occur in no other application of mathematical analysis to practical subjects, arising out of the great magnitudes of the numbers concerned, which defeat the ordinary processes of arithmetical and logarithmic calculation, by exhausting the patience of the computer, and require special methods of approximate evaluation to bring them within the compass of human industry. These methods form a conspicuous feature of the general subject, and have furnished scope for very extraordinary displays of mathematical talent and invention. That very large numbers will inevitably be concerned in questions where numerous and independent contingencies may take place, and in any order or mode of combination, will be apparent to any one who considers the astonishing fecundity of such combinations numerically estimated, when the combining elements are many. For example, the number of possible “hands” at whist (regard being had to the trump) is 1,270,027,119,200.

The Author commences by stating that he has in the course of his investigations found the theory of probabilities in a far from perfect state. In many cases, when he has wished to make use of the theory of probabilities, he has searched in vain for demonstrated formulae capable of immediate application, and he has generally been obliged to begin by working out the formulae independently for himself.

In an article “Countering a Counter-intuitive Probability” [4], Lynn E. Rose discusses a question in conditional probability, claiming that the following problem posed by Copi [1] is usually incorrectly solved:Remove all cards except aces and kings from a deck, so that only eight cards remain, of which four are aces and four are kings. From this abbreviated deck, deal two cards to a friend. If he looks at his cards and announces (truthfully) that his hand contains an ace, what is the probability that both his cards are aces? If he announces instead that one of his cards is the ace of spades, what is the probability then that both his cards are aces? (These two probabilities are not the same!) ([1], p. 433)

In the contribution to the joint Royal Institute of Navigation/Royal Ocean Racing Club meeting on Safety of Navigation under Sail (p. 467), an expression was quoted for the random probability of collision in the case of a large, relatively fast ship passing through a concentration of smaller craft. It was not appropriate to discuss the derivation of the expression in the particular circumstances of the meeting but the methodology may be of some general interest and it is therefore outlined in this note.

The problem to be considered arose in connexion with applications for plant breeder's rights. An applicant has to submit material for scrutiny by the authority, one aspect considered being uniformity. The applicant may wish to conduct a preliminary test, of the same type as the authority's, to allow a calculation of the probability of passing the regulation test. In both tests only a sample is examined and the population frequency of off-types remains unknown. It is assumed here that the authority sets a critical frequency (e. g. 0–01) and specifies a size of sample (e.g. 300). The result is regarded as a failure if the observed frequency exceeds the critical frequency (e.g. if more than three off-types are observed in a total of 300). Given that this is the test procedure it is possible to calculate the conditional probability of passing the regulation test, knowing the result of a preliminary test.