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 Randomness Reichenbach (1934/1949) is credited with having ﬁrst We argue that the apparent inconsistency between peo- suggested that mathematical novices will be unable to ple’s intuitions about chance and the normative predic- produce random sequences, instead showing a tendency tions of probability theory, as expressed in judgments about randomness and coincidences, can be resolved by to overestimate the frequency with which outcomes alter- focussing on the evidence observations provide about the nate. Subsequent research has provided support for this processes that generated them rather than their likelihood. claim (reviewed in Bar-Hillel & Wagenaar, 1991; Tune, This argument is supported by probabilistic modeling of 1964; Wagenaar, 1972), with both sequences of numbers sequence and number production, together with two experiments that examine judgments about coincidences. (eg. Budescu, 1987; Rabinowitz, Dunlap, Grant, & Cam- pione, 1989) and two-dimensional black and white grids (Falk, 1981). In producing binary sequences, people al- People are notoriously inaccurate in their judgments ternate with a probability of approximately 0.6, rather about randomness, such as whether a sequence of heads than the 0.5 that is seen in sequences produced by a ran- and tails like ¢£¢£¤¥¢£¤£¤¥¤£¢ is more random than the se- dom generating process. This preference for alternation quence ¢£¢£¢£¢¥¢£¢£¢¥¢ . Intuitively, the former sequence results in subjectively random sequences containing less seems more random, but both sequences are equally runs – such as an interrupted series of heads in a set of

likely to be produced by a random generating process coin flips – than might be expected by chance (Lopes, ¤ that chooses ¢ or with equal probability, such as a fair 1982). coin. This kind of question is often used to illustrate how our intuitions about chance deviate from the normative Theories of subjective randomness standards set by probability theory. Our intuitions about A number of theories have been proposed to account for coincidental events, which seem to be defined by their the accuracy of Reichenbach’s conjecture. These theo- improbability, have faced similar criticism from statisti- ries have included postulating that people develop a con- cians (eg. Diaconis & Mosteller, 1989). cept of randomness that differs from the true definition The apparent inconsistency between our intuitions of the term (eg. Budescu, 1987; Falk, 1981; Skinner, about chance and the formal structure of probability the- 1942), and that limited short-term memory might con- ory has provoked attention from philosophers and mathe- tribute to people’s responses (Baddeley, 1966; Kareev, maticians, as well as psychologists. As a result, a number 1992; 1995; Wiegersma, 1982). Most recently, Falk and of definitions of randomness exist in both the mathemat- Konold (1997) suggested that the concept of randomness ical (eg. Chaitin, 2001; Kac, 1983; Li & Vitanyi, 1997) can be connected to the subjective complexity of a se- and the psychological (eg. Falk, 1981; Lopes, 1982) lit- quence, characterized by the difficulty of specifying a erature. These definitions vary in how well they satisfy rule by which a sequence can be generated. This idea our intuitions, and can be hard to reconcile with proba- is related to a notion of complexity based on descrip- bility theory. In this paper, we will argue that there is tion length (Li & Vitanyi, 1997), and has been considered a natural relationship between people’s intuitions about elsewhere in psychology (Chater, 1996). chance and the normative standards of probability theory. The account of randomness that has had the strongest Traditional criticism of people’s intuitions about chance influence upon the wider literature of cognitive psychol- has focused on the fact that people are poor estimators ogy is Kahneman and Tversky’s (1972) suggestion that of the likelihood of events being produced by a particu- people may be attempting to produce sequences that are lar generating process. The models we present turn this “representative” of the output of a random generating question around, asking how much more likely a set of process. For sequences, this means that the number of events makes a particular generating process. This ques- elements of each type appearing in the sequence should tion may be far more useful in natural inference situa- correspond to the overall probability with which these el- tions, where it is often more important to reason diagnos- ements occur. Random sequences should also maintain tically than predictively, attempting to infer the structure local representativeness, such that subsequences demon- of our world from the data we observe. strate the appropriate probabilities.

Formalizing representativeness any sequence of the same length. However, comput-

§ ¨ A major challenge for a theory of randomness based ing P ¦ x regular requires specifying the probability of upon representativeness is to express exactly what it the observed outcome resulting from a generating pro-

means for an outcome to be representative of a random cess that involves regularities. While this probability is

§ ¨ generating process. One interpretation of this statement hard to deﬁne, it is in general easy to compute P ¦ x hi , is that the outcome provides evidence for having been where hi might be some hypothesised regularity. In the produced by a random generating process. This interpre- case of sequences of heads and tails, for instance, hi

tation has the advantage of submitting easily to formal- might correspond to a particular probability of observ-

¦¢£¢¥¤£¢£¤¥¤£¤£¢§ ¨ ization in the language of probability theory. ing heads, P ¦¢¨ p. In this case P hi is

4 4

If we are considering two candidate processes by p ¦ 1 p . Using the calculus of probability, we can ob-

§ ¨ which an outcome could be generated – one random, tain P ¦ x regular by summing over a set of hypothesized

and one containing systematic regularities – the total ev- regularities, ,

idence in favor of the random generating process can be

§ ¨ ¦ § ¨ ¦ § ¨ P ¦ x regular ∑ P x hi P hi regular (4)

assessed by the logarithm of the ratio of the probabilities

h H

of these processes i

§ ¨

where P ¦ hi regular is a prior probability on hi. In all

§ ¨ P ¦ random x

log (1) applications discussed in this paper, we make the simpli-

¦ § ¨

§ ¨ ©

P ¦ regular x fying assumption that P hi regular is uniform over all

hi . However, we stress that this assumption is not

§ ¨ ¦ § ¨ where P ¦ random x and P regular x are the probabili- necessary for the models we create, and the prior may ties of a random and a regular generating process respec- in fact differ from uniformity in some realistic judgment tively, given the outcome x. contexts. This quantity can be computed using the odds form of Bayes’ rule Random sequences

For the case of binary sequences, such as those that might

§ ¨ ¦ § ¨ ¦ ¨ P ¦ random x P x random P random

(2) be produced by ﬂipping a coin, possible regularities can

§ ¨ ¦ § ¨ ¦ ¨ © P ¦ regular x P x regular P regular be divided into two classes. One class assumes that ﬂips are independent, and the regularities it contains are asser-

in which the term on the left-hand side of the equation is tions about the value of P ¦¢¨ . The second class includes called the posterior odds, and the first and second terms regularities that make reference to properties of subse- on the right-hand side are called the likelihood ratio and quences containing more than a single element, such prior odds, respectively. Of the latter two terms, the spe- as alternation, runs, and symmetries. Since this second cific outcome x influences only the likelihood ratio. Thus class is less well defined, it is instructive to examine the the contribution of x to the evidence in favour of a ran- account that can be obtained just by using the first class

dom generating process can be measured by the loga- of regularities.

¦¢¨ rithm of the likelihood ratio,

Taking to be all values of p P 0 1 , we

1 H T

§ ¨ ¦ ¨ ¦ § ¨

have P ¦ H T random 2 and P H T regular

¦ § ¨

1 H T

random(x) log (3) 0 p ¦ 1 p dp, where H T are the sufﬁcient statistics

¦ § ¨

P x regular © of a particular sequence containing H heads and T tails. This method of assessing the weight of evidence for a Completing the integral, it follows from (3) that

particular hypothesis provided by an observation is often H T

¨ ¦ ¨¥ ¦ ¨

random ¦ H T log H f H T (5)

H T

¨ ¦ ¨ given above is called a Bayes factor (Kass & Raftery, where f ¦ H T is log2 log H T 1 , a ﬁxed 1995). The Bayes factor for a set of independent obser- function of the total number of ﬂips in the sequence. This vations will be the sum of their individual Bayes factors, result has a number of appealing properties. Firstly, it is and the expression has a clear information theoretic in- maximized when H T, which is consistent with Kah- terpretation (Good, 1979). The above expression is also neman and Tversky’s (1972) original description of the closely connected to the notion of minimum description representativeness of random sequences. Secondly, the length, connecting this approach to randomness with the ratio involved essentially measures the size of the set of ideas of Falk and Konold (1997) and Chater (1996). sequences sharing the same number of heads and tails.

A sequence like ¢£¢£¢¥¢£¢£¢£¢¥¢ is unique in its composition, Deﬁning regularity

whereas ¢£¢£¤£¢¥¤£¤£¤¥¢ has a composition much more com- Evaluating the evidence that a particular outcome pro- monly obtained by ﬂipping a coin eight times.

vides for a random generating process requires com-

§ ¨ ¦ § ¨ puting two probabilities: P ¦ x random and P x regular . The Zenith radio data The first of these probabilities follows from the defi- Having defined a framework for analyzing the subjective nition of the random generating process. For exam- randomness of sequences, we have the opportunity to de-

1 8

¨ ¦ ¨ ple, P ¦¢£¢£¤¥¢£¤£¤¥¤£¢§ random is 2 , as it would be for velop a speciﬁc model. One classic data set concerning the production of random sequences is the Zenith radio extent to which ¢ results in a more random outcome than

data. These data were obtained as a result of an attempt ¤ , assessed over the subsequences starting one step back, by the Zenith corporation to test the hypothesis that peo- two steps back, and so forth, ple are sensitive to psychic transmissions. On several

occasions in 1937, a radio program took place during k 1

¨ ¦ ¨

which a group of psychics would transmit a randomly Lk ∑ random ¦ Hi 1 Ti random Hi Ti 1 (6)

© © © generated binary sequence to the receptive minds of their i 1 listeners. The listeners were asked to write down the se- where the Hi Ti are the tallies of heads and tails counting quence that they received, one element at a time. The back i steps © in the sequence. We can then convert this binary choices included heads and tails, light and dark, quantity into a probability using a logistic function, to black and white, and several symbols commonly used in give a probability distribution for the kth response, Rk: tests of psychic abilities, and all sequences contained a

total of ﬁve symbols. Listeners then mailed in their re- 1 ¢¨ P ¦ Rk λ

k sponses, which were analyzed. These responses demon- 1 e strated strong preferences for particular sequences, but 1 (7)

there was no systematic effect of the actual sequence that λ

k 1 Ti 1

1 ∏ was transmitted (Goodfellow, 1938). The data are thus a i 1 Hi 1 rich source of information about response preferences for

k 1 λ

¨ ∏ ¦ T 1

random sequences. The relative frequencies of the differ- i 1 i (8)

k 1 λ k 1 λ

¨ ¦ ¨

ent sequences, collapsed over choice of ﬁrst symbol, are ∏ ¦ T 1 ∏ H 1 i 1 i i 1 i shown in the upper panel of Figure 1. The λ parameter scales the effect that Lk has on the resulting probability. The probability of the sequence as a Zenith Radio Data 0.2 whole is then the product of the probabilities of the Rk, and the result deﬁnes a probability distribution over the 0.15 set of binary sequences of length k. This distribution is 0.1 shown in the lower panel of Figure 1 for k 5. Probability This simple model provides a remarkably good ac- 0.05 count of the response preferences people demonstrated 0 in the Zenith radio experiment. There is one clear dis-

crepancy: the model predicts that the sequence ! " ,

Randomness model ¤£¢¥¤£¢£¤ 0.2 equivalent to ¢£¤¥¢£¤£¢ or , should occur far more often than in the data. We can explain people’s avoidance 0.15 of this sequence by the fact that alternation itself forms a

0.1 regularity, which could easily be introduced into the hy-

Probability pothesis space. More striking is the account the model

0.05 gives of the different frequencies of sequences with less £ ¥ ! 0 apparent regularities, such as £ £ £ and . Ex- cluding the discrepant data point, the model gives a 00000 00001 00010 00011 00100 00101 00110 00111 01000 01001 01010 01011 01100 01101 01110 01111

parameter-free ordinal correlation rs 0 97, and with

λ 0 6 has a linear correlation r 0 95. Interestingly,

the model predicts alternation, for sequences that are Figure 1: The upper panel shows the original Zenith radio otherwise equally representative, with a probability of data, representing the responses of 20,099 participants, from 1

λ . With the value of λ used in fitting the Zenith radio # 1 2 Goodfellow (1938). The lower panel shows the predictions of data, the resulting predicted probability of alternation is the randomness model. Sequences are collapsed over the initial 0.6, consistent with previous findings (eg. Falk, 1981). choice, represented by 0. Pick a number Modeling random sequence production Research on subjective randomness has focused almost One of the most important characteristics of the Zenith exclusively on sequences, but sequences are not the only radio data is that people’s responses were produced se- stimuli that excite our intuitions about chance. In partic- quentially. In producing each element of the sequence, ular, random numbers loom larger in life than in the lit- people had knowledge of the previous elements. Kah- erature, although there have been a few studies that have neman and Tversky (1972) suggested that in producing investigated response preferences for numbers between such sequences, people pay attention to the local repre- 0 and 9. Kubovy and Psotka (1976) reported the fre- sentativeness of their choices – the representativeness of quency with which people produce numbers between 0 each subsequence. and 9 when asked to pick a number, aggregated across To capture this idea, we define Lk to be the local rep- several studies. These results are shown in the upper

resentativeness of choosing ¢ as the kth response – the panel of Figure 2. People showed a clear preference for the number 7, which Kubovy and Psotka (1976, p. 294) Kubovy and Psotka (1976) explained with reference to the properties of the num- 0.4

bers involved – for example, 6 is even, and a multiple 0.3 of 3, but it is harder to ﬁnd properties of 7. This ex- planation is suggestive of the kinds of regular generat- 0.2 Probability ing processes that could be involved in producing num- 0.1 bers. Shepard and Arabie (1979) found that similarity 0 judgments about numbers could be captured by proper- 0 1 2 3 4 5 6 7 8 9

ties like those described by Kubovy and Psotka (1976), Randomness model such as being even numbers, powers of 2, or occupying special positions such as endpoints. Taking the arithmetic properties of numbers to consti-

tute hypothetical regularities, we can specify the quan- ¨ tities necessary to compute random ¦ x . Our hi are sets of numbers that share some property, such as the set of even numbers between 0 and 9. For any hi, we deﬁne 0 1 2 3 4 5 6 7 8 9

1 Number

$ $

¦ § ¨ § §

P x hi for x hi and 0 otherwise, where hi is the hi size of the set. This means that observations generated from a regularity are uniformly sampled from that regu-

Figure 2: The upper panel shows number production data

§ ¨ larity. Setting P ¦ hi regular to give equal weight to all hi,

from Kubovy and Psotka (1976), taken from 1,770 participants

§ ¨ we can compute P ¦ x regular . This model can be applied to the data of Kubovy choosing numbers between 0 and 9. The lower panel shows the and Psotka (1976). Since there are ten possible re- transformed predictions of the randomness model.

§ ¨ sponses, we have P ¦ x random . Taking hypothet-

ical regularities of multiples of 2 ( 0 2 4 6 8 ¡ ), multi-

ples of 3 ( 3 6 9 ¡ ), multiples of 5 ( 0 5 ), powers of 2

¡ ¡ ¡

( 2 4 8 ¡ ), and endpoints ( 0 1 9 ), we obtain the

2. Randomness also needs to be included in so that

periences (Falk & MacGregor, 1983) might facilitate the ¨ random ¦ x is deﬁned when x is not in any other regu- under-estimation of the probability of events. larity. Its inclusion is analogous to the incorporation of a noise process, and is in fact formally identical in this Not just likelihood )*)+) case. The order of the model predictions is a parameter The above analyses reﬂect the same bias that made it

free result, and gives the ordinal correlation rs 0 99.

difﬁcult to construct a probabilistic account of random- Applying a single parameter power transformation to the

0 ' 98 ness: the notion that people’s judgments reﬂect the likeli-

¦ ¦ ¨&¨

predictions, y% y min y , gives r 0 95.

hood of particular outcomes. Subjectively, coincidences are events that seem unlikely, and are hence surprising Coincidences when they occur. However, just as with random se- The surprising frequency with which unlikely events quences, sets of events that are equally likely to be pro- tend to occur has drawn attention from a number of duced by a random generating process differ in the de- psychologists and statisticians. Diaconis and Mosteller gree to which they seem to be coincidences. Follow- (1989), in their analysis of such phenomena, define a co- ing Diaconis and Mosteller’s suggestion that the Birth- incidence as ‘ a surprising concurrence of events, per- day Problem provides a domain for the investigation of ceived as meaningfully ( & related, with no apparent causal coincidences, consider the kinds of coincidences formed connection’ (p. 853). They go on to suggest that the by sets of birthdays. If we meet four people and find out “surprising” frequency of these events is due to the flex- that their birthdays are October 4, October 4, October 4, ibility that we allow in identifying meaningful relation- and October 4, this is a much bigger coincidence than ships. Together with the fact that everyday life provides if the same people have birthdays May 14, July 8, Au- a vast number of opportunities for coincidences to oc- gust 21, and December 25, despite the fact that these sets cur, our willingness to tolerate near misses and to con- of birthdays are equally likely to be observed by chance. sider each of a number of possible concurrences mean- The way that these sets of birthdays differ is that one of ingful contributes to explaining the frequency with which them contains an obvious regularity: all four birthdays coincidences occur. Diaconis and Mosteller suggested occur on the same day. that the surprise that people show at the solution to the Birthday Problem – the fact that only 23 people are re- Modeling coincidences quired to give a 50% chance of two people sharing the Just as sequences differ in the amount of evidence they same birthday – suggests that similar neglect of combi- provide for having been produced by a random gener- natorial growth contributes to the underestimation of the ating process, sets of birthdays differ in how much evidence they provide for having been produced by a pro- one week across a month boundary, 4 birthdays in the cess that contains regularities. We argue that the amount same calendar month, 4 birthdays with the same calendar of evidence that an event provides for a regular generat- dates, and 2 same day, 4 same day, and 4 same date with ing process will correspond to how big a coincidence it an additional 4 unrelated birthdays, as well as 4 same seems, and that this can be computed in the same way as week with an additional 2 unrelated birthdays. These for randomness, dates were delivered in a questionnaire. Each participant

was instructed to rate how big a coincidence each set of

§ ¨ P ¦ x regular

coincidence(x) log (9) dates was, using a scale in which 1 denoted no coinci-

¦ § ¨, P x random dence and 10 denoted a very big coincidence. The results of the experiment and the model predic- To apply this model we have to deﬁne the regulari- tions are shown in the top and middle panels of Figure 3 ties . For birthdays, these regularities should corre- respectively . Again, the ordinal predictions of the model spond to relationships that can exist among dates. Our

are parameter free, with rs 0 94. Applying the transfor-

model of coincidences used a set of regularities that re- 0 ' 48

¦ ¦ ¨(¨

mation y% y min y , gives r 0 95. The main

ﬂected proximity in date (from 1 to 30 days), belonging discrepancies between the model and the data are the to the same calendar month, and having the same cal- four birthdays that occur in the same calendar month, and endar date (eg. January 17, March 17, September 17, the ordering of the random dates. The former could be December 17). We also assumed that each year con- addressed by increasing the prior probability given to the sists of 12 months of 30 days each. Thus, for a set of

regularity of being in the same calendar month – clearly

¦ § ¨

n birthdays, X x1 xn ¡ , we have P X random this was given greater weight by the participants than by

©( & ( (©

1 n

¨ ¦ § ¨ ¦ 360 . In defining P X regular , we want to respect the model. Explaining the increase in the judged coin- the fact that regularities among birthdays are still strik- cidence with larger sets of unrelated dates is more diffi- ing even when they are embedded in noise – for in- cult, but may be a result of opportunistic coincidences: stance, February 2, March 26, April 3, June 12, June as more dates are provided, participants have more op- 12, June 12, June 12, November 22 still provides strong portunities to identify complex regularities or find dates evidence for a regularity in the generating process. To of personal relevance. This process can be incorporated allow the model to tolerate noisy regularities, we can into the model, at the cost of greater complexity.

§ ¨ introduce a noise term into P ¦ X h . The probabil- i How big a coincidence? ity calculus lets us integrate out unwanted parameters, 10 so the introduction of a noise process need not result 5 in adding a numerical free parameter to the model. In Rating

1 0

§ ¨ ¦ § ¨ ¦ § ¨ ¦ α α α

α α

§ ¨ ¦ § ¨ P ¦ X hi ∏ P x j hi , and, taking a uniform

x j X

§ ¨ ¦ § ¨ prior on α, P ¦ X hi is simply ∏ P x j α hi dα, 0 x X

α 1 5 $

α $ -¦ ¨ Rating 1 x j hi

360 hi

§ ¨ P ¦ x j α hi α (10)

the entire year with probability α, and uniformly from 2 random 4 random 6 random 8 random 2 same day 2 in days 4 same day 4 same week 4 same month 4 same date 2 same day 4 same day 4 same date 4 same week

α (with random dates)

the regularity with probability ¦ 1 . The resulting

§ ¨

P ¦ X hi can then be substituted into (4), and taking a uni-

§ ¨ ¦ § ¨ form distribution for P ¦ hi regular gives P X regular . Figure 3: The top panel shows the judged extent of coinci- How big a coincidence? dence for each set of dates. The middle panel is the predictions The model outlined above makes strong predictions of the coincidences model, subjected to a transformation de- about the degree to which different sets of birthdays scribed in the text. The bottom panel shows randomness judg- should be judged to constitute coincidences. We con- ments for the same stimuli. ducted a simple experiment to examine these predictions. The participants were 93 undergraduates from Stanford Relating randomness and coincidences University, participating for partial course credit. Four- Judgments of randomness and coincidences both reflect teen potential relationships between birthdays were ex- the evidence that a set of observations provides for hav- amined, using two sets of dates. Each participant saw ing been produced by a particular generating process. one set of dates, in a random order. The dates reflected: Events that provide good evidence for a random gener- 2, 4, 6, and 8 apparently unrelated birthdays, 2 birthdays ating process are viewed as random, while events that on the same day, 2 birthdays in 2 days across a month provide evidence for a generating process incorporating boundary, 4 birthdays on the same day, 4 birthdays in some regularity seem like coincidences. By examining (3) and (9), we see that these phenomena are formally Falk, R. and Konold, C. (1997). Making sense of randomness: identified as inversely related: coincidences are events Implicit encoding as a bias for judgment. Psychological Re- that deviate from our notions of randomness. view, 104:301–318. Falk, R. and MacGregor, D. (1983). The surprisingness of co- We conducted a further experiment to see if this re- incidences. In Humphreys, P., Svenson, O., and Vari, A., ed- lationship was borne out in people’s judgments. Partici- itors, Analysing and aiding decision processes, pages 489– pants were 120 undergraduates from Stanford University, 502. North-Holland, New York. participating for partial course credit. The dates were the Good, I. J. (1979). A. M. Turing’s statistical work in World same as those used previously, and delivered in similar War II. Biometrika, 66:393–396. format. Each participant was instructed to rate how ran- Goodfellow, L. D. (1938). A psychological interpretation of the results of the Zenith radio experiments in telepathy. Journal dom each set of dates was, using a scale in which 1 de- of Experimental Psychology, 23:601–632. noted not at all random and 10 denoted very random. Hintzman, D. L., Asher, S. J., and Stern, L. D. (1978). Inci- The results of this experiment are shown in the bot- dental retrieval and memory for coincidences. In Gruneberg, tom panel of Figure 3. The correlation between the ran- M. M., Morris, P. E., and Sykes, R. N., editors, Practical as- domness judgments and the coincidence judgments is pects of memory, pages 61–68. Academic Press, New York. Kac, M. (1983). What is random? American Scientist, 71:405–

r 0 94, consistent with the hypothesis that randomness and coincidences are inversely linearly related. The 406. Kahneman, D. and Tversky, A. (1972). Subjective probabil- main discrepancy between the two data sets is that the ity: A judgment of representativeness. Cognitive Psychology, addition of unrelated dates seems to affect randomness 3:430–454. judgments more than coincidence judgments. Kareev, Y. (1992). Not that bad after all: Generation of random sequences. Journal of Experimental Psychology: Human Per- Conclusion ception and Performance, 18:1189–1194. Kareev, Y. (1995). Through a narrow window: Working mem- The models we have discussed in this paper provide ory capacity and the detection of covariation. Cognition, a connection between people’s intuitions about chance, 56:263–269. expressed in judgments about randomness and coinci- Kass, R. E. and Raftery, A. E. (1995). Bayes factors. Journal dences, and the formal structure of probability theory. of the American Statistical Association, 90:773–795. This connection depends upon changing the way we Kubovy, M. and Psotka, J. (1976). The predominance of seven model questions about probability. Rather than consider- and the apparent spontaneity of numerical choices. Journal of Experimental Psychology: Human Perception and Perfor- ing the likelihood of events being produced by a partic- mance, 2:291–294. ular generating process, our models address the question Li, M. and Vitanyi, P. (1997). An introduction to Kolmogorov of how much more likely a set of events makes a par- complexity and its applications. Springer Verlag, London. ticular generating process. This is a structural inference, Lopes, L. (1982). Doing the impossible: A note on induction drawing conclusions about the world from observed data. and the experience of randomness. Journal of Experimental Framed in this way, people’s judgments are revealed to Psycholgoy, 8:626–636. accurately approximate the statistical evidence that ob- Rabinowitz, F. M., Dunlap, W. P., Grant, M. J., and Campione, J. C. (1989). The rules used by children and adults to gen- servations provide for having been produced by a partic- erate random numbers. Journal of Mathematical Psychology, ular generating process. The apparent inaccuracy of our 33:227–287. intuitions may thus be a result of considering normative Reichenbach, H. (1934/1949). The theory of probability. Uni- theories based upon the likelihood of events rather than versity of California Press, Berkeley. the evidence they provide for a structural inference. Shepard, R. and Arabie, P. (1979). Additive clutering: Repre- sentation of similarities as combinations of discrete overlap- ping properties. Psychological Review, 86:87–123. References Skinner, B. F. (1942). The processes involved in the repeated Baddeley, A. D. (1966). The capacity of generating informa- guessing of alternatives. Journal of Experimental Psychology, tion by randomization. Quarterly Journal of Experimental 39:322–326. Psychology, 18:119–129. Tune, G. S. (1964). Response preferences: A review of some Bar-Hillel, M. and Wagenaar, W. A. (1991). The perception of relevant literature. Psychological Bulletin, 61:286–302. randomness. Advances in applied mathematics, 12:428–454. Wagenaar, W. A. (1972). Generation of random sequences by Budescu, D. V. (1987). A Markov model for generation of ran- human subjects: A critical survey of literature. Psychological dom binary sequences. Journal of Experimental Psychology: Bulletin, 77:65–72. Human perception and performance, 12:25–39. Wiegersma, S. (1982). Can repetition avoidance in randomiza- Chaitin, G. J. (2001). Exploring randomness. Springer Verlag, tion be explained by randomness concepts? Psychological London. Research, 44:189–198. Chater, N. (1996). Reconciling simplicity and likelihood principlesin perceptual organization. Psychological Review, 103:566–581. Acknowledgments Diaconis, P. and Mosteller, F. (1989). Methods for studying This work was supported by a Hackett studentship to the coincidences. Journal of the American Statistical Association, first author and funds from Mitsubishi Electric Research 84:853–861. Laboratories. The authors thank Persi Diaconis for inspi- Falk, R. (1981). The perception of randomness. In Proceed- ration and conversation, Tania Lombrozo, Craig McKen- ings of the fifth international conference for the psychology of mathematics education, volume 1, pages 222–229, Grenoble, zie and an anonymous reviewer for helpful comments, France. Laboratoire IMAG. and Kevin Lortie for finding the leak.