Subjects were requested to choose between gambles, where the outcome of one gamble depended on a single elementary event, and the other depended on an event compounded of a series of such elementary events. The data supported the hypothesis that the subjective probability of a compound event is systematically biased in the direction of the probability of its components resulting in overestimation of conjunctive events and underestimation of disjunctive events. Studies pertaining to this topic are discussed.

1. Stumpers, as defined here, are riddles so simple that kids can solve and appreciate them – yet so elusive that even intelligent adults can be completely stumped by them. Example: Alex is Bobbie’...

Recently, Nathan (1986) criticized Bar-Hillel and Falk's (1982) analysis of some classical probability puzzles on the grounds that they wrongheadedly applied mathematics to the solving of problems suffering from "ambiguous informalities". Nathan's prescription for solving such problems boils down to assuring in advance that they are uniquely and formally soluble--though he says little about how this is to be done. Unfortunately, in real life problems seldom show concern as to whether their naturally occurring formulation is or is not ambiguous, (...) does or does not allow for unique formalization, etc. One step towards dealing with such problems intelligently is to recognize certain common cognitive pitfalls to which solvers seem vulnerable. This is discussed in the context of some examples, along with some empirical results. (shrink)

Rutherford criticizes the way some people have analyzed the 2-children problem, claiming that slight nuances in the problem’s formulation can change the correct answer. However, his own data demonstrate that even when there is a unique correct answer, participants give intuitive answers that differ from it systematically — replicating the data reported by those he criticizes. Thus, his critique reduces to an admonition to use care in formulating and analyzing this brainteaser — which is always a good idea — but (...) contributes little what is known, analytically or empirically, about the 2-children problem. (shrink)

Rutherford criticizes the way some people have analyzed the 2-children problem, claiming that slight nuances in the problem's formulation can change the correct answer. However, his own data demonstrate that even when there is a unique correct answer, participants give intuitive answers that differ from it systematically — replicating the data reported by those he criticizes. Thus, his critique reduces to an admonition to use care in formulating and analyzing this brainteaser — which is always a good idea — but (...) contributes little what is known, analytically or empirically, about the 2-children problem. (shrink)

Recently, Nathan criticized Bar-Hillel and Falk's analysis of some classical probability puzzles on the grounds that they wrongheadedly applied mathematics to the solving of problems suffering from “ambiguous informalities”. Nathan's prescription for solving such problems boils down to assuring in advance that they are uniquely and formally soluble—though he says little about how this is to be done. Unfortunately, in real life problems seldom show concern as to whether their naturally occurring formulation is or is not ambiguous, does or (...) does not allow for unique formalization, etc. One step towards dealing with such problems intelligently is to recognize certain common cognitive pitfalls to which solvers seem vulnerable. This is discussed in the context of some examples, along with some empirical results. (shrink)

The concepts of supportive evidence and of relevant evidence seem very closely related to each other. Supportive evidence is clearly always relevant as well. But must relevant evidence be defined as evidence which is either supportive or weakeking? In an explicit or implicit manner, this is indeed the position of many philosophers. The paradox of ideal evidence, however, shows us that this is to restrictive. Besides increasing or decreasing the probability attached to some hypothesis, evidence can alter or interact with (...) the background assumptions underlying the hypothesis.In most circumstances, the (post hoc) relevance of evidence can indeed be judged by its effect on the confidence one attaches to hypotheses. Occasionally, as in the circumstances described by Example I, and more generally called “the Paradox of Ideal Evidence”, the relevance of evidence to an hypothesis can only be understood by appeal to a broader sense. (shrink)

We offer a case-study in irrationality, showing that even in a high stakes context, intelligent and well trained professionals may adopt dominated practices. In multiple-choice tests one cannot distinguish lucky guesses from answers based on knowledge. Test-makers have dealt with this problem by lowering the incentive to guess, through penalizing errors (called formula scoring), and by eliminating various cues for outperforming random guessing (e.g., a preponderance of correct answers in middle positions), through key balancing. These policies, though widespread and intuitively (...) appealing, are in fact ‘‘irrational’’, and are dominated by alternative solutions. Number-right scoring is superior to formula scoring, and key randomization is superior to key balancing. We suggest that these policies have persisted since all stake-holders – test-makers, test-takers and test-coaches – share the same faulty intuitions. (shrink)

Foundations of Set Theory discusses the reconstruction undergone by set theory in the hands of Brouwer, Russell, and Zermelo. Only in the axiomatic foundations, however, have there been such extensive, almost revolutionary, developments. This book tries to avoid a detailed discussion of those topics which would have required heavy technical machinery, while describing the major results obtained in their treatment if these results could be stated in relatively non-technical terms. This book comprises five chapters and begins with a discussion of (...) the antinomies that led to the reconstruction of set theory as it was known before. It then moves to the axiomatic foundations of set theory, including a discussion of the basic notions of equality and extensionality and axioms of comprehension and infinity. The next chapters discuss type-theoretical approaches, including the ideal calculus, the theory of types, and Quine's mathematical logic and new foundations; intuitionistic conceptions of mathematics and its constructive character; and metamathematical and semantical approaches, such as the Hilbert program. This book will be of interest to mathematicians, logicians, and statisticians. (shrink)

One of the tasks with which communication engineers are presented is that of devising a mechanism by which a significant sequence of words, a message, produced by somebody, the sender of the message, is reproduced at some other place, with the shortest practical time lag. The reproduction must be such that the receiver of the message will be able to understand what the sender meant by his message, at least, if he knows the sender's language. The following illustration is typical: (...) A writes on a sheet of paper “I love you” and wishes that B, 3000 miles away, should become aware of the full content of this message, with little delay and at a low cost. There will be institutions, in a capitalistic society, which will compete with each other in providing A, for a price, with the required service. Those companies which perform these services most satisfactorily, i.e., with an overall better combination of faithfulness, time lag, and cost, will get the job. The executives of these companies will hire engineers and put them to work on improving this overall combination. (shrink)