Complementary event

Abstract: Brams mentions John A. Ferejohn's resolution of the paradox by reformulating the situation in such a way that the Being's choices "to put the money in B2" or "not to put" are changed to two "states of the world," namely, "the Being guesses correctly" and "the Being guesses incorrectly." Formulated in this way, the situation becomes a decision problem under risk without a dominating strategy for the chooser. It appears, consequently, that maximization of utility dictates the choice of only B2 over the choice of both boxes. (For simplicity, we regard the utility of money to be a linear function of the amount of money. If not, the amounts of money can be calibrated to make the payoffs reflect the actual utilities.) Note, however, that the states of the world, "Being guesses correctly" and "Being guesses incorrectly," are not independent of what the chooser does. Therefore in computing expected utilities, not the probabilities of these two states as such must be used, but rather the conditional probabilities with respect to the chooser's choices. Let E be the event "The chooser takes only B2"; E the complementary event, "He takes both boxes"; F the event "the Being predicts E"; F the complementary event, "the Being predicts R." Finally, let C be the event, "the Being guesses correctly." Then Pr(C I E) = Pr(F I E); for to predict correctly when E occurs means to predict E when E occurs. This is the relevant probability associated with the upper-left cell in the matrix shown in Brams's Figure 2.

Abstract: The subjective probability model represents one of the few quantitative approaches that examines the impact of arguments on a person's beliefs (McGuire, 1960; Wyer, 1974; Wyer & Goldberg, 1970). The subjective probability model remains an accurate predictor of beliefs based on numerous investigations testing the veracity of the basic model (Hample 1977; 1978; 1979). The model retains a high degree of predictive accuracy despite tests using longer chains involving more complex arguments (Allen, 1987; Allen & Burrell, 1992; Allen & Kellermann, 1988). The importance of the model lies in the ability to evaluate arguments and argumentative strategies. For example, the controversy in the academic debate community over high impact/low probability arguments within academic debate was tested using this model (Allen & Kellermann, 1988). The results indicated that judges who based decisions on such arguments acted rationally and with prudence when considering the available evidence. The evidence indicates that naive persons listening to such arguments judge them as possible and acceptable when provided the available evidence. This paper examines another untested extension of the model. Consider an argument that says that some event C is caused by two events (A and B). The current data testing the subjective probability model considers a single causal sequence of one event leading to another. The mathematical logic of the model, based on the Total Probability Theorem, extends to the more complex case of argument where multiple events cause some single outcome. This experiment tests whether the model continues to maintain accuracy when considering such arguments. Further, this experiment provides a persuasive message on the topic and determines if the model accurately handles change generated as the result of a persuasive message. SUBJECTIVE PROBABILITY MODEL The subjective probability model explains the workings of a causal argument. Causal arguments suggest an explanation for the nature of cause and effect among elements of a system. The typical arguer will suggest that one event occurs and that event subsequently causes another event, which in turn causes another event, etc. The application of subjective probability model usually involves arguments about public policies where one event leads to another event. Consider the following example of a sequential argument: The North American Free Trade Agreement (Event A) caused corporations to relocate in Mexico for cheaper labor (Event B) and subsequently hundreds of thousands of persons became unemployed in the United States (Event C). This simple causal argument suggests that if event A occurs then event B takes place and ultimately event C. Such arguments are commonplace in the public forum where arguments about actions and the consequences of those actions form a constant part of the argumentative landscape. The subjective probability model argues that the estimation of the occurrence of any event is equivalent to the sum of the probability of the occurrence of causal events multiplied by the probability that the causal events in fact result in the event under consideration. For a simple chain event where event A is said to cause event B the equation is: P(B) = P(A)P(B/A) + P([sim]A)P(B/[sim]A) This equation reads that the estimate of the probability of event B occurring is equivalent to the probability that event A will occur multiplied by the probability that event B will happen given event A added to the probability of A not occurring multiplied by the probability that if A does not occur B still takes place. The subjective probability model is an application of the Total Probability Theorem (see Wyer & Goldberg, 1970 for a more complete explanation of the mathematical issues). The theorem states that given an event B and an exhaustive set of mutually exclusive events Ai, the probability of the event B is provided by: P(B) = [sigma]P([A. …

Abstract: Multiple source information forms one of the key components of data fusion. Such information may emanate from various mechanical sensor sources such as radar or Doppler systems, or it may derive from human-based sources, such as via expert opinion expressed through natural language. In general, each unit information is associated with degree of uncertainty/certainty which is traditionally determined through the use of probability. Thus, one can evaluate probabilistically any desired logical combination of events for use in decision-making. On the other hand, information uncertainty is provided in a way that there appears to be no single underlying Boolean event whose probability evaluation matches the prescribed uncertainty. Such uncertainty is often expressed in the form of given functions of probability evaluations of contributing simpler events. For example, when these functions are simple arithmetic divisions with arguments being pairs of events, each numerator argument event being a subevent of the denominator argument event, the quantitative uncertainty corresponding to each unit of information then, becomes a conditional probability. But, in general, the standard development of probability theory and statistics has not produced a way to represent conditional probabilities as single event probability evaluations, so that standard statistical decision-making techniques cannot be used in systematic sound way here. Recently, probability theory has been expanded to address the problem in the form of conditional event algebra. Conditional Event Algebra (CEA) is a relatively new logic system which rigorously extends standard probability theory to include events which are contingent such as rules and conditionals. The" if ...then" is modeled as Boolean elements, and yet compatible with conditional probability quantitative value. The principle theory and application of conditional event algebra is presented. We also introduce the idea of relational event algebra which is more general than conditional event algebra.

Abstract: This chapter focuses on events and probabilities. The notion of the probability of an event is approached by three different methods. One method is to repeat an experiment or game many times under identical conditions and compute the relative frequency with which an event occurs. In the second way of approaching the notion of probability, a minimal list of axioms is set down, which assumes certain properties of probabilities. From this minimal set of assumptions, the further properties of probability are deduced and applied. The third method for arriving at the notion of probability is limited in application; however, it is extremely useful. The probability of the event is defined to be the number of “equally likely” ways, in which the event can occur divided by the total number of possible “equally likely” outcomes. The number of equally likely ways in which the event can occur must be from among the total number of equally likely outcomes. An event is simply a collection of certain elementary events. Different events are different collections of elementary events.