Why the distinction between single-event probabilities and frequencies is important for psychology (and vice versa)
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Gigerenzer, Gerd


Please note:
This paper is a preprint of an article published in G. Wright & P. Ayton (Eds.). (1994). Subjective probability (pp. 129-161). Chichester [u.a.]: Wiley, therefore may be minor differences between the two versions.
The copyright of this electronic version remains with the author and the Max Planck Institute for Human Development.



Some years ago in Stanford I was lunching with a motley group of colleagues, mostly psychologists and economists, all interested in judgment under uncertainty. We gnawed our way through our sandwiches and through the latest embellishments of the prisoners' dilemma, trading stories of this or that paradox or stubborn irrationality. Finally one economist from MIT concluded the discussion with the following dictum: "Look," he said with conviction, "either reasoning is rational or it's psychological."

This forked opposition between the rational and the psychological has haunted me ever since. Frege scholars will hear in it an echo of the nineteenth-century debate between the logician Frege and the psychologist Wundt over the status of the "laws of thought;" the economists and psychologists seated at the picnic table with me that afternoon had in mind the more recent findings of the "heuristics and biases" research program in cognitive psychology (e.g., Tversky & Kahneman 1974, 1983). Certainly anyone acquainted only with this aspect of contemporary psychology - and it remains among the best publicized, both to colleagues in other disciplines and to the public at large - could easily have come to think that psychology was about revealing and explaining human irrationality. The conjunction fallacy, the base-rate fallacy, the overconfidence bias - this was the gloomy litany of sins people seemed to commit routinely and incorrigibly against reason. According to the exponents of the "heuristics and biases" program, human beings were programmed to be systematically, stubbornly irrational when making judgments under uncertainty - at least, most of the time. (Experimental subjects were not dazzling at logical thinking either, but that is another story and another research program.) No wonder the psychology of reasoning had become nearly synonymous with the investigation of the irrational (you get a taste from Table 1).

Table 1
A sample of conclusions from the heuristics and biases program

In making predictions and judgments under uncertainty, people do not appear to follow the calculus of chance or the statistical theory of prediction. Instead, they rely on a limited number of heuristics which sometimes yield reasonable judgments and sometimes lead to severe and systematic errors.
Daniel Kahneman & Amos Tversky, 1973, p. 237

It appears that people lack the correct programs for many important judgmental tasks. ...we have not had the opportunity to evolve an intellect capable of dealing conceptually with uncertainty.
Paul Slovic, Baruch Fischhoff & Sarah Lichtenstein, 1976, p. 174

The biases of framing and overconfidence just presented suggest that individuals are generally affected by systematic deviations from rationality.
Max Bazerman & M.A. Neale, 1986, p. 317

The genuineness, the robustness, and the generality of the base-rate fallacy are matters of established fact.
Maya Bar-Hillel, 1980, p. 215

[Overconfidence bias] has proved so robust that it is hard to acquire much insight into the psychological processes producing it.
Baruch Fischhoff, 1988, p. 172

[We are] a species that is uniformly probability-blind, from the humble janitor to the Surgeon General ... We should not wait until A. Tversky and D. Kahneman receive a Nobel prize for economics. Our self-deliberation from cognitive illusions ought to start even sooner.
Massimo Piattelli-Palmarini, 1991, p. 35

What exactly did it mean to be irrational, according to the psychologists of the heuristics and biases program? Let me use a well-known example, the Linda Problem . Assume you are a subject in a psychological experiment. In front of you is a text problem and you begin to read:

Linda is 31 years old, single, outspoken and very bright. She majored in philosophy. As a student, she was deeply concerned with issues of discrimination and social justice, and also participated in antinuclear demonstrations. Which of these two alternatives is more probable?

(a) Linda is a bank teller

(b) Linda is a bank teller and active in the feminist movement

Which alternative would you choose? Assume you chose (b), just as most subjects - 80% to 90% - in previous experiments did. Tversky and Kahneman (1983) argue: (b) is the conjunction of two facts, namely that Linda is a bank teller and is active in the feminist movement, whereas (a) is one of the conjuncts. Because the probability of a conjunction cannot be greater than that of one of its conjuncts, the correct answer is (a), not (b). Therefore, your judgment is recorded as an instance of a celebrated reasoning error, known as the conjunction fallacy. Tversky, Kahneman, and others have shown that this type of judgment is highly stable across experimental manipulations. By analogy to stable visual illusions, stable reasoning errors, such as the conjunction fallacy, have been labeled cognitive illusions. The standard conclusion is that the mind does not possess the proper statistical algorithms, but relies on non-statistical quick-and-dirty algorithms, such as the representativeness heuristic. That is, the mind assesses the probability by calculating the similarity between the description of Linda and each of the alternatives, and chooses the alternative with the highest similarity. Judging probability by similarity has been termed the representativeness heuristic.

This alleged demonstration of human irrationality in the Linda Problem has been widely publicized in psychology, philosophy, economics and beyond. Stephen J. Gould (1992, p.469) puts the message clearly:

I am particularly fond of [the Linda] example, because I know that the [conjunction] is least probable, yet a little homunculus in my head continues to jump up and down, shouting at me - 'but she can't just be a bank teller; read the description' ... Why do we consistently make this simple logical error? Tversky and Kahneman argue, correctly I think, that our minds are not built (for whatever reason) to work by the rules of probability.

In what follows I will argue that Gould should have had more trust in the rationality of his homunculus.

Two aspects of the standards of rationality versus irrationality assumed by this and other celebrated experiments cry out for closer inspection.

1. The distinction between single-event probabilities and frequencies. In the supposed demonstrations of the conjunction fallacy, the base rate fallacy, and the overconfidence bias, rationality is identified not simply with probability theory, but with some particular interpretation of probability theory, often a narrow version of Bayesianism. The Linda Problem applies probability theory to a single event - whether Linda is a bank teller - rather than to frequencies. Does probability theory apply to single events?

This is a controversial matter amongst probabilists, who have long and heatedly debated the merits of subjective Bayesian versus objective frequentist interpretations of probability. The influential Bayesian Leonard J. Savage (1954), for instance, introduced his notion of personal probability with every-day reasoning about singular events: "I personally consider it more probable that a Republican president will be elected in 1996 than that it will snow in Chicago sometime in the month of May, 1994. But even this late spring snow seems to me more probable than that Adolf Hitler is still alive." (p.27). Savage's proposal challenged the frequentist schools which were then dominant, as they are in most statistics departments today. Savage was quite explicit about the deviant character of his proposal, when he added: "Many, after careful consideration, are convinced that such statements about probability to a person mean precisely nothing, or at any rate that they mean nothing precisely." (p.27).

The mathematician Richard von Mises (1957) was one of those many. In his view, a reference class (collective) has to be defined first, and then the probability of an event is the relative frequency of this event in its class. One of his examples is the probability of death at age 41, as determined from the data of insurance companies. The class is "all men insured before reaching the age of forty after complete medical examination and with the normal premium." The number of deaths at age 41 was 940 out of 85,020, which corresponds to a relative frequency of about .011. This probability is attached to a class, but not to a particular person or event. Every particular person is always a member of many different classes, whose relative frequencies of death may have different values. Therefore, von Mises concluded: "It is utter nonsense to say, for instance, that Mr. X, now aged forty, has the probability 0.011 of dying in the course of the next year." (p.17-18).

By now it should be clear that according to a strong frequency view of probability (e.g., Neyman, 1977; von Mises, 1957), what has been labeled the conjunction fallacy is not an error in probabilistic reasoning. In this view, probability theory is about frequencies and simply doesn't apply to single events.

2. Content-independent rationality. There is a peculiar indifference in this standard of rationality to background knowledge: rationality here means the deployment of formal algorithms (or rules, such as the conjunction rule) which are content-independent. That is, it is assumed that they can and should be applied to tasks with different specific contents, provided the formal structure remains constant. From this point-of-view, rationality is not bound to any specific domain, and knowledge ideally is irrelevant to proper reasoning.

In the Linda Problem, for instance, whatever you know about bank tellers and feminists is assumed to be entirely irrelevant; indeed, you need not read the description of Linda at all - it is irrelevant to content-independent rationality. Hence there is little analysis of how the content of a problem cues the understanding of the term "probable" in single-event statements or questions, such as in "which is more probable?" "Probable" can refer to typical, prototypical, frequent, credible, to the weight of evidence, to a plausible causal story, or to what "may in view of present evidence be reasonably expected to happen," as the Oxford English Dictionary informs us. Most of these meanings do not obey the laws of probability. For instance, judgments of typicality do not follow the conjunction rule. Betty Friedan may count as a typical feminist writer, but not as a typical writer. Such psychological considerations, however, are not part of the content-independent rationality that defines right and wrong reasoning in the heuristics and biases program.

These two issues are not independent. For instance, imagine I am a person of the sort who insists that every single-event statement involving the term "probable," as in the Linda problem, must obey the laws of probability theory rather than, say, the guidelines of the Oxford English Dictionary. (I take the statements in Table 1 to epitomize this conviction.) Then I would be uninterested in how content (add physical and social context, goals, if you want) determines what is reasonable in a given situation.

I will focus in this chapter on the distinction between single-event probabilities and frequencies, and will say little about the role of content in understanding what is rational (on the latter see Gigerenzer, 1991; Gigerenzer & Hug, 1992).

In the first part, drawing on recent work in the history of probability, I will show that the distinction between single-event probabilities and frequencies was dependent on theories of mind: the meaning of probability changed when theories of mind changed. In the second part, drawing on recent experimental work, I will show that apparently stable cognitive illusions are dependent on the distinction between single-event probabilities and frequencies: cognitive illusions disappear when single-event probabilities are changed into frequencies. Thus, I argue that the distinction between single-event probabilities and frequencies is not just the province of philosophers and mathematicians, but of direct relevance for psychology, and vice versa.

1. How theories of psychology shaped the meaning of probability

According to legend, probability is one of the few seminal ideas that have an exact birthday. In 1654, precisely three hundred years before Savage's treatise, the now famous correspondence between Blaise Pascal and Pierre Fermat first cast the calculus of probability in mathematical form. Ian Hacking (1975) argued that the probability that emerged so suddenly was Janus-faced from the very beginning. One face was aleatory, concerned with observed frequencies (e.g., co-occurrences between fever and disease, comets and death of kings); the other face was epistemic, concerned with degrees of belief or opinion warranted by authority. In his view, the 20th century duality between objective frequencies and subjective probabilities existed then as it exists now. Barbara Shapiro (1983) and Lorraine Daston (1988), however, have argued that probability in the 17th and 18th centuries had more than Janus's two faces. It included physical symmetry (e.g., the physical construction of dice, now called "propensity"); frequency (e.g., how many people of a given age died annually); strength of argument (e.g., evidence for or against a judicial verdict); intensity of belief (e.g., the firmness of a judge's conviction of the guilt of the accused); verisimilitude and epistemological modesty, among others. Over the centuries, probability also conquered new territory and created further meanings, such as in quantum physics, and lost old territory, such as the probability of causes (Daston, 1988). Rather than Janus's two faces, probability seems more like a group of visages loosely assembled in a family portrait, with some members joining over time and others dropping out.

The unity: frequencies and subjective beliefs

The puzzling fact about the Enlightenment probabilists is the ease with which they slid from one meaning of probability to the next - and this holds independently of whether you see probability as Janus-faced or more like a family portrait. This ease created the apparent paradox that competing present-day interpretations of probability could claim the same work as their ancestor. Jakob Bernoulli's Ars conjectandi (1713), for instance, has been variously claimed to anticipate the 20th century subjective interpretation, Rudolf Carnap's logical interpretation, and the extreme frequentist interpretation of Jerzy Neyman and Richard von Mises as well (Hacking, 1975, p.15-16).

The solution to this puzzle lies in the intimate link between psychology and probability. Daston (1988, ch. 4) argues that only with hindsight does it seem that Bernoulli and other classical probabilists vacillate between objective and subjective interpretations. Whereas today these interpretations look incompatible to many, the classical probabilists were able to reconcile the subjective and objective facets of probability on the basis of the theories of mind advanced by John Locke, David Hartley, and David Hume. The following account of how associationist psychology shaped and incorporated ideas of probability is a condensed version of Daston's (1988) detailed study.

Philosophers like Hartley and Hume, and mathematicians like Condorcet and Laplace, treated associationist psychology and mathematical probability as kindred topics. As in Locke's associationism, Hume held that the mind unconsciously and automatically tallied frequencies and proportionated degree of belief (for Hume, the vivacity of an idea). And Hume insisted that the psychological mechanism that converted frequency into belief was finely tuned: "When the chances or experiments on one side amount to ten thousand, and on the other to ten thousand and one, the judgment gives the preference to the latter, upon account of that superiority." (Hume, 1739/1975, p.141). Despite his reservation about the validity of induction, Hume made probabilistic thinking the de facto standard of reasonableness. Hume linked frequency with belief, but his account contained almost no reference to the mathematical theory of probability. David Hartley's (1749) work did. He combined elements from Locke's sketch of associationism and Newton's physiological speculations concerning the vibratory basis of sensations, and worked it into a full-blown associationism that connected the laws of mind with the laws of probability. Repeated associations created cerebral vibrations until grooves of mental habit were etched in the brain. Through this physiological mechanism, human judgment, when undeflected by strong emotion or passion, imitated the law of large numbers.

The list of psychological mechanisms underlying the mapping of objective frequencies into subjective belief, postulated from Locke to Hartley to Laplace, seems surprisingly familiar to a contemporary psychologist: observed frequencies are transformed into degrees of belief through "traces," "vibrations," "interior images," and "impressions." All these mechanisms postulated the passive, automatic and unconscious mapping of experienced frequencies into subjective probabilities. Being built up from frequencies, degrees of belief were rational. The Enlightenment empiricists had taken due notice of the distortion of rational belief through passion and interest, but they believed these were corrigible aberrations.

These psychological theories were the backbone of what is now known as the classical interpretation of probability, from 1660 to 1840, and they explain some of its central features. First, classical probability conflated subjective belief and objective frequencies, based on associationist psychology. Second, probabilities were epistemic, a figment of human ignorance and therefore subjective, not part of the physical world. Classical probabilists, from Jakob Bernoulli through Laplace, were arch determinists (Daston, 1992). God, or Laplace's secularized demon, could dispense fully with probability. However, we humans are, as John Locke put it, most of the time condemned to live in the twilight of probability rather than in the noonday sun of certainty. Although the world itself is deterministic, human cognition is inherently probabilistic and empirical in its working - a view that was revived two centuries later in Egon Brunswik's (1955) functional probabilism. Third, the mapping of frequencies into subjective probabilities is rational, and so were subjective probabilities, unless disrupted by passion or interest. The Enlightenment probabilists cherished the fiction of the hommes éclairés, an elite of educated people who can prevent such disruptions from affecting their beliefs. Probability theory mirrored their reasoning, and it provided a tool for those unfortunates in need of help staying clear from these disruptions. Human reasoning and probability theory were two sides of the same coin. In Laplace's famous phrase, probability theory was nothing more than "good sense reduced to a calculus."

By the time Siméon-Denis Poisson (1837) published his major work on probability, the classical interpretation was under attack on several fronts. The psychological theories postulating mechanisms that guaranteed the proportioning of belief to frequencies had given way to psychological theories that emphasized the illusionary nature of human belief. Etienne de Condillac (1754) was one of the first to express misgivings about the reliability of the link between frequency and belief. In his psychology wishful thinking became the rule rather than the exception. Condillac was preoccupied with pathological associations caused by experiences early in life, by prejudice, or by brain consistency. He held, for instance, that young girls were prone to confuse chimeras for realities, because their brains were soft and even faint associations left permanent impressions in a soft medium. Condillac and his followers shifted the associationist psychology of Hume and Hartley to a psychology in which needs, wants, and temperaments (and other sources of pathologies) determined how the mind distributed attention, which in turn organized experience (Daston, 1988). The unity between frequency and belief was slowly eroded. What psychology had given to probability, it now took away. Poisson was the first to distinguish clearly in print, in 1837, between the subjective and the objective meaning of probability.

Although I am focusing in this chapter on the relation between psychology and probability, there is a broader intellectual and social context in which the rise and the fall of the classical interpretation of probability is embedded. The French revolution and its aftermath shook the confidence of the probabilists in the existence of a single shared standard of reasonableness. The consensus and the values of the intellectual and political elites fragmented and disappeared, as did l'homme éclairé, the fiction of the reasonable man who embodied this consensus (Gigerenzer et al., 1989, ch.1).

The divorce; frequencies versus subjective belief

Subjective belief and objective frequencies began as equivalents and ended up as diametric opposites. Poisson had distinguished the two, and the political economist and philosopher Antoine Cournot (1843) seems to have been the first who went one step further and eliminated subjective belief from the subject matter of mathematical probability: mathematical probability was not a measure of belief. Only then did it become evident that the classical interpretation of probability had been an interpretation. Classical probability was a form of "mixed mathematics," a term stemming from Aristotle's explanation of how optics and harmonics mixed the forms of mathematics with the matter of light and sound. Classical probability theory had no existence independent of its subject matter - the beliefs of reasonable men. The modern view that a mathematical theory might exist independently of a particular subject matter - the distinction between formal theory and application - was foreign to mixed mathematics. Arguably, mathematical probability did not free itself from its particular applications until very recently, when in 1933 A. N. Kolmogoroff presented his axiomatization of probability.

The new associationist psychology which focused on illusions had, by the early 19th century, provided the arguments for severing subjective probabilities and objective frequencies, and ironically, for severing associationist psychology from probability theory, too. By about 1840, l'homme éclairé had given way to l'homme moyen. Probability was no longer about mechanical rules of rational belief embodied in an elite of reasonable men, but about the properties of the average man (l'homme moyen), the embodiment of mass society, if not of mediocrity. Adolphe Quetelet's (1835) social physics determined the statistical distributions of suicide, murder, marriage, prostitution, height, weight, education, and almost everything else in Paris, and compared these with the distributions in London or Brussels. The means of these distributions defined the fictional average man in each society. The means and rates of moral behaviors, such as suicides and crimes in Paris or in London, proved to be strikingly stable over the years, and this was cited as evidence that moral phenomena are governed by the laws of a society rather than by the free decisions of its individuals. In 19th century France, statistics became known as "moral science."

The new focus on mass phenomena had a tremendous impact on pioneer sociologists such as Herbert Spencer and Emile Durkheim, and it shaped demography, insurance, epidemology, Prussian bureaucracy, the debates on free will, Francis Galton's enthusiasm for the normal curve and Gustav Theodor Fechner's statistical aesthetics, inter alia (Hacking, 1990; Stigler, 1986). Quetelet's model of human behavior as erratic and unpredictable at the individual level, but governed by statistical laws and predictable at the level of society, was independently adopted by James Clerk Maxwell and Ludwig Boltzmann to justify, by analogy, their statistical interpretation of the behavior of gas molecules (Porter, 1986). By this strange route, physics became revolutionized through the analogy with statistical laws of society.

Throughout most of the 19th and 20th century, the "probabilistic revolution" (Krüger, Daston & Heidelberger, 1987; Krüger, Gigerenzer & Morgan, 1987) was about frequencies: from the kinetic theory of gas to quantum statistics, and from population genetics to the Neyman-Pearson theory of hypothesis testing. The urn model of classical probability now concerned these mass phenomena, excluding subjective degrees of belief such as single-event probabilities. Joseph Bertrand in his Calcul des probabilités (1889), for instance, criticized Laplace's applications of Bayes' theorem to calculate degrees of belief: We believe the sun will rise tomorrow because of "the discovery of astronomical laws and not by renewed success in the game of chance" (p.xliv).

As is well known, subjective probability has regained acceptance in the second half of this century with the pioneering work of Bruno de Finetti and Frank Ramsay in the 1920s and 1930s, and of Leonard Savage in the 1950s. The reasonable man, once exiled from probability theory, had his comeback. Economists, psychologists, and philosophers now struggle again with the issue of how to codify "reasonableness" in mathematical form - the same issue once abandoned by mathematicians as a thankless task. Before the 1970s, the return of subjective probability still provoked a particularly lively debate between frequentists and subjectivists (now called "Bayesians"). Today, both sides pretend to know each other's arguments all to well and seem to have stopped listening. Frequentists dominate statistics and the experimental sciences, subjectivists dominate theoretical economics and artificial intelligence. The territory has been divided up. As Glenn Shafer (1989) complained, "conceptually and institutionally, probability has been balkanized" (p.15).

To summarize: Theories of psychology have been important in shaping the meaning of probability, and therewith the subject matter of probability theory. In particular, the assocationist psychology of Locke, Hume and Hartley provided the ground for not distinguishing objective frequencies and subjective degrees of belief - from the inception of probability theory circa 1650 to roughly 1840. The turn of associationist psychology towards illusions dethroned the reasonable man of classical probability theory and made the distinction between degrees of reasonable belief and frequencies obvious. After this conceptual transformation, psychology was dissociated from probability theory, too.

2. How the distinction between single-events and frequencies affects cognitive illusions

Psychologists like precise birthdays too. Textbooks celebrate 1879 as the beginning of what is referred to as scientific psychology, when Wilhelm Wundt devoted some space at the University of Leipzig for conducting experiments. For Wundt, the experimental method was a means to study elementary cognitive processes, such as attention and perceptual thresholds, but not (what he believed to be) deeply culture-bound processes such as thinking (Danziger, 1990). For these and other reasons, such as the dominance of American behaviorism, probabilistic reasoning was only occasionally a topic for psychologists in the first half of this century.

The classical probabilists would have felt a strong sense of déjá vu upon learning about some theoretical developments in the second half of the 20th century. Around 1950, Jean Piaget in Geneva revived the reasonable man of classical probability theory. In Piaget and Inhelder's (1951/1975) experimental work, the formal laws of probability are the laws of the adolescent and adult mind. Errors in probabilistic reasoning were characteristic only during ontogenetic development, until by the age of fourteen or so, when formal probabilistic reasoning emerges. Take for instance the law of large numbers. In 1739 Jacob Bernoulli had written in a letter to Leibniz that the law of large numbers is a rule that "even the stupidest man knows by some instinct of nature per se and by no previous instruction" (see Gigerenzer et al, 1989, p.29). More than two centuries later, Piaget and Inhelder concluded that even twelve to thirteen year olds intuitively apply the law of large numbers and understand the reasons for the law (p.207).

Locke, Hartley, and Hume had assumed that the mind unconsciously tallies frequencies and converted them into degrees of belief. Hasher and Zacks (1979) concluded from their experiments that frequencies are one of the few kinds of information (the others being word meaning and spatial and temporal location) that are monitored automatically - that is, without intention and attention, and without interfering with other tasks. Moreover, what is now called automatic frequency processing seems to be generally accurate, a conclusion independently arrived at by others (e.g., Brehmer & Joice, 1988). The thesis that objective frequencies eventually shape degrees of belief, has now been experimentally demonstrated (Hasher, Goldstein & Toppino, 1977). Locke, Hartley, and Hume would have been enthusiastic about these experimental findings. The reasonable man is back, dressed in modern fashion: less elite (everyone is a reasonable intuitive statistician) and confirmed by numerous experimental results.

The déjá vu, however, goes beyond the recreation of the reasonable man. Around 1970, much of cognitive and social psychology turned away from the rational intuitive statistician and focused on illusions (Kahneman, Slovic & Tversky, 1982; Nisbett & Ross, 1980). One-and-a-half centuries earlier, associationist psychology had turned to illusions, and the reasonable man had crumbled along with the classical interpretation of probability. Now again illusions were used to destroy belief in the rational homo sapiens, and to challenge economists' rational homo economicus. Now as then, illusions were no longer the exception, but the rule.

Here the historical similarities end. The old challenge was that passion and wishful thinking almost always interfere with the rational laws of thought. Freud's attack on human rationality is a well-known variation on that old theme. The unconscious wishes and desires of the Id are a steady source of intrapsychical conflict that manifests itself in all kinds of irrational beliefs, fears, and behaviors. The new challenge, however, does not invoke passion or wishful thinking to interfere with otherwise rational reasoning. This challenge is stronger: the human mind does not posses the proper statistical algorithms. Poor reasoning is seen as a straightforward consequence of the laws of human reasoning, which are a-statistical, simple "rule-of-thumb" heuristics. The mind is a poor intuitive statistician whether or not passion and wishful thinking compound this state of affairs.

Ironically, the departure point of the unreasonable man that emerged two decades after Piaget's revival of the reasonable man, was Savage's neo-Bayesianism. In the 1960s, Ward Edwards and his colleagues at the University of Michigan made two related proposals. First, Edwards, Lindman & Savage (1963) attempted to persuade experimental psychologists to turn Bayesian and to dispense with frequentist hypothesis testing. Second, Edwards (1968) proposed to study empirically whether intuitive reasoning follows Bayesian statistics. The first proposal fell still-born from the press, the second became a raging success.

Experimenters already had their frequentist statistics, a curious and confused mishmash between Fisher's significance testing and Neyman-Pearson hypotheses testing (Gigerenzer, 1993). This was generally presented as the sine qua non of scientific method. The textbooks did not tell their readers that they were teaching a shotgun marriage between Fisher and Neyman-Pearson. Rather, the textbooks created the illusion that statistics is statistics is statistics. Since the 1950s, statistical inference had become a mechanical ritual in psychology and beyond, enforced by journal editors and internalized by researchers as the guardian of objectivity and scholarly morality. Bayesianism, by contrast, looked subjective and, above all, unnecessary.

Thus, in the 1970s and 1980s, Bayesianism became a rational yardstick for the subjects in psychological experiments, but not for the experimenters who analyzed their subjects. Subjects were judged rational if their inferences from data to hypotheses followed Bayes' theorem, otherwise their judgments were recorded as an error in reasoning, such as the base rate fallacy (see below). However, when experimenters made inferences from data to hypotheses - here, whether subjects are Bayesians - they did not use Bayes' theorem. They used, as they had been taught for two decades before Edwards' proposal, frequentist statistics. But the most commonly used kind of frequentist statistics, R. A. Fisher's significance testing, does not use prior probabilities or base rates. This was not recorded as an error in reasoning, although it had all the characteristics of the base rate fallacy. Nor do I know of a single experimenter who noticed and remarked on that amazing double standard. The split between Bayesians and frequentists not only divides disciplines today, it can also go right through the same person (Gigerenzer, 1993).

Edwards soon seemed to have become dissatisfied with pointing out discrepancies between subjects' reasoning and Bayes' formula, and no interesting and rich theory of how subjects actually do reason had emerged. He turned to the task of designing tools that help people reason the Bayesian way. In the 1970s, Amos Tversky and Daniel Kahneman took over Edwards' second proposal and turned it into what is now known as the heuristics and biases program.

The heuristics and biases program arrived at a view of human rationality (Table 1) diametrically opposed to that of classical probability theory. Yet this modern program neglects the distinction between single-event probabilities and frequencies just as the classical probabilists ignored the distinction between subjective degrees of certainty and objective chances.

I will now use the distinction between single-event probabilities and frequencies to unearth the reasonableness buried under the perspective of the heuristics and biases program.

Representation of information: single-event probabilities versus frequencies

My point here is precisely not to champion one side over another - frequentism over Bayesianism, or vice versa - but to point out a connection between the single-event probabilities/frequencies distinction and a second distinction, that between algorithms and information representation.

Much ink has been spilled in debates about mental algorithms: is the mind equipped with the right statistical algorithms or only with suboptimal algorithms based on rules of thumb such as the representativeness heuristic? However, our discussion about the cognitive processes responsible for probabilistic reasoning would be incomplete if it remained at the level of algorithms, optimal or otherwise. Algorithms need information, and information needs representation. This distinction between algorithms and information representation is central to David Marr's (1982) analysis of visual information processing systems.

For example, take numerical information. This information can be represented by the Arabic numeral system, by the binary numeral system, by Roman numerals, and by other symbol systems. These different representations can be mapped one-to-one onto each other, and are in this sense formally equivalent representations. But they are not necessarily equivalent for calculating algorithms. The algorithms programmed into my pocket calculator work well when I feed them Arabic numerals but not binary numbers. The human mind seems to have or at least acquire analogous preferences for one form of representation: contemplate for a moment long division in Roman numerals.

Let me now return to the distinction between single-event probabilities and frequencies. Instead of squabbling over which one captures the "real" meaning of probability, let us instead regard them as two different representations of probability information. Finer distinctions can be made, but this will suffice for a start.

An evolutionary speculation links these two distinctions. Assume that some capacity or algorithm for statistical reasoning has been built up through evolution by natural selection. To what information representation would such an algorithm be adapted? Certainly not percentages and single event probabilities (as is assumed in many experiments on human reasoning), since these took millennia of literacy and numeracy to evolve as tools for communication. Rather, in an illiterate and innumerate world, the representation would be frequencies of events, sequentially encoded as experienced - for example, 3 out of 20 as opposed to 15% or p = .15. Such a representation is couched in terms of discrete cases.

Note that bumblebees, birds, rats, and ants all seem to be good intuitive statisticians, highly sensitive to changes in frequency distributions in their environments, as recent research in foraging behavior indicates (Gallistel, 1990; Real & Caraco, 1986). (One wonders, reading that literature, why birds and bees seem to do so much better than humans.)

In short, the proper functioning of a mental algorithm depends on the way in which information is represented. So, to analyze probabilistic reasoning, we must attend to the difference between, at least, the frequency and the single-event representation of probability. If evolution has favored one of these forms of representation, then it will be frequencies which prelinguistic organisms could observe and act on.

Attending to this distinction suffices to make several apparently stable cognitive illusions disappear.

How to make the conjunction fallacy disappear

Now we apply the distinction between single-event and frequency information representation to the Linda Problem. We only change the format from single event to a frequency representation, leaving everything else as it was.

Linda is 31 years old, single, outspoken and very bright. She majored in philosophy. As a student, she was deeply concerned with issues of discrimination and social justice, and also participated in antinuclear demonstrations.

There are 100 people who fit the description above. How many of them are:

(a) bank tellers

(b) bank tellers and active in the feminist movement

Subjects are now asked for frequency judgments rather than for single-event probabilities. If the mind solves the Linda Problem by using a representativeness heuristic, changes in information representation should not matter, because they do not change the degree of similarity. The description of Linda is still more representative of (or similar to) the conjunction "teller and feminist" than of "teller." Subjects therefore should still exhibit the conjunction fallacy.

However, if there is some statistical algorithm in the mind that is adapted to frequencies as information representation, then something striking should happen to this stable cognitive illusion. Violations of the conjunction rule should largely disappear.

Table 2
How to make the conjunction fallacy disappear
Linda problem
Conjunction violations (%)
Single-event versions
Tversky & Kahneman (1983)
Which is more probable?
Probability ratings
Probability ratings T*
Fiedler (1988)

Probability ranking, Exp. 1
Probability ranking, Exp. 2

Hertwig & Gigerenzer (1993)

Probability ranking

Frequency versions
Fiedler (1988)

How many out of 100?
How many out of X?

Hertwig & Gigerenzer (1993)

How many out of 200?
How many out of 200? (poll version)

Note: The various versions of the Linda problem are (i) which is more probable (see text, n = 142), (ii) probability ratings on a nine-point scale (n = 119), (iii) probability ratings using the alternative "Linda is a bank teller whether or not she is active in the feminist movement" (T*) instead of "Linda is a bank teller" (T) (n = 75), (iv) hypothetical betting, i.e. subjects were asked "if you could win $10 by betting on an event, which of the following would you choose to bet on?" (n = 60). Fiedler asked subjects to rank order T, T&F, and other alternatives with respect to their probability. In his first frequency version the population size was always 100, in the second it varied (n = 44 and 23, in Experiments 1 and 2, respectively). Hertwig & Gigerenzer asked subjects to rank order T, T&F, and F, with respect to their probability (single-event version, n = 24), or estimate the frequency of T, T&F, F (frequency version, n = 25). The poll version (n = 24) read "200 women with the following characteristics have been selected for a poll: They are 30 years old, ...[followed by Linda's characteristics]."


The experimental evidence available confirms this prediction. Klaus Fiedler (1988) reported that the number of conjunction violations in the Linda problem dropped from 91% in the original, single-event representation to 22% in the frequency representation. The same result was found, when he replaced "there are 100 persons" by some odd number such as "there are 168 persons." The drop in the number of conjunction violations here was from 83% to 17%. Hertwig and Gigerenzer (1993) used three alternatives: F (Linda is active in the feminist movement), T&F (Linda is a bank teller and active in the feminist movement) and T(Linda is a bank teller). In the single-event task, subjects rank-ordered F, T&F and T with respect to their probability; in the frequency task, they estimated the frequency of T, T&F and F ("how many out of 200?"). The percentage of conjunction violations dropped from 88% in the single-event task to 20% and 17%, respectively, in two frequency tasks.

Hertwig and Gigerenzer, and Fiedler reported similar results for other reasoning tasks from which the conjunction fallacy has been inferred as a stable cognitive illusion. Tversky and Kahneman (1983) had reported a similar case in their original paper, but maintained the claim that people commit a fallacy when choosing the conjunction in the single-event case.

To summarize: The philosophical and statistical distinction between single events and frequencies clarifies that judgments hitherto labeled instances of the "conjunction fallacy" cannot be properly called reasoning errors in the sense of violations of the laws of probability. The conceptual distinction between single event and frequency representations suffices to make this allegedly stable cognitive illusion disappear. The conjunction fallacy is not the only cognitive illusion that is subject to this argument.

How to make the base-rate fallacy disappear

In the 1960s, Ward Edwards and his colleagues designed probability revision problems to find out whether their subjects were Bayesians. Many of these problems used the tried-and-true urns-and-balls problems, and the major finding was that subjects exhibited conservatism, that is, that they seemed to give too much weight to the base rates. From the 1970s on, however, Tversky, Kahneman, and many of their followers claimed that reasoning deviates from Bayes' rule in the opposite direction: subjects ignore base rates - the so-called base-rate fallacy.

Recently, some researchers have weakened their claims about the generality and robustness of the base-rate fallacy, but some of the fundamental confusions with which this stimulating research was burdened from the very start have survived (Gigerenzer & Murray, 1987, ch.5).

The two confusions I point out are both instances of blurring single-event probabilities with frequencies. One confusion was between the Bayesian notion of a person's prior probability and the frequentist concept of a base rate. Tversky and Kahneman (e.g., 1974; Kahneman & Tversky, 1973) started out using the terms "neglect of base rates" or "insensitivity to base rates" interchangeably with the term "neglect of prior probabilities" or "insensitivity to prior probabilities." However, priors and base rates are different things. Priors are subjective degrees of belief that may be informed by objective base rates, but need not be identical. (Similarly, the subjective likelihoods that enter Bayes' theorem and the "individuating" information presented by the experimenter need not be identical; see Birnbaum, 1983; Schum, 1990). This confusion, however, was necessary to argue that if a subject does not give much weight to whatever base rate information the experimenter has presented, this counts as a demonstration of a fallacy, i.e., that the subject does not reason by Bayesian principles. Whether or not the mind actually reasons by Bayesian principles, this confusion between a base rate and a subjective prior has prevented us from drawing adequate conclusions from experimental work.

The second and related confusion is between normative theories of the subjective and of the frequentist variety. For instance, when subjects seemed not to pay much attention to base rate information, Kahneman and Tversky (1973, p. 243) asserted: "The failure to appreciate the relevance of prior probability in the presence of specific evidence is perhaps one of the most significant departures of intuition from the normative theory of prediction." But which normative theory? They had Bayesianism in mind, and a narrow version thereof - e.g., one that conflated base rates with priors (Gigerenzer, Hell & Blank, 1988). But what if intuition were measured against the frequency view?

I will now apply the distinction between single-event and frequency information representation to the base-rate fallacy . Here is an observation to start with.

Some researchers tend to change the representation of a problem from single-event probabilities to frequencies when they turn away from their subjects and explain the correct solution to their readers. An early example is Hammerton (1973, p. 252) who used single-event probabilities to communicate information to his subjects:

1. A device has been invented for screening a population for a disease known as psylicrapitis. 2. The device is a very good one, but not perfect. 3. If someone is a sufferer, there is a 90% chance that he will be recorded positively. 4. If he is not a sufferer, there is still a 1% chance that he will be recorded positively. 5. Roughly 1% of the population has the disease. 6. Mr. Smith has been tested, and the result is positive. The chance that he is in fact a sufferer is:______.

Hammerton seems to have been surprised that his subjects gave a median response of 85% (which is close to the 90% hit rate) despite the 1% base rate. Such judgments have been labeled by others the base-rate fallacy. When the author explained the correct answer to his readers, he switched, without comment, into a frequency representation:

Out of every 100 persons tested, we expect 1 to have the disease; and the device is nearly certain to say that he has. Also, out of that 100, we expect the machine to say that 1 healthy person has the disease. Thus, in the long run, out of every 100 persons tested, we expect 2 positive results, one of which will be correct and the other incorrect. Therefore the odds on any positive result being valid are roughly even.

The frequency format could be easily digested by Hammerton's readers. You can "see" that the relative frequency is one-to-one (i.e., 50%), and not 85%. Hammerton's subjects, however, were tested and failed on a single-event representation.

Here is a second example. In a fascinating article on mammography, Eddy (1982) reports that he asked 100 physicians questions of the following kind:

The prevalence of breast cancer is 1% (in a specified population). A mammography gives a positive result in 80% of women with breast cancer, and in 10% of women without breast cancer. What is the probability that a woman who tests positive actually has breast cancer? _____%

Eddy (1982) reports that 95 out of 100 physicians estimated the probability p(cancer|positive) to be about 70% to 80%. However, if one applies Bayes' theorem to the information given, p(cancer|positive) is only .08 (or 8%). The judgment of these 95 physicians once more looks like an instance of the base-rate fallacy. College students, physicians, writers of medical textbooks (Eddy, 1982), staff at the Harvard Medical School (Casscells et al., 1978) - all seem to have equally great difficulties with problems of this kind. Reasoning about single-event probabilities does not seem to come naturally to them.

Let us now perform a thought-experiment with the mammography problem. Change the information representation in the mammography problem from single-event probabilities to frequencies:

Imagine 100 people (think of a 10x10 grid). We expect that 1 woman has cancer and a positive mammography. Also, we expect that there are 10 more women with a positive mammography but no cancer. Thus we expect 11 people with a positive mammography. How many women with a positive mammography will actually have breast cancer?

With frequencies, you immediately "see" that only 1 out of 11 people who test positive will have cancer. The relative frequency, or probability, is about .09. The base-rate fallacy disappears if the information is represented in frequencies. Let us now turn from the thought-experiment to real experiments.

Casscells et al.(1978) gave 60 staff and students of the Harvard Medical School the following problem, cast in single-event probabilities (except for the base rate):

If a test to detect a disease whose prevalence is 1/1000 has a false positive rate of 5%, what is the chance that a person found to have a positive result actually has the disease, assuming you know nothing about the person's symptoms or signs?

If one inserts these numbers into Bayes' theorem, the posterior probability that the person actually has the disease is .02 (assuming that the test correctly diagnoses every person who has the disease - a piece of missing information).

Most of the staff and students at Harvard Medical School were hopelessly lost - almost half estimated this probability as .95, not .02. Only 11 participants answered .02. Note the amount of variability in the physicians' judgments about the probability of the disease! The modal answer of .95 was taken to be another instance of the base-rate fallacy, or, base-rate neglect, as Tversky and Kahneman (1982) called it: the base rate of the disease (1/1000) is neglected, and the judgment is based only (or mainly) on the characteristics of the test (here: the false positive rate). This seemed yet another proof of the stability of the base-rate fallacy.

But I will now apply to the Harvard Medical School problem the same frequency-representation procedure I applied to the preceding problems. If there is some kind of algorithm for statistical reasoning that works on frequency representations, changing the information representation in the Harvard Medical School problem from single-event probabilities and percentages to frequencies should make the base-rate fallacy disappear. Consequently, the large variability in judgments should also disappear.

Cosmides and Tooby (1993) have tested this prediction in a series of experiments with more than 400 Stanford undergraduates. They constructed a dozen or so variations of this medical problem, substituting step-by-step frequencies for single-event probabilities. In the original single-event version, the Stanford undergraduates gave almost the same low percentage of .02 answers as the staff and students at Harvard Medical School, 12% compared to 18% (Table 3). The original single-event version was somewhat ambiguous, because the true positive rate was not specified and Stanford undergraduates might not know what the term "false positive rate" means. Therefore, Cosmides and Tooby constructed a purified single-event version in which these ambiguities were eliminated:

The prevalence of disease X is 1/1000. A test has been developed to detect when a person has disease X. Every time the test is given to a person who has the disease, the test comes out positive. But sometimes the test also comes out positive when it is given to a person who is completely healthy. Specifically, 5% of all people who are perfectly healthy test positive for the disease.

What is the chance that a person found to have a positive result actually has the disease, assuming that you know nothingabout the person's symptoms or signs? ____ %

This made the percentage of .02 answers go up to 36%, but not higher. Rather dramatic effects were obtained, however, when the single-event format was changed into a frequency format. There were two major changes, the format of the information (first paragraph of the single-event version) and that of the task (second paragraph). To change the format of the information from single events to frequencies, (1) all probability information was expressed in frequencies such as "50 out of 1000" instead of 5%, and (2) a reference class ("Americans") was added on which these frequencies are defined. Nothing else was changed:

One out of 1000 Americans has disease X. A test has been developed to detect when a person has disease X. Every time the test is given to a person who has the disease, the test comes out positive. But sometimes the test also comes out positive when it is given to a person who is completely healthy. Specifically, out of every 1000 people who are perfectly healthy, 50 of them test positive for the disease.

To transform the task from estimating a single-event probability to estimating a frequency, the second paragraph of the single-event version was replaced by the following question:

How many people who test positive for the disease will actually have the disease? _____ out of _____.

If our minds were not built to reason statistically, but only equipped with crude heuristics (consult Table 1), then the distinction between single events and frequencies should not matter. But it does. Table 3 shows how one can make almost everybody (or almost nobody, or anything in-between) find the answer that corresponds exactly with applying Bayes' theorem to the information given - that is, a probability of .02 or a frequency "1 out of 51," respectively. If both the information and the task were in terms of frequencies, this percentage was over seventy; if only one of the two was represented by frequencies, the percentage was in-between the single-event and the frequency versions. If the frequency format was combined with asking the subjects to construct a pictorial frequency representation (i.e., to represent each person by a square, and mark those who do have the disease and those who test positive), then the percentage reached 92%.

Cosmides and Tooby's experimental variations, both in number and in detail, go beyond what I have summarized here. But my summary suffices to make the same point made with regard to the conjunction fallacy. The philosophical distinction between single-event probabilities and frequencies seems to be as important for the untutored mind as it is for probability theory. It can make apparently stable cognitive illusions disappear.

These results have direct implications for teaching statistical reasoning.

Natural sampling of frequency information

So far I have dealt with situations in which frequency information comes in one package, as in textbook problems or in newspapers. In many natural environments, and for animals or people in an illiterate world, however, frequencies must be sequentially learned through experience. How does an algorithm vary if we move from the standard single-event probability textbook problem to a corresponding ecological situation, in which the structure of the environment is sequentially learned through experience? Here is another thought-experiment.

Let us transpose the above Medical Diagnosis Problem to a non-literate society where physicians have to rely on their experience alone. Assume you are a physician. Your tribe has been afflicted for one year by a previously unknown and fatal disease. Everyone suspected of having the new disease is sent to you. You were lucky to discover one symptom that seems to signal the outbreak of the disease. What would it mean to be a Bayesian physician in this non-literate society?

You would encounter all information sequentially, as discrete cases that add up to frequencies. This information gathering is sometimes called natural sampling (Kleiter, 1993), a concept corresponding to Brunswik's (1955) representative sampling. So far you have seen 30 people suspected of having the disease. Ten of these turned out to have the disease, 20 did not. Of the 10 persons afflicted by the disease, 8 showed the symptom; of the 20 persons not afflicted, only 4 had the symptom. Now they bring in number 31. She has the symptom. What mental algorithm do you need in order to calculate the Bayesian posterior probability that she actually has the disease?

It turns out that in natural sampling this algorithm is quite simple - indeed, much simpler than required in those studies from which the base rate fallacy has been concluded. The algorithm needs only two absolute frequencies: the number a of people with symptom and disease, and the number b of people with symptom and no disease. These frequencies are a = 8 and b = 4, respectively. The algorithm to calculate the relative frequency f(D|S) of people with disease D among those who have the symptom S is:

ggwtdsp__06.gif (355 Byte)

If you are a Bayesian and want to calculate from the frequencies monitored so far the posterior probability p(D|S) that patient number 31 has the disease, your mental algorithm is just as simple:

ggwtdsp__04.gif (268 Byte)

Compare now this algorithm to that algorithm needed in the standard probability revision tasks of the heuristics and biases program. In the latter, the information is presented in terms of three single-event probabilities (forget for a moment the confusion between base rates and subjective priors): the prior probability p(D), and the likelihoods p(S|D) and p(S|-D). For this representation of information, Bayes' theorem is:

ggwtdsp__05.gif (691 Byte)

The information (corresponding to the natural sampling condition) would be represented as p(D) = .33, p(S|D) = .80, and p(S|-D) = .20. Inserting these numbers into Bayes' theorem results in the following calculation:

p(D|S) = .33 x .80 /(.33 x .80 + .67 x .20) = .67

The result is the same as in natural sampling, but the calculation is much more difficult.

The general point I want to make is that the way information is represented in an experiment, or encountered in a natural environment, can require reasoning algorithms of different complexity. Even if these algorithms are mathematically equivalent, as they are in the thought-experiment just presented, they can be computationally and psychologically different. Specifically, if information is encoded through natural sampling of frequencies - as opposed to the typical laboratory studies which present three single-event probabilities - the following differences arise:

(1) In natural sampling, memory needs to monitor only two kinds of information, the frequencies a and b. No attention need be paid to the base rates themselves.

(2) In natural sampling, Bayes' rule reduces to a simple algorithm.

(3) Frequency information, naturally sampled, carries more information than single-event probabilities. Absolute frequencies contain information about the sample size (e.g., "3 out of 20," as opposed to p = .15), which allows for computing the precision (so-called second-order probabilities) of the information.

I know of very few studies that used natural sampling instead of displaying three single-event probabilities. Christensen-Szalanski and Beach (1982) represented the information in a medical diagnosis problem (similar to those described before) both in the single-event probability format, as usual, and by natural sampling. In the single-event version the usual results were obtained, from which the base rate fallacy has been concluded. In the natural sampling condition, subjects were shown, one by one, 100 slides. Each slide contained information about one patient: whether or not the patient had pneumonia, and whether or not the test result was positive. As in the single-event version, the task was to estimate p(pneumonia|positive). The mean estimate in the natural sampling condition was .22, almost identical with the actual relative frequency f(pneumonia|positive) = 6/(6 + 19) = .24. (Although the means were very close, there was still considerable individual variability in estimates, perhaps due in part to individual differences in monitoring the actual frequencies.)

Here is a second study. One of the best publicized demonstrations of the base rate fallacy outside of the realm of medical diagnosis problems is Tversky and Kahneman's (1982) Cab Problem:

A cab was involved in a hit-and-run accident at night. Two cab companies, the Green and the Blue, operate in the city. You are given the following data:

(i) 85% of the cabs in the city are Green and 15% are Blue.

(ii) a witness identified the cab as Blue. The court tested the reliability of the witness under the same circumstances that existed on the night of the accident and concluded that the witness correctly identified each one of the two colors 80% of the time and failed 20% of the time.

What is the probability that the cab involved in the accident was Blue rather than Green?

Tversky and Kahneman reported that the modal and median response of several hundred subjects was .80, whereas Bayes' theorem gives only .41. The median response is identical with the witness' hit rate - just as in some of the medical diagnosis problems - and this has been interpreted to mean that subjects neglect base rates. Bar-Hillel (1980; 1983) has tried many variations, such as presenting the base rates before or after the other information, and concluded that the base rate fallacy was a robust phenomenon. Tversky and Kahneman (1980) suggested that the reason for this is that base rates tend not be used unless they are seen as causal: "The proportions of Blue and Green cabs does not induce a differential propensity to be involved in accidents and this information is therefore neglected." (p.70).

Note that the problem is presented in a single-event probability format. If base rates are neglected because they are not "causal," then the distinction between single-event probabilities and frequencies should not matter, since frequencies do not induce a "differential propensity to be involved in accidents" either.

Gigerenzer and Schlotterbek (1993) displayed the information in the Cab problem by means of natural sampling. In analogy to the study by Christensen-Szalanski and Beach, 100 incidents of hit-and-run accidents were shown, one by one, using a computer display. In each case the subjects could see whether the cab was blue or green, and what the witness reported. After they had seen all 100 incidents, subjects were given a frequency task, "how many of the cabs reported as 'blue' were actually blue?" We also asked our subjects (after they had seen all 100 cases) for their perceived four conjoint frequencies (blue cabs and report "blue," blue cabs and report "green," and so on). This allowed us to control for individual differences in perceived frequencies. Each subject's response to the frequency task was compared with the actual relative frequency f(blue|"blue"), which was 12 out of 29, or .41, and with the corresponding individual relative frequency, calculated from the subject's reported conjoint frequencies.

Subjects performance was no less impressive than in the Christensen-Szalanski and Beach study. Half of the subjects' responses to the frequency task hit exactly either the true frequency (12 out of 29) or the corresponding number calculated from their perceived conjoint frequencies. Perceived conjoint frequencies were in very good correspondence with true frequencies, with a slight overestimation of the smallest frequency (blue cabs and report "green") and underestimation for the largest frequency (green cabs and report "green"). Neither of these two frequencies, however, is needed to solve the frequency task.

To summarize: The examples given, including sequential frequency processing, show that the distinction between single-event probabilities and frequencies is relevant to understanding how the mind reasons about a class of problems that are often termed Bayesian probability revision problems.

Thus far, we have seen how to make two cognitive illusions, the conjunction fallacy and the base rate fallacy, largely disappear. I will now turn to a third prominent illusion.

How to make overconfidence bias disappear

Confidence in one's knowledge has been typically studied with questions of the following kind:

Which city has more inhabitants?

(a) Hyderabad

(b) Islamabad

How confident are you that your answer is correct?

50%, 60%, 70%, 80%, 90%, 100%

Imagine you are an experimental subject: Your task is to choose one of the two alternatives. Suppose you choose Islamabad, as most subjects in previous studies did. Then you are asked to state your confidence, or subjective probability, that your answer "Islamabad" is correct. 50% confident means guessing, 100% confident means that you are absolutely sure that Islamabad is the larger city. From a large sample of questions, the experimenter counts how many answers in each of the confidence categories were actually correct.

The major finding of some two decades of research is the following (Lichtenstein, Fischhoff & Phillips, 1982): In all the cases where subjects said, "I am 100% confident that my answer is correct," the relative frequency of correct answers was only about 80%; in all the cases where subjects said, "I am 90% confident" the relative frequency of correct answers was only about 75%, when subjects said "I am 80% confident" the relative frequency of correct answers was only about 65%, and so on. Values for confidence were systematically higher than relative frequencies. This systematic discrepancy has been interpreted as an error in reasoning and has been named overconfidence bias. Quantitatively, overconfidence bias is defined as the difference between mean confidence and mean percentage correct.

Is overconfidence bias really a "bias" in the sense of a violation of probability theory? Let me rephrase the question: has probability theory been violated if one's degree of belief (confidence) in a single event (i.e., that a particular answer is correct) is different from the relative frequency of correct answers in the long run? From the point of view of the frequency interpretation, the answer is "no." In this view, probability theory is about frequencies; it does not apply to single-event judgments like confidences. Therefore, no statement about confidences can violate the laws of probability. Even for Bayesians, however, the answer is not "yes." The issue here is not internal consistency, but the relation between subjective probability and external (objective) frequencies, which is a more complicated issue and depends on conditions such as independence. In particular, if there is no feedback after each answer, as in this research, and if the true answers for a series of questions are dependent, one cannot expect that one's average degree of belief matches the relative frequency of correct answers. Take, for instance, predictions of the following type: "Will there be snowfall on December 24, 1999, in downtown Boston? Yes/No." "Will there be snowfall on December 24, 1999, at Logan (Boston) airport? Yes/No." "Will there be snowfall on December 24, 1999, in Cambridge, Mass.? Yes/No." And so on. Assume, after careful consideration, your probability that there will be snow is .7 in each case. Nevertheless, you cannot expect that your single-event confidences match the relative frequencies in the long run, because the outcomes are dependent. If it snows in downtown Boston, it will most likely snow in all places, and you appear to be underconfident; otherwise you will appear overconfident.

For these various reasons, a discrepancy between confidence in single events and relative frequencies in the long run should not be simply labeled an "error" in statistical and probabilistic reasoning - contrary to the claims in the heuristics-and-biases literature. It only looks that way from the perspective of a narrow interpretation of probability theory that blurs the fundamental distinction between single events and frequencies.

However, for the last two decades or so, most researchers took it for granted that any systematic difference between confidence and frequency is a reasoning error, a regrettable deviation from rationality. And they assumed that their task is to explain this discrepancy by some deficiency in our mental or motivational programming, such as a "confirmation bias" (Koriat, Lichtenstein & Fischhoff, 1980), "insensitivity to item difficulty" (von Winterfeldt & Edwards, 1986, p. 128), and the tendency of humans in the Western world to overestimate their intellectual powers (Dawes, 1980). Similar to other "cognitive illusions," overconfidence bias has been suggested as an explanation for human disasters of many kinds, including deadly accidents in industry (Spettell & Liebert, 1986), errors in the legal process (Saks & Kidd, 1980) and systematic deviations from rationality in negotiation and management (Bazerman & Neale, 1986).

Many experiments have demonstrated the stability of the overconfidence phenomenon despite various "debiasing methods," such as warning subjects about overconfidence prior to the experiment, or providing monetary incentives. We even used a bottle of French champagne as an incentive, but to no avail. Edwards and von Winterfeldt (1986, p.656) concluded in a tone of regret: "Can anything be done? Not much."

I will now apply the distinction between single-event probabilities and frequencies to the overconfidence bias. Take the same kind of general-knowledge questions that have been used before to demonstrate the overconfidence bias. But now let our subjects make frequency judgments. After our subjects answered 50 general-knowledge questions of the Hyderabad-Islamabad type, in the usual format, they also had the opportunity to judge frequencies: "How many of these 50 questions do you think you answered correctly?"

If confidence in one's knowledge were truly biased due to confirmation bias, wishful thinking, or other deficits in cognition, motivation, or personality, then the difference between a single-event and a frequency representation should not matter. Overestimation should remain stable, as it does with warnings and bribes.

Table 4 shows the results of two experiments (Gigerenzer, Hoffrage & Kleinbölting, 1991). If one calculates, for each subject, the difference between mean confidence (averaged over 50 questions) and the relative frequency of correct answers, one finds, as usual, a stable positive difference that has been called the overconfidence bias. For instance, the value +13.8 is such a difference (multiplied by 100, and averaged across the 80 subjects in the first experiment). But the interesting issue is how the frequency estimates compare with the actual frequencies.

When we compared subjects' estimated frequencies with their true frequencies, there was no overestimation. Frequency judgments were quite accurate. In both experiments, the mean differences were even slightly negative, indicating a tendency towards underestimation. For instance, the figure -2.4 means that in the actual set of 50 questions, the estimated frequency of correct answers was, on the average, 1.2 lower than the true frequency. Subjects missed the true frequency by an average of only about 1 correct answer in a set of 50 questions.

Note that the very same subjects appear to be overestimating their knowledge, if one blurs the distinction between single-event probabilities and frequencies. You may think that this difference between single-event and frequency judgment is simply due to subjects having second thoughts about their performance at the end of the experiment. We have checked this. When the sequence "confidence judgments - frequency judgment" was repeated again and again (by presenting several sets of 50 questions in a sequence), subjects consistently gave different values for confidence and frequency.

This chapter is not the place to pursue the question of how to model these striking judgments. We have developed the theory of probabilistic mental models (Gigerenzer et al., 1991), which explains this and related phenomena by an algorithm that infers both confidences and frequencies from frequency information, that is, from frequency information based on different reference classes.

To summarize: I have argued that the discrepancy between mean confidence and relative frequency of correct answers, known as "overconfidence bias," is not an error in probabilistic reasoning. It only seems to be from a narrow normative perspective, in which the distinction between single-event confidence and frequencies is blurred. If we ask our subjects about frequencies instead of single-event confidences, we can make this stable phenomenon disappear. The philosophical distinction is much more effective with our subjects than money and French champagne.

The striking effect of frequency representations on apparent violations of probability theory, as reported in this chapter, seems to generalize to so-called violations of utility theory as well. For instance, Keren and Wagenaar (1987) showed that standard violations such as the "certainty effect" and the "possibility effect"(Kahneman & Tversky, 1979) largely disappear when a single gamble is changed into a repeated gamble (see also Keren, 1991; Lopes, 1981; Montgomery & Adelbratt, 1982). It also seems to generalize to a class of phenomenon known as the "illusion of control" (Langer, 1975), which diminishes if single-event estimates are replaced by judgments about a series of events (Hogarth, Koehler & Gibbs, 1993).

3. Conclusions

Probability theory and psychology have been historically intertwined since the Enlightenment. The psychological theories of Locke, Hume, and Hartley provided the grounds for the classical interpretation of probability, in particular for the assumption that the mind unconsciously tallies frequencies and converts them into rational degrees of belief. This created the fiction of the reasonable man (l'homme éclairé) and the blurring of the distinction between objective frequencies and subjective probabilities. When, by the early 19th century, psychological theories had shifted to illusions, the reasonable man dissolved and the difference between frequencies and degrees of belief became clear. The reasonable man gave way to the average man (l'homme moyen), and the frequency interpretation of probability emerged and became dominant. When the subjective interpretation - Bayesianism and subjective utility theory - regained influence in the second half of this century, these modern versions of Enlightenment rationality often did not distinguish between single-event probabilities and frequencies, nor between single and repeated gambles -just as classical probability theory had not. And many psychologists, following in their footsteps, did not make this distinction either and found the human mind overflowing with cognitive illusions. Conflating single-event probabilities and frequencies now served the fiction of the irrational man.

Much of the current view is condensed in my economist colleague's dictum "either reasoning is rational or it's psychological." Rationality is now defined in terms of formal algorithms or axioms, and psychology is called upon to explain the irrational. However, algorithms work on information, and information needs representation. To discuss rationality in terms of algorithms alone, good or bad ones, is incomplete if one does not pay attention to the kind of information representation these algorithms were designed to work upon. Consequently, one cannot simply conclude from what looks like bad performance, or cognitive illusion, that there are poor algorithms. This non sequitur has been a basic flaw in the heuristics and biases program. When information is represented in terms of frequencies rather than single-event probabilities, apparently stable cognitive illusions tend to disappear.


Author's Note

This chapter is based in part on work supported by the Fonds zur Förderung der wissenschaftlichen Forschung (P 8842MED), Austria. I am grateful to Lorraine Daston, Dan Goldstein, Ralph Hertwig, Cheryce Kremer, Jim Magnuson, and Peter Sedlmeier for many helpful comments.



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Table 3
How to make the base-rate fallacy disappear: The Harvard Medical School problem (see Cosmides & Tooby, 1993)
Representation of the problem
Anwers in accordance with Bayes's rule (in %)
Original single-event format
(Casscells et al., 1987
Single-event format, replication
Information in frequency format, task in single-event format
Information in single-event format, task in frequency format
Information and task in frequency format
Information and task in frequncy format, pictorial representation


Table 4
How to make the overconfidence bias disappear (see Gigerenzer et al., 1991)
Difference between
Experiment 1
(n= 80)
Experiment 2
(n= 97)
Mean confidence and true relative frequency of correct answers ("overconfidence bias")
Estimated frequency and true frequency of correct answers

Note: To make values for frequency and confidence judgments comparable, all frequencies were transformed to relative frequencies. Values shown are differences multiplied by a factor of 100.
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