Randomness and Coincidences: Reconciling Intuition and Probability Theory
Randomness and Coincidences: Reconciling Intuition and Probability Theory Thomas L. Griffiths & Joshua B. Tenenbaum Department of Psychology Stanford University Stanford, CA 94305-2130 USA gruffydd,jbt @psych.stanford.edu Abstract We argue that the apparent inconsistency between peo- ple’s intuitions about chance and the normative predic- tions of probability theory, as expressed in judgments about randomness and coincidences, can be resolved by focussing on the evidence observations provide about the processes that generated them rather than their likelihood. This argument is supported by probabilistic modeling of sequence and number production, together with two ex- periments that examine judgments about coincidences. People are notoriously inaccurate in their judgments about randomness, such as whether a sequence of heads and tails like is more random than the se- quence . Intuitively, the former sequence seems more random, but both sequences are equally likely to be produced by a random generating process that chooses or with equal probability, such as a fair coin. This kind of question is often used to illustrate how our intuitions about chance deviate from the normative standards set by probability theory. Our intuitions about coincidental events, which seem to be defined by their improbability, have faced similar criticism from statisti- cians (eg. Diaconis & Mosteller, 1989). The apparent inconsistency between our intuitions about chance and the formal structure of probability the- ory has provoked attention from philosophers and mathe- maticians, as well as psychologists. As a result, a number of definitions of randomness exist in both the mathemat- ical (eg. Chaitin, 2001; Kac, 1983; Li & Vitanyi, 1997) and the psychological (eg. Falk, 1981; Lopes, 1982) lit- erature. These definitions vary in how well they satisfy our intuitions, and can be hard to reconcile with proba- bility theory. In this paper, we will argue that there is a natural relationship between people’s intuitions about chance and the normative standards of probability theory. Traditional criticism of people’s intuitions about chance has focused on the fact that people are poor estimators of the likelihood of events being produced by a particu- lar generating process. The models we present turn this question around, asking how much more likely a set of events makes a particular generating process. This ques- tion may be far more useful in natural inference situa- tions, where it is often more important to reason diagnos- tically than predictively, attempting to infer the structure of our world from the data we observe. Randomness Reichenbach (1934/1949) is credited with having first suggested that mathematical novices will be unable to produce random sequences, instead showing a tendency to overestimate the frequency with which outcomes alter- nate. Subsequent research has provided support for this claim (reviewed in Bar-Hillel & Wagenaar, 1991; Tune, 1964; Wagenaar, 1972), with both sequences of numbers (eg. Budescu, 1987; Rabinowitz, Dunlap, Grant, & Cam- pione, 1989) and two-dimensional black and white grids (Falk, 1981). In producing binary sequences, people al- ternate with a probability of approximately 0.6, rather than the 0.5 that is seen in sequences produced by a ran- dom generating process. This preference for alternation results in subjectively random sequences containing less runs – such as an interrupted series of heads in a set of coin flips – than might be expected by chance (Lopes, Theories of subjective randomness A number of theories have been proposed to account for the accuracy of Reichenbach’s conjecture. These theo- ries have included postulating that people develop a con- cept of randomness that differs from the true definition of the term (eg. Budescu, 1987; Falk, 1981; Skinner, 1942), and that limited short-term memory might con- tribute to people’s responses (Baddeley, 1966; Kareev, 1992; 1995; Wiegersma, 1982). Most recently, Falk and Konold (1997) suggested that the concept of randomness can be connected to the subjective complexity of a se- quence, characterized by the difficulty of specifying a rule by which a sequence can be generated. This idea is related to a notion of complexity based on descrip- tion length (Li & Vitanyi, 1997), and has been considered elsewhere in psychology (Chater, 1996). The account of randomness that has had the strongest influence upon the wider literature of cognitive psychol- ogy is Kahneman and Tversky’s (1972) suggestion that people may be attempting to produce sequences that are “representative” of the output of a random generating process. For sequences, this means that the number of elements of each type appearing in the sequence should correspond to the overall probability with which these el- ements occur. Random sequences should also maintain local representativeness, such that subsequences demon- strate the appropriate probabilities.