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Page 1: Searching for Patterns in Random Sequences - CVL …vitallongevity.utdallas.edu/cnl/wp-content/uploads/2014/04/Wolford... · Searching for Patterns in Random Sequences George Wolford,

Searching for Patterns in Random Sequences

George Wolford, Sarah E. Newman, Dartmouth CollegeMichael B. Miller, University of California at Santa Barbara and Dartmouth College

Gagan S. Wig, Dartmouth College

Canadian Journal of Experimental Psychology, 2004, 58:4, 221-228

Abstract In a probability-guessing paradigm, participantspredict which of two events will occur on each trial.Participants generally frequency match even though fre-quency matching is nonoptimal with random sequences.The optimal strategy is to guess the most frequent event,maximizing. We hypothesize that frequency matchingresults from a search for patterns, even in randomsequences. Using both callisotomy patients and patientswith frontal brain damage, Wolford, Miller, and Gazzaniaga(2000) found frequency matching in the left hemisphere butmaximizing in the right hemisphere. In this paper, we showthat a secondary task that competes for left hemisphereresources moves the participants toward maximizing butthat a right-hemisphere task preserves frequency matching.We also show that a misunderstanding of randomness con-tributes to frequency matching.

People experience difficulty in estimating andunderstanding uncertainty in many situations(Kahneman, Slovic, & Tversky, 1982). We believe thatthe search for patterns is an important factor in peo-ple’s behaviour in uncertain situations. An examinationof people’s behaviour in a probability-guessing taskmay help us understand the way they deal with uncer-tainty. We provide evidence from probability-guessingexperiments that people look for patterns in sequencesof events, even when informed that there are no pat-terns and that this search is related to their understand-ing of uncertainty.

In a standard probability-guessing situation, thereare at least two stimuli (e.g., lights, optimal routes towork, etc.) that are presented on some schedule. Withtwo stimuli, the probabilities of the two stimuli are pand (1-p). We are most interested in the case in whichp is greater than .50. Participants are asked to predictwhich stimulus will be presented on each trial.Frequency matching refers to guessing the most fre-quent stimulus with probability p. In a case in which p = .70, frequency matching yields correct answers58% of the time (0.7.0.7 + 0.3.0.3). Maximizing, alwaysguessing the stimulus that has the highest probability

of appearing, yields correct answers 70% of the time(0.7.1.0 + 0.3.0.0). Maximizing is superior to frequencymatching as long as p ≠ .5 and the sequence is trulyrandom.

When asked to predict which of two things willoccur over a series of trials, most humans exhibit fre-quency matching. That is, they make predictions inproportion to the frequency with which the eventshave occurred in the past (Estes, 1961). Frequencymatching has intrigued scientists since the 1930s(Humphreys, 1939) for at least two reasons: It is anonoptimal strategy for random sequences and mostother species exhibit maximizing, the optimal strategy(Hinson & Staddon, 1983). We propose that peopleexhibit frequency matching because they look for pat-terns in sequences even when told that the sequencesare random. If there were a real pattern in the data,then any successful hypothesis about that patternwould result in frequency matching. If people weresearching for patterns, then frequency matching wouldnot be a strategy, per se, but a consequence of thesearch for patterns.

Yellott (1969) provided a striking demonstration ofthe extent to which people seek patterns in sequences.In his experiment, a light was flashed to either the leftor the right on each trial and subjects had to predictwhich light would appear. Participants experiencedmany trials and p varied across blocks. Subjects’ pre-dictions matched the frequency of the actual presenta-tions (frequency matching), changing when the fre-quency changed. In the final 50-trial block, the contin-gencies changed without the participants’ knowledge.The light appeared wherever the subject predicted itwould. In other words, if the subject guessed left, itwas left, etc. Subjects continued to frequency matchduring these last 50 trials. When Yellott stopped theexperiment and asked subjects for their impressions,they overwhelmingly responded that there was a fixedpattern to the light sequences and that they had finallyfigured it out. They proceeded to describe elaboratesequences of right and left choices that resulted intheir responses always being correct. These verbalreports support the contention that subjects had been

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222 Wolford, Newman, Miller, and Wig

searching for fixed sequences all along and werefooled into thinking they had succeeded.

Wolford, Miller, and Gazzaniga (2000) provided evi-dence that this search for patterns might be occurringin the left hemisphere. They reasoned that the searchfor patterns might be related to a proposed neuralmodule, the interpreter, that was proposed byGazzaniga and his colleagues (Gazzaniga, 1989, 1995;Metcalfe, Funnell, & Gazzaniga, 1995). The interpreteris thought to play the role of trying to make sense outof the information that it confronts (Gazzaniga, 1989,1995; Metcalfe et al., 1995). Using split-brain patients,who have had their corpus callosum severed to treatintractable epilepsy thus preventing interhemispherictransfer of information at the cortical level, Gazzaniga(1989) provided evidence that the interpreter is locatedin the left hemisphere. To test our hypothesis aboutthe relationship between searching for patterns andthe left hemisphere, we carried out a probability-guessing paradigm in two split-brain patients and inseveral patients with unilateral damage to the frontalregion of one of the hemispheres. We hypothesizedthat if processes housed in the left hemisphere wereresponsible for the search for patterns, then we shouldobserve frequency matching in the left hemisphereand maximizing in the right hemisphere. That is whatwe found. The patients with unilateral cortical damagereplicated the findings with the split-brain patients.

In the present paper, we use competing-task para-digms in participants with intact brains to corroboratethe previous findings with split-brain patients. We usedone competing task that we presumed would engageprimarily left-hemisphere resources and one that wepresumed would engage primarily right-hemisphereresources. Based on our previous work, we predictedthat the left-hemisphere task would interfere with fre-quency matching but the right-hemisphere task wouldnot.

In a second experiment, we capitalized on people’smisunderstanding of random sequences (Falk &Konold, 1997). We altered the conditional probabilitiesbetween events to create sequences that conform topeoples’ common, but incorrect, views about random-ness to see if those sequences yield different patternsof behavior than truly random sequences.

Experiment 1In Experiment 1, participants engaged in a probabil-

ity-guessing task in the presence of distracting tasks.Our previous research suggested that the neuralprocesses responsible for frequency matching are inthe left hemisphere. Therefore, we wanted one dis-tracting task that would compete primarily for left-hemisphere processes and a second distracting task

that should compete primarily for right-hemisphereprocesses. Our prediction was that subjects wouldcontinue to frequency match in the presence of aright-hemisphere distracting task but would maximizewith a left-hemisphere distracting task.

For the left-hemisphere task we chose a variant ofthe three-back verbal working memory task. Resultsfrom neuro-imaging studies show asymmetric activa-tion of structures in the left hemisphere for verbalworking memory tasks (Awh et al., 1996; Coull, Frith,Frackowiak, & Grasby, 1996). Dunbar and Sussman(1995) used a similar verbal working memory task tosimulate frontal-brain damage in participants withoutdamage. Participants in the verbal working memorygroup had to maintain three digits in memory at alltimes while guessing whether the target would occurat the top or bottom of the screen. Our prediction wasthat the participants who had to engage in this distract-ing task would be unable to devote the resourcesrequired to look for patterns and might switch to maxi-mizing.

For the right-hemisphere task, we had participantsengage in a same/different or one-back task using ran-domly constructed polygons. Results from neuro-imag-ing studies show asymmetric activation of structures inthe right hemisphere for visual-spatial working memo-ry tasks (Smith et al., 1995).

MethodParticipants

Thirty Dartmouth undergraduates served in the firstexperiment, 10 in each of three conditions. Subjectsreceived credit toward the introductory psychologycourse as well as the possibility of monetary rewardbased on performance.

Design and ProcedureParticipants were randomly assigned to one of three

distractor-task conditions: a verbal working-memorytask, a visual-spatial working-memory task, or a no-distractor task. The distractor task was interwoven witha probability-guessing paradigm in which the more fre-quent choice appeared 75% of the time, and the lessfrequent 25%. Participants were informed that the tar-get in the probability-guessing task was chosen ran-domly. They were paid $.01 for each correct predictionand were instructed to be as accurate as they could onboth tasks.

Verbal working-memory task. The verbal working-memory task required subjects to remember the lastthree digits they saw. Each trial began with the presen-tation of a digit in the middle of the screen. The digitserved as a cue to make a guess in the probability-

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guessing task. After each prediction, a new digitappeared. Participants were instructed to maintain thelast three digits in memory, updating the set with theappearance of each new digit. Each digit was chosenrandomly from the set of 10 digits. Participants wererandomly probed four times per block and asked totype in the last three digits they had seen. Participantswere told that errors on the memory task wouldreduce the earnings on the probability-guessing task.

Visual-spatial task. Each trial of the visual-spatialtask began with the presentation of a six-sided poly-gon in the centre of the screen. The participants had tojudge whether the polygon was the same as or differ-ent than the one the trial before, by hitting the appro-priate key. The polygon was a blue outline that hadsix vertices. The original polygon on Trial 1 was gener-ated randomly, subject to certain size constraints, andcentred on the screen. On each subsequent trial thepolygon was altered randomly with probability .50using the computer language’s random function. Ontrials in which the polygon was altered, the changewas accomplished by moving a single, randomly cho-sen vertex 30 pixels (680 x 480 screen) in one of thefour cardinal directions (N, E, S, W). After responding

same or different to the polygon task, a 500-ms pausewas inserted and then the participant made a predic-tion for the probability-guessing paradigm. Participantswere told that errors on the polygon task wouldreduce the payoff from the probability-guessing task.

Guessing task. The experimenter advised the partici-pants that one of their tasks was to predict whether acoloured square would appear above or below a fixa-tion cross (+) presented in the middle of the screen.Participants were informed that the sequence of upsand downs was completely random. To indicate theirprediction, the subjects would press the appropriatebutton on the keyboard. The correct buttons werelabeled top or bottom. Upon predicting either top orbottom, a coloured square would then appear either onthe top or bottom of the screen, providing immediatefeedback regarding the accuracy of their prediction.

The experiment was carried out on iMacs with thescreen resolution set to 640 x 480. The experimentconsisted of five blocks of 100 trials. After every 100trials the participants received additional feedbackindicating the accuracy of their predictions and theaccuracy on the distracting task for that block of 100trials.

Figure 1. Probability-guessing behaviour as a function of the presence and type ofcompeting task in Experiment 1. Points represent the probability of choosing the mostfrequent alternative. Error bars represent standard errors.

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224 Wolford, Newman, Miller, and Wig

ResultsThe primary results are displayed in Figure 1. The

points in the figure represent the probability of choos-ing the most likely alternative. Overall, the participantsin the Control and Polygon Groups approached fre-quency matching and the participants in the 3-BackGroup approached maximizing. The linear contrast forthe interaction between Group and Block reflects thegrowing separation between the conditions in laterblocks and was significant, F(1,27) = 4.31, p = .02. Theomnibus interaction of Group and Block was signifi-cant with a multivariate F(8,50) = 3.13, p = .006. Usingt-tests with Bonferroni corrections, the means of the 3-Back Group were significantly higher than the meansof the other two groups during the last two blocks.The Control and Polygon Groups differed on Blocks 3and 4, but the Polygon task was actually closer to fre-quency matching than the Control Group throughout.Finally, to look at maximizing in individual subjects,we defined maximizing as choosing the most frequentalternative 95% or more of the time in the last twoblocks. Using that definition, zero participants maxi-mized in the Control Group, one participant maxi-mized in the Polygon Group, and seven participantsmaximized in the 3-Back Group. Those differencesyield χ2(2) = 14.66, p = .0008.

Looking at performance on the secondary tasks,participants were slightly more accurate on the 3-Backtask than the polygon task (.85 versus .75). That per-formance is shown by block in Table 1. The accuracyscores were quite variable in the 3-Back task, so thedifference between tasks was not significant, t(18) =1.633, p = .12. In neither task was accuracy on thecompeting task correlated with the degree of maximiz-ing, yielding correlations of .14 and .09 between accu-racy on competing task and performance on probabili-ty-guessing averaged over the final two blocks.

We interpret the results to indicate that participantscontinue searching for patterns in the presence of thepolygon task, but participants do not do so in thepresence of the 3-Back task.

Experiment 2Our proposal is that people exhibit frequency

matching because they are searching for patterns in thesequence even though they are told the sequence israndom. The results of the first experiment supportsthat notion. We think that one reason people search forpatterns in the face of contrary information is that ran-dom sequences do not look random. Several investiga-tors have shown that when people are asked to con-struct random sequences they include too few longruns and too many alternations (Falk & Konold, 1997;Lopes, 1982; Lopes & Oden, 1987). We reasoned that ifthe sequences were constructed to look more like theaverage person’s stereotype of a random sequence,then participants might abandon the search for patternsand move toward maximizing. In the third experiment,we made the sequences look “more random” by alter-ing the conditional probabilities to break up runs.

MethodParticipants

Thirty Dartmouth undergraduates served in the sec-ond experiment, 15 in each of two conditions. Subjectsreceived credit toward the introductory psychologycourse as well as the possibility of monetary rewardbased on performance as in the previous experiments.

Design and ProcedureParticipants were randomly assigned to one of two

conditions: a control condition and an altered-condi-tional condition. In the control condition, we used theprobability-guessing task described in Experiment 1with no competing tasks. The most frequent alternativeoccurred on 70% of the trials.

For the control group, each trial was independentof the preceding one so the probability of the mostfrequent alternative was 70% independent of whichstimulus appeared previously. In the altered condition-al group, the probability of the most frequent stimuluson the first trial was .70. On all subsequent trials, theprobability of the most frequent stimulus was condi-tionalized on the preceding trial. The probability ofpresenting the most frequent alternative was .60 if themost frequent one had just occurred versus .93 if theother alternative had just occurred. Those conditionalsyield a marginal probability of .70 for the top, but theprobability of an alternation is increased from .42 to.56 compared to the truly random condition in whichthe conditionals equal the marginals. The alteredsequences were still “random” in the sense that thelocation of the next alternative was not perfectly pre-dictable. Regardless of which condition they wereassigned to, participants were told that the boxeswould appear randomly at the top or the bottom.

TABLE 1Performance on the Secondary Tasks by Block

–––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––Task

–––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––Block N-Back (s.e.) Polygon (s.e.)

1 0.780 (0.079) 0.746 (0.029)2 0.833 (0.071) 0.742 (0.025)3 0.900 (0.067) 0.746 (0.019)4 0.913 (0.066) 0.753 (0.035)5 0.827 (0.070) 0.748 (0.027)

–––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––

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ResultsThe results are shown in Figure 2. We predicted that

the group with altered conditionals would movetoward maximizing, but the control group would fre-quency match as usual. Specifically, we predicted thatthe amount of difference between the two groupswould increase across blocks. This prediction was test-ed and confirmed with a linear contrast on the interac-tion term. The contrast yielded an F(1,28) = 4.04, p =.05. The groups were significantly different by Block 5,t(28) = 2.16, p = .04.

The differences between the groups are even morestriking if you look at the probability of respondingwith the most frequent stimulus on Trial N, condition-alized on which stimulus occurred on Trial N-1. Figure3 shows the difference between those two conditionalsfor each of the two groups. As shown in Figure 3, par-ticipants in the Control Group are about 20% to 25%more likely to respond to the most frequent one afterjust seeing the most frequent one than after seeing theleast frequent one. The actual probabilities of thosetwo conditional probabilities are identical in theControl Group. In contrast, the participants in theAltered Conditional Group respond less differently as afunction of the preceding stimulus, especially in the

first few hundred trials, even though there really is a33% difference between the two conditionals in thatgroup. The main effect of Group was highly signifi-cant, F(1,28) = 17.68, p < .001. Neither the main effectof Block nor the interaction of Block and Group wassignificant although the interaction approached signifi-cance, F(4,66.2) = 2.379, p < .079. To reiterate, in thegroup in which the conditionals are actually the same,participants predict a big difference, but in the condi-tion in which they are very different, participants pre-dict less difference and the difference goes in oppositedirections in the two groups. This difference is consis-tent with our belief that participants see the sequenceswith altered conditionals as random and are less likelyto look for patterns, but participants in the controlgroup do not believe the sequences are random andtherefore search for clues to uncover the “true”sequence.

General DiscussionBased on our previous research with both split-

brain patients and patients with selective frontallesions, we hypothesized that frequency matching in aprobability-guessing task is the result of searching forpatterns and that the search for patterns asymmetrical-

Figure 2. Probability-guessing behaviour for a control condition versus a condition inwhich the conditional probabilities have been altered to appear more random fromExperiment 2. Error bars represent standard errors.

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226 Wolford, Newman, Miller, and Wig

ly involves structures in the left hemisphere. By usingcompeting tasks that drew differentially on the twohemispheres, we found some support for that earlierhypothesis. In Experiment 1, participants did maintainfrequency matching when the competing task primari-ly involved right-hemisphere resources, butapproached maximizing when the task primarilyinvolved left-hemisphere resources.

Our proposal that participants regularly search forpatterns is supported by the comments of the partici-pants as in Yellott (1969). During debriefing, many ofour subjects mentioned that they were searching forpatterns.

In the second experiment we found support for thecontention that people’s misunderstanding of random-ness contributes to the search for (and belief in) pat-terns. Specifically, those that received altered condi-tionals that conformed to the common (and incorrect)perception of randomness were found to approachmaximizing.

Note that in an important sense, there was morepredictability in the sequences with the altered condi-tionals. H is a measure of uncertainty or information(Attneave, 1959) computed according to Equation 1:

H = -∑πlog2π (1)

We computed the value of H on pairs of trials foreach condition. Pairs are required to capture the differ-ence in conditionals. The value of H for normalsequences in which each trial is independent is 1.76and the value of H for the group with altered condi-tionals is 1.67. Lower values of H indicate less uncer-tainty and thus more predictability. The maximal possi-ble uncertainty with only two alternatives would be tohave each one have an independent probability of .5and that would yield an H of 2.0. The value of H orsome similar measure is relevant because we are argu-ing that participants will give up the search for struc-ture in the sequences with altered conditionals eventhough those sequences have more structure by objec-tive measures.

Probability-guessing experiments tend to elicit fairlyregular functions over blocks (Estes, 1961). Whetherthe subject is frequency matching or maximizing, he orshe has to learn what the frequencies are. The subjecthas no information about frequencies in Trial 1.Therefore, the mean probability of guessing a particu-lar outcome during that block represents some combi-nation of learning about the frequencies and laterstrategies.

We believe that people are constantly searching forpatterns in sequences. Such a search is of great utility

Figure 3. The probability of a “most frequent” response following a most frequent outcome onthe preceding trial minus the probability of a “most frequent” response following a least fre-quent outcome on the preceding trial from Experiment 2. Error bars represent standard errors.

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if there are patterns, but is counterproductive if thereare not. We believe that the search for patterns insequences that are truly random impacts the use andunderstanding of uncertainty, often to the detriment ofhuman behaviour.

This research was supported, in part, by NationalInstitute of Neurological Disease Grant NS17778 – Section 3.

Correspondence should be addressed to GeorgeWolford, Department of Psychological and Brain Sciences,Dartmouth College, 6207 Moore Hall, Hanover, NH 03755(E-mail: [email protected]).

ReferencesAttneave, F. (1959). Applications of information theory to

psychology. New York: Holt, Rinehart and Winston, Inc.Awh, E., Jonides, J., Smith, E. E., Schumacher, E. H.,

Koeppe, R. A., & Katz, S. (1996). Dissociation of storageand rehearsal in verbal working memory: Evidence frompositron emission tomography. Psychological Science, 7,25-31.

Coull, J. T., Frith, C. D., Frackowiak, R. S. J., & Grasby, P. M.(1996). A fronto-parietal network for rapid visual infor-mation processing: A PET study of sustained attentionand working memory. Neuropsychologia, 34, 1085-1095.

Dunbar, K., & Sussman, D. (1995). Toward a cognitiveaccount of frontal lobe function: Simulating frontal lobedeficits in normal subjects. Annals of the New YorkAcademy of Sciences, 769, 289-304.

Estes, W. K. (1961). A descriptive approach to the dynamicsof choice behavior. Behavioral Science, 6, 177-184.

Falk, R., & Konold, C. (1997). Making sense of randomness:Implicit encoding as a basis for judgment. PsychologicalReview, 104, 301-318.

Gazzaniga, M. S. (1989). Organization of the human brain.

Science, 245, 947-952.Gazzaniga, M. S. (1995). Principles of human brain organi-

zation derived from split-brain studies. Neuron, 14, 217-228.

Gopnik, A. (1998). Explanation as orgasm. Minds andMachines, 8, 101-118.

Hinson, J. M., & Staddon, J. E. R. (1983). Matching, maximiz-ing and hill-climbing. Journal of the ExperimentalAnalysis of Behavior, 40, 321-331.

Humphreys, L. G. (1939). Acquisition and extinction of ver-bal expectations in a situation analogous to condition-ing. Journal of Experimental Psychology, 25, 294-301.

Kahneman, D., Slovic, P., & Tversky, A. (1982). Judgmentunder uncertainty: Heuristics and biases. New York:Cambridge University Press.

Lopes, L. L. (1982). Doing the impossible: A note on induc-tion and the experience of randomness. Journal ofExperimental Psychology: Learning, Memory, andCognition, 8, 626-636.

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Wolford, G., Miller, M. B., & Gazzaniga, M. (2000). The lefthemisphere’s role in hypothesis formation. Journal ofNeuroscience, 20, RC64.

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Sommaire

Les personnes qui observent une série d’événe-ments semblent vouloir y déceler une tendanceexplicative qui les aiderait à prédire des événementsultérieurs. Elles consacreraient beaucoup de temps à larecherche de tendances même en présence d’une sériealéatoire dont la nature leur a été signalée. Dans leprésent article, nous offrons la preuve que des struc-tures neurales de l’hémisphère gauche président à larecherche de tendances et que l’incompréhension ducaractère aléatoire influence la propension à vouloirdéceler des tendances. Nous faisons appel à un para-digme de probabilités arbitraires pour illustrer nosaffirmations.

Les participants qui se fondent sur un paradigme de

probabilités arbitraires prédisent lequel de deux événe-ments se produira à chaque essai. Règle générale, ilsse reportent à la fréquence (prédisent un événementdonné selon la fréquence à laquelle il s’est produit lorsd’essais antérieurs), bien que l’appariement parfréquence soit non optimal face à des séquences aléa-toires. La stratégie optimale consiste à deviner sansexception l’événement le plus fréquent, en maximisantles résultats. Nous posons que l’appariement parfréquence découle de la recherche de tendances,même dans des séquences aléatoires. Wolford, Milleret Gazzaniaga (2000) ont étudié à la fois des person-nes qui avaient subi une calosotomie et d’autres dontle lobe frontal avait été atteint et ont constaté que l’ap-

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pariement par fréquence se produisait dans l’hémis-phère gauche, tandis que la maximisation se situaitdans l’hémisphère droit. Nous démontrons, dans leprésent article, qu’une tâche secondaire qui vise àmobiliser les ressources de l’hémisphère gauche(mémoire de travail verbale) pousse les participants àmaximiser, tandis qu’une tâche qui fait appel à l’hémis-phère droit (mémoire de travail spatiale) n’occasionneaucun changement à l’appariement par fréquence. Nosrésultats correspondent à nos constatations antérieures,selon lesquelles l’appariement par fréquence dépendde structures de l’hémisphère gauche.

Les gens semblent interpréter erronément le carac-

tère aléatoire. Invités à produire une séquence aléa-toire de piles et de faces, ils retiennent trop d’alter-nances et trop peu de longues suites (Lopes et Oden,1987). La seconde expérience a consisté à composerdes séquences conformes aux attentes des gens quantà leur caractère aléatoire. En modifiant les probabilitésconditionnelles, nous avons conçu des séquences dontles probabilités marginales étaient identiques à cellesde séquences véritablement aléatoires, à ceci prèsqu’elles comportaient plus d’alternances et moins delongues suites. Les participants se sont rapprochés dela maximisation face aux séquences aux probabilitésconditionnelles modifiées.

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