Cognition 120 (2011) 372–379

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Cognition

journal homepage: www.elsevier .com/ locate/COGNIT

Combinatorics and probability: Six- to ten-year-olds reliably predictwhether a relation will occur

Michel Gonzalez a,⇑, Vittorio Girotto a,b

a Laboratoire de Psychologie Cognitive, CNRS and Université de Provence, 3 Place Victor Hugo, 13331 Marseille, Franceb Dipartimento delle Arti e del Disegno Industriale, Università IUAV di Venezia, Dorsoduro 2206, 30123 Venezia, Italy

a r t i c l e i n f o a b s t r a c t

Article history:Available online 4 November 2010

Keywords:Probabilistic cognitionChildren’s predictionsExtensional reasoningCombinatoricsBinary relations

0010-0277/$ - see front matter � 2010 Elsevier B.Vdoi:10.1016/j.cognition.2010.10.006

⇑ Corresponding author. Address: Universite de PCharles, Laboratoire de Psychologie Cognitive, 3 PlacD, 13331 Marseille Cedex 3, France.

E-mail address: michel.gonzalez@univ-provence

Young children are able to judge which of two possibilities is more likely to occur whenthese possibilities are characterized by a simple property, like color (‘‘Is it more likely todraw a red chip or a blue chip?’’). Here we ask whether they can do so when the possibil-ities concern a relation between simple properties (‘‘Is it more likely to draw two chips ofthe same color or two different colored chips?’’). Three studies show that from the age ofsix children are able to predict the occurrence of a relation on the basis of its probability,and that from the age of nine their performance reaches adult levels. These results corrob-orate the theory of naive extensional reasoning, and are inconsistent with the hypothesisthat children need the help of instruction to reason correctly about relations.

� 2010 Elsevier B.V. All rights reserved.

1. Introduction

Probability theory and combinatorics are two con-nected branches of mathematics. Historically, the calculusof probability emerged as a method for solving games ofchance asking for complex combinatorial analyses (e.g.,Franklin, 2001). Even some of the simplest probabilityproblems require reasoning about combinations of possi-bilities. Consider two boxes: one containing a white anda black token, the other one containing a red and a greentoken. Suppose that one randomly takes a token from eachbox and asks you to evaluate the chances of getting a whiteand a red token. To provide a suitable answer, you need tocombine the two colors in the first box with the two colorsin the other box. Thus, you may conclude that the chancesof getting a white and a red token are 1/4 because whiteand red is one of the four possible combinations.

. All rights reserved.

rovence – Centre St.e Victor Hugo – case

.fr (M. Gonzalez).

Recent studies have shown that children and even infantsform expectations of an event on the basis of its probability(Denison & Xu, 2010; Girotto & Gonzalez, 2008; Teglas, Gir-otto, Gonzalez, & Bonatti, 2007). In these studies, a simpleproperty, like the color of a chip, characterized an event.But can children form expectations of this sort when a rela-tion between simple properties characterizes an event? Thispaper presents an account that explains how children pre-dict uncertain events. Following this account, we argue thatan elementary combinatorial capacity allows one to predictthat a relation between two tokens will hold on the basis ofits possibilities. We then report three studies that show thatfrom the age of about six children exhibit such a capacity.

Consider a container in which three red balls and oneblue ball are bouncing. Twelve-month-olds expect thatone of the red balls, rather than the blue ball, should exitfrom the container (Teglas et al., 2007). Preschoolers makesuitable predictions on the basis of these expectations. Forexample, they answer correctly ‘‘a red ball’’ to the question:‘‘If one ball exits, will you get a red ball or the blue ball?’’(e.g., Goldberg, 1966; Reyna & Brainerd, 1994). Theseresults show that children make predictions extensionally,by considering the possible ways in which events mayoccur (Johnson-Laird, Legrenzi, Girotto, Sonino-Legrenzi,

M. Gonzalez, V. Girotto / Cognition 120 (2011) 372–379 373

& Caverni, 1999). Thus, they predict the occurrence of a redoutcome because there are three possibilities to get a redball, and just one possibility to get the blue one.

Children’s extensional reasoning has been documentedwith tasks in which probabilities depend on simple proper-ties. For example, children are presented with a set of ob-jects (e.g., balls, chips), which vary along some dimension(e.g., their color). One of the objects is chosen at random.In such a situation, the higher the proportion of objectshaving a given property (e.g., being red), the higher theprobability of choosing one of them. When children haveto predict whether that property will hold for a randomlychosen object, their responses agree with an elementaryprinciple of classical probability, according to which thechances of an event equal the proportion of possibilitiesthat it occurs. Children seem to apply such a principle byextensionally considering the possibilities that a givenevent occurs or does not occur. Their evaluations are notnecessarily based on a precise counting of the number ofpossibilities attributed to each outcome. Indeed, anapproximate comparison of possibilities allows childrento conclude which outcome is likely to occur, even whenthey face large sets of possibilities (Davies, 1965; Denison& Xu, 2010). These results are in line with the finding thatyoung children can make approximate comparisons ofquantities before knowing the numerical symbolic system(e.g., Barth, Le Mont, Lipton, & Spelke, 2005). In sum, chil-dren are able to extensionally compare the chances of twocompeting events without making an explicit and preciseenumeration of the possibilities favoring each of them.

In principle, an extensional evaluation of chances can beapplied to any sort of event defined by a finite set of possi-bilities. Consider a set of two red, two blue, two green, andtwo yellow tokens. Suppose that one randomly choosestwo tokens, and you have to make a prediction about theircolor relation, that is, to predict whether they will have thesame color or two different colors. You might answer byconsidering all the possible pairs of tokens, and the result-ing combinations of colors. Thus, you will count 24 mixed-color pairs and 4 same-color pairs. Alternatively, you mightconsider only a sample of possibilities. For example, youmight focus on one token, and notice that most of the othertokens have a different color. In this way, you might con-clude that it is more likely to get a mixed-color pair. In-deed, for each given token there is only one possibility toget a token of the same color, but six possibilities to get atoken of a different color. Notice that such a conclusion isthe same as the one you might draw by considering all pos-sible pairs of tokens: There are six times more chances toget a mixed-color pair than a same-color one. Are childrenable to consider a variety of possibilities in order to predictwhether a given relation will hold between two randomlychosen tokens? The scant available evidence suggests anegative answer. According to Piaget (1950; Piaget & Inh-elder, 1951/1975), before adolescence, children lack thelogical abilities necessary to systematically combine possi-bilities. Indeed, Piaget and Inhelder reported that, given abag containing some chips (e.g., six white, five red, fourblue, three green, two red, one pink), children failed to an-swer the question ‘‘If you take out two chips at a time,without looking, what colors will you most likely get?’’.

Post-Piagetian studies showed the difficulty to teachchildren the basic combinatorial procedures, and how toapply them to the solution of probability problems (e.g.,Fischbein, 1975).

Do failures of this sort really demonstrate that childrenare unable to reason about the possibilities of a relation?Consider the above-described Piagetian task. Childrenhad to predict one specific outcome (i.e., the colors oftwo chips). To answer suitably, they had to consider thevarious possible outcomes (i.e., the twenty possible combi-nations of colors), and to count the possibilities favoringeach of them. Moreover, the suitable prediction consistedin indicating an outcome that was unlikely to occur, giventhat the most likely outcome (i.e., white and red) had lessthan a 15% chance to occur. These specific requirements ofthe Piagetian tasks might have hidden children’s ability tomake correct predictions when they have to consider arelation. Thus, the possibility that children can reason cor-rectly about a relation between simple properties remainsuntested.

To provide a proper test of children’s ability, in the re-ported studies we have used novel tasks. Consider, forexample, the four couple task described above. Unlike thetraditional Piagetian tasks, it asks to predict if a given rela-tion (i.e., same color) will occur. Moreover, the suitableprediction consists in identifying an event that has at leasta 50% chance of occurring. Thus, one does not need to as-sess the exact number and the proportion of the possibili-ties in which the relation holds. Considering a sample ofpossibilities suffices. The common principle of our tasksis to present children ranging from five to ten years ofage, a set of objects, and to ask them to predict whethertwo randomly chosen objects will be in a given relation.If children turn out to succeed, we will conclude that theyare capable of predicting the occurrence of a relation onthe basis of its possibilities.

2. Experiment 1

Experiment 1 examined the ability of children aged5–10 to reliably predict the same color relation betweentwo chips randomly chosen from a set containing chipsof various colors. We used two series of tasks. In one series,there were k chips of the same color and one chip of a dif-ferent color. Hence, out of a total of k(k + 1)/2 possibilities,there were k(k � 1)/2 possibilities to randomly choose twotwin-chips and k possibilities to randomly choose two dif-ferently colored chips. Therefore, in the tasks in which kwas greater than 3, the chances of choosing two twin-chipswere higher than those of choosing two differently coloredchips. Participants, however, could make a reliable predic-tion without making a precise enumeration of all the pos-sibilities. For example, they could notice that, when k wasgreater than 3, each chip having the predominant colorneighbored several – actually k � 1 – twin-chips, and onlyone differently colored chip. In the other series of tasks,there were chips of k colors, each color represented bytwo chips. Hence, out of the total of k(2k � 1) possibilities,there were k possibilities to randomly choose two twin-chips and 2k(k � 1) possibilities to randomly choose two

374 M. Gonzalez, V. Girotto / Cognition 120 (2011) 372–379

differently colored chips. Therefore, the chances of choos-ing two differently colored chips were higher than thoseof choosing two twin-chips. Participants, however, couldmake a correct prediction without making a precise enu-meration of the different color pairs. For example, theycould notice that each chip had just one twinned chip,and several – actually 2(k � 1) – differently colored chips.

2.1. Method

2.1.1. ParticipantsIn all experiments, the children were native Italian

speakers from northern Italian public kindergartens andschools. They were from middle class background. Signedconsent was obtained from parents. In Experiment 1, wetested 133 kindergarten, first, third and fifth grade chil-dren: 26 (seven girls) five-year-olds (mean years M = 5.8,standard deviation SD = 0.2); 38 (15 girls) six-year-olds(M = 6.7, SD = 0.3); 34 (18 girls) eight-year-olds (M = 8.8,SD = 0.3); 35 (14 girls) ten-year-olds (M = 10.7, SD = 0.3).We also tested 25 Italian undergraduates, between 19and 23 years of age.

2.1.2. MaterialEach task used a different set of chips. In three tasks,

one chip of a given color and, respectively, 3, 7 and 15chips of another color composed the set. We call thesetasks 3 vs. 1, 7 vs. 1, and 15 vs. 1, respectively. In threeother tasks, the set was composed of, respectively, 2, 4and 8 pairs of twin-chips. Each twin-pair represented a dif-ferent color. We call these tasks 2 by 2, 4 by 2, and 8 by 2,respectively.

2.1.3. ProcedureIn all experiments, each child was tested individually, in

a quiet room, and the average duration of the testing ses-sion was about 15 min. In Experiment 1, children youngerthan 8 years of age received a familiarization task. Theexperimenter presented children with four red and fourblue chips, and asked them to indicate two same coloredchips and then two differently colored chips. If childrenfailed, they were corrected. Then, all children performedthe six experimental tasks. In each task, the experimenterput a set of chips on a table, asked children to examine itand then put it in an opaque bag. The children’s task wasto draw two chips at the same time without looking insidethe bag. Before the children drew the chips, the experi-menter said: ‘‘Do you think you will draw two chips ofthe same color or two chips of two different colors? If youranswer turns out to be right, you’ll win a piece of candy.’’After the draw, the experimenter provided the reward tothe children whose bet turned out to be correct. In allexperiments, the order of presentation of the tasks wasvaried in such a way that each task was presented in allpossible ranks to the same number of children.

In all experiments, the adult participants tackled thesame tasks as children did, but were tested collectively.In Experiment 1, in each task, the experimenter showedan opaque bag containing some chips and then presentedits content in a picture projected on a large screen. Beforethe experimenter randomly drew two chips from the bag,

the participants had to provide a written answer to thequestion: ‘‘Do you think I will draw two chips of the samecolor or two chips of two different colors?’’ The partici-pants wrote their prediction for each task. At the end ofthe task sequence, the experimenter drew two chips fromeach bag. The participant with the highest number of cor-rect predictions received a 15-euro notepad.

2.2. Results and discussion

We scored the performance of each participant as afunction of the expected value of his or her answers. Ineach task, the probability of obtaining a reward equalsthe probability that the prediction is correct and, thus,the proportion of expected reward. In the tasks 15 vs. 1,7 vs. 1, and 3 vs. 1, the probability of obtaining two twin-chips is 6/7, 3/4 and 1/2, respectively. In the tasks 2 by 2, 4by 2, and 8 by 2, this probability is 1/3, 1/7 and 1/15,respectively. In order to optimize the expected number ofrewards, one should predict same color in the first twotasks, and different colors in the last three tasks. These pre-dictions yield an expected number of rewards equal to6/7 + 3/4 + 1/2 + 2/3 + 6/7 + 14/15 = 4.56. On this basis,one can attribute an expected value V to each individualpattern of prediction. By considering the 64 possible pat-terns of prediction, the distribution of V ranges between1.44 (Vmin) and 4.56 (Vmax), it is symmetrical, and its mean(Vmean) equals 3. Vmean is the mean expected value of ran-domly distributed predictions. We computed an index Qof the quality of a given prediction pattern by normalizingthe expected value V: Q = 2(V � Vmean)/(Vmax � Vmin). Qranges between �1 (worse quality, when V = Vmin),and + 1 (best quality, when V = Vmax), and is equal to 0when V = Vmean, that is, when V equals the expected valueof random predictions.

Apart from five-year-olds (mean Q = 0.12, standarddeviation SD = 0.51), children performed better thanchance, at least at p = 1 � 10�9 level, using one-samplet-tests of the hypothesis that the mean Q equals 0: six-year-olds (mean Q = 0.53, SD = 0.39), eight-year-olds(mean Q = 0.85, SD = 0.23), and ten-year-olds (meanQ = 0.88, SD = 0.24). In the last two groups, children’s per-formance reached adult levels (mean Q = 0.80, SD = 0.29).From the age of six, the rate of same color predictionsfollowed the same tendency as the probability of such anoutcome, as indicated by Fig. 1. Optimal answers consistedin predicting that the same color relation held when itsprobability was greater than 1/2, and that it did not holdwhen its probability was smaller than 1/2. From the ageof eight, children’s predictions were close to optimality(see Fig. 1).

In all experiments, to motivate children, we asked themto make proper bets, rather than disinterested judgments.Did the feedback they received after each prediction affecttheir performance? The sequence of tasks differed acrosschildren, but approximately the same number of childrenreceived a given task in each possible rank. Thus, we com-puted a separate Q for the first three and the last threetasks. In Experiment 1, for older children (eight- and ten-year-olds), the initial Q was similar to the final one (0.87vs. 0.86, respectively). For younger children (five- and

Fig. 1. Percentage of the same color predictions by task, in each age groupabove five years. The dotted line indicates the probability of the samecolor outcome.

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six-year-olds), the initial Q was lower than the final one(0.25 vs. 0.47, respectively), t(63) = 2.24, one-tailedp = 0.01. This result cannot be attributed to probabilitylearning, because each task was presented only once andthe probability of choosing two twin-chips differed acrosstasks. We attribute this result to a familiarization effect, inwhich younger children learned how to deal with a seriesof novel tasks. The same conclusion holds for the followingexperiments.

In sum, from the age of six years, children tended to cor-rectly predict the occurrence of the same color relation be-tween two tokens. From the age of eight, their predictionsreached adult levels.

3. Experiment 2

In Experiment 1, children might have considered justthe distribution of the colors in the set, rather than the col-or relation between pairs of chips. For example, based onthe fact that in the 15 vs. 1, 7 vs. 1, and 3 vs. 1 tasks therewas a predominant color, children might have inferred thatthe pair was likely to have this color. Likewise, based on

123456

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3 pieces 4 pieces 6 pieces

Fig. 2. Materials used in Experiments 2 and 3. The rectangles were cut in, resrepresented a portion of a triangle.

the fact that in the 4 by 2, and 8 by 2 tasks there was a vari-ety of colors, children might have inferred that the pair waslikely to have two different colors. To rule out this explana-tion, in Experiment 2 children had to consider a relationthat did not depend on the distribution of the propertiesof the tokens. Consider the triangles depicted in Fig. 2. Eachtriangle is drawn inside a rectangle. Since the rectanglesare divided in strips, one can obtain 3, 4, 6 or 10 piecesof triangle, respectively. Imagine that one takes randomlytwo pieces of a triangle.

Do you think that one will get two pieces that toucheach other or two pieces that do not touch each other?You might answer by enumerating all the possible pairsof pieces: Given a k-piece triangle, there are k � 1 pairsof contiguous pieces out of a total of k(k � 1)/2 possiblepairs. Therefore, the proportion of these pairs equals 2/k,and the probability of getting one of them equals 2/3, 1/2, 1/3 and 1/5 given a triangle of 3, 4, 6 and 10 pieces,respectively. To answer correctly, however, one does notneed to enumerate all the possible pairs of pieces. Like inthe chip tasks used in Experiment 1, an approximate com-parison based on the application of some simple heuristicsyields suitable predictions. For example, each piece has oneor at most two contiguous pieces, regardless of the numberof pieces in the triangle. By contrast, the higher the numberof pieces in a triangle, the higher the number of not contig-uous pieces. Therefore, the chances of getting two contigu-ous pieces decrease as a function of the total number ofpieces in the triangle. Now, unlike the chip tasks, one can-not solve the triangle tasks without taking into account therelation between two elements, because, unlike the color,the contiguity is not an inherent property of a single object.Hence, if children can solve the triangle tasks, we will con-clude that they are able to reason about the possibilities ofa relation. To this aim, in Experiment 2 five- to ten-year-olds had to make predictions about the contiguity relationbetween two pieces of a triangle.

3.1. Method

3.1.1. ParticipantsWe tested 306 kindergarten, first, second, third, fourth,

and fifth grade children: 31 (15 girls) five-year-olds(M = 5.6, SD = 0.3); 37 (16 girls) six-year-olds (M = 6.6,SD = 0.3); 64 (27 girls) seven-year-olds (M = 7.4, SD = 0.3);

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pectively, 3, 4, 6 and 10 equal-sized pieces of cardboard, each of which

376 M. Gonzalez, V. Girotto / Cognition 120 (2011) 372–379

57 (26 girls) eight-year-olds (M = 8.5, SD = 0.3); 50 (22girls) nine-year-olds (M = 9.6, SD = 0.3); 67 (28 girls) ten-year-olds (M = 10.5, SD = 0.3). We also tested another sam-ple of 37 undergraduates between 19 and 24 years of age.

3.1.2. MaterialThe tasks concerned, respectively 3, 4, 6 and 10 equal-

sized pieces of cardboard, each of which represented a por-tion of a triangle (see Fig. 2). The pieces measured about1 � 10 cm each in the 10-piece triangle, and 2 � 10 cmeach in all other triangles. In each triangle, the pieces werenumbered sequentially from the top of the triangle, begin-ning with the number 1. Each triangle was coloreddifferently.

3.1.3. ProcedureIn the familiarization task, the experimenter showed a

5-piece triangle, pointed out two contiguous pieces andtwo non-contiguous pieces and asked the children whetherthese pieces touched each other. Then the experimenterasked the children to indicate two pieces that touched eachother, and two pieces that did not touch each other. If thechildren failed, the experimenter corrected them. Then,children performed four experimental tasks. In each task,the experimenter showed a triangle, mixed its pieces up,and then put them in an opaque bag. Children’s task wasto draw two pieces at the same time, without looking in-side the bag. Before they drew the pieces, the experimentersaid, ‘‘Do you think that you will get two pieces that toucheach other or two pieces that do not touch each other? Ifyour answer turns out to be right, you’ll win a piece of can-

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Fig. 3. Percentage of the contiguous predictions (black lines) elicited in Experimlines indicate the percentage of bigger than predictions elicited in Experiment 3

dy.’’ After the draw, the experimenter provided the rewardto the children whose bet turned out to be correct. For theadult participants, the procedure was the same as in Exper-iment 1, except that they had to bet on whether the exper-imenter was going to draw two contiguous or two notcontiguous pieces.

3.2. Results and discussion

We discarded the answers of one eight-year-old andthree nine-year-old boys who did not follow the instruc-tions. As in Experiment 1, we computed the expected valueV. In the triangle tasks, V ranges between 1.37 (Vmin) and2.63 (Vmax), and its mean (Vmean) equals 2. Thus, we derivedthe normalized quality-index Q, that is, Q = 2(V � Vmean)/(Vmax � Vmin), which ranges between �1 and +1.

Five-year-olds did not perform better than chance (meanQ = 0.06). The performance of six-year-olds (mean Q = 0.18,SD = 0.67) tended to depart from chance, t(36) = 1.60, p = .06(one-tailed). From the age of 7 years, children performedbetter than chance at least at p = 1 � 10�6 level: seven-year-olds (mean Q = 0.39, SD = 0.57), eight-year-olds (meanQ = 0.63, SD = 0.52), nine-year-olds (mean Q = 0.64,SD = 0.55), and ten-year-olds (mean Q = 0.67, SD = 0.50). Inthe last three groups, children’s performance reached adultlevels (mean Q = 0.68, SD = 0.58). From the age of seven, therate of predictions on two contiguous pieces followed thesame tendency as the probability of such an outcome, asindicated by Fig. 3.

In sum, from the age of about seven, children make suit-able predictions of the occurrence of a relation that did not

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ents 2 (circle mark) and 3 (square mark), by task and age group. The grey.

M. Gonzalez, V. Girotto / Cognition 120 (2011) 372–379 377

depend on the distribution of a simple property among thetokens.

1 Various studies suggest a logarithmic relation between a quantity andits perception, in accordance with Weber’s law (e.g., Halberda & Feigenson,2008).

4. Experiment 3

Did children actually consider the contiguity relationwhen they tackled the tasks of Experiment 2, or did theysimply apply some superficial, non-combinatorial heuris-tic, like using a global feature of the set of pieces, withoutconsidering the relation? As indicated, the probability thattwo pieces are contiguous is inversely proportional to thenumber of pieces of the triangle. One might argue that inExperiment 2 children did not answer by taking into con-sideration the relation of contiguity between each pair ofpieces, but by simply considering the total number ofpieces in the set. If this interpretation is correct, it followsthat children are not able to distinguish different sorts ofrelations, and will always base their predictions on thenumber of pieces. To test this interpretation, in Experiment3 children had to consider two different relations. Theexperimenter randomly chose a piece of a triangle and,without showing it, asked children to choose anotherpiece. Before children actually chose their piece, they hadto predict whether it would touch the experimenter’sone, as well as whether it would be bigger or smaller thanthe latter. The chances that the children’s piece touches theexperimenter’s one are inversely proportional to the totalnumber of pieces. By contrast, the probability that the chil-dren’s piece is bigger than the experimenter’s does not de-pend on the number of the pieces: it is always 50%. Ifchildren do not actually consider the nature of the relation,the contiguity and the size relations should elicit the samesort of answers. By contrast, if children consider the spe-cific possibilities of a relation, their contiguity answersshould differ from their size answers. To test these predic-tions, in Experiment 3 six- to ten-year-olds received thesame triangles as in Experiment 2, but had to make predic-tions about both the contiguity and the size relations.

4.1. Method

4.1.1. ParticipantsWe tested 194 first, second, third, fourth, and fifth grade

children: 44 (17 girls) six-year-olds (M = 6.6, SD = 0.3); 45(22 girls) seven-year-olds (M = 7.5, SD = 0.3); 29 (15 girls)eight-year-olds (M = 8.5, SD = 0.3); 33 (14 girls) nine-year-olds (M = 9.5, SD = 0.3); 43 (23 girls) ten-year-olds(M = 10.6, SD = 0.4). We also tested another sample of 32undergraduates between 21 and 27 years of age.

4.1.2. Material and procedureThe material was the same as in Experiment 2 (see

Fig. 2). In the familiarization task, the experimentershowed a 5-piece triangle, and asked the children whethertwo given pieces touched each other, and whether one ofthem was bigger than the other one. Then the experi-menter asked the children to indicate two pieces thattouched each other, and one piece that was bigger than an-other one. If the children failed, the experimenter correctedthem. Then, all children performed four experimental

tasks. In each task, the experimenter randomly chose onepiece of a triangle and, without showing it to the child,said, ‘‘Now, you’ll draw another piece, without looking in-side. Do you think that you will get one piece that willtouch or will not touch my piece? And do you think thatyou will get one piece that will be bigger or smaller thanmy piece? For each answer that turns out to be right, you’llwin a piece of candy.’’ After the draw, the experimenterprovided the reward to the children whose predictionsturned out to be correct. For half the children, the first pre-diction concerned the contiguity relation and the secondone the size relation. For the other half, the reverse held.For the adult participants, the procedure was the same asin Experiment 2, except that they had to predict whetheror not they were going to draw a piece that touched theexperimenter’s one, and whether or not they were goingto draw a piece that was bigger than the experimenter’sone.

4.2. Results and discussion

We scored the predictions about the contiguity relationby computing a quality-index Q, as in Experiment 2. Thequality of six-year-olds’ predictions (mean Q = 0.16,SD = 0.64) tended to be above chance level (one-tailedp = 0.06). From the age of 7 years, children performed bet-ter than chance at least at p = 1 � 10–3 level: seven-year-olds (mean Q = 0.30, SD = 0.57), eight-year-olds (meanQ = 0.48, SD = 0.64), nine-year-olds (mean Q = 0.67,SD = 0.45), and ten-year-olds (mean Q = 0.69, SD = 0.43).In the last two groups, children’s performance reachedadult levels (mean Q = 0.70, SD = 0.54). From the age of se-ven, the rate of predictions on two contiguous pieces fol-lowed the same tendency as the probability of such anoutcome, as indicated by Fig. 3.

Suppose that children took into account only the mostsalient characteristic of the material, that is, the numberof pieces of the rectangle. It follows that their contiguityand size predictions should both depend on this variable.The results rebut such a prediction. For each participant,we computed the point-biserial correlation between thelogarithm of the number of pieces of the triangle and chil-dren’s predictions1. For the contiguity predictions, the cor-relation diminished regularly with age, ranging from �0.10to �0.57. These correlations were negative because contigu-ity predictions decreased when the number of pieces in-creased. For the bigger than predictions, the correlation didnot change with age: it ranged from �0.03 to 0.16, remain-ing always close to 0. At all ages, the correlations betweennumber of pieces and contiguity prediction turned out tobe smaller than those between number of pieces and sizeprediction: mean difference ranging from 0.21 to 72, one-tailed p < 0.05. These findings indicate that contiguity andsize predictions largely diverged, as shown by Fig. 3.

The results of Experiment 3 confirmed and extendedthose obtained in Experiment 2: from the age of about se-ven, children make correct predictions of the occurrence of

378 M. Gonzalez, V. Girotto / Cognition 120 (2011) 372–379

a contiguity relation. From the age of about nine, their per-formance reaches adult levels. Finally, they consider thenature of the relation, by making different predictionsabout the contiguity and the size relations.

5. General discussion

Our results provide the first evidence that, from the ageof about six, children suitably expect the occurrence of arelation between two events, by considering their possibleproperties. They do so when they have to consider the pos-sibility that two randomly chosen tokens have the samecolor (Experiment 1) or that two pieces of a drawing toucheach other (Experiment 2). Children’s predictions agreewith those prescribed by combinatorics, cannot be attrib-uted to the application of shallow heuristics (Experiment3), and improve with age (reaching adult levels as earlyas the age of 9).

These results conflict with the claim that children can-not deal with tasks that ask for a combinatorial treatmentof possibilities. Indeed, previous studies appeared to showthat children lack the logical abilities necessary to enumer-ate combinations of possibilities in a systematic way (Pia-get & Inhelder, 1951/1975), and are even unable to learnhow to do it (Fischbein, 1975). Why did our results divergefrom the previously reported ones? Unlike the Piagetiantests, our tasks did not require children to count the possi-ble outcomes of a given process, but simply to predictwhether a given relation would hold between two tokens.Thus, an approximate comparison, rather than a preciseassessment of the proportion of possibilities in which therelation held, sufficed to produce a reliable prediction.For example, children could consider a small sample ofpossibilities: They could focus on any token of the initialset (a chip, a piece of triangle) and consider its relationwith the other ones. In the case of the chips, this heuristicallows one to easily estimate whether most of the otherchips have the same color or a different one. Likewise, inthe case of the pieces of a triangle, it allows one to easilyestimate whether among the other pieces there are morepieces that touch or do not touch the focal piece. In manycases, this sort of simplified treatment of possibilitiesyields a correct prediction.

Of course, there are cases in which a sampling processdoes not guarantee reliability. Indeed, even adults some-times fail to predict the occurrence of a relation, especiallyin problems that elicit a biased representation of the possi-bilities. For example, in Experiment 1, the task 3 vs. 1 pre-sented three chips of the same color and one chip of adifferent color. In this case, the chances of getting twotwin-chips are the same as those of getting two chips oftwo different colors. However, in all age groups, includingthe adult one, a majority of participants predicted theoccurrence of two twin-chips. We attribute this result toa tendency to focus on one of the chips of the predominantcolor. Each of these chips neighbors two chips of the samecolor, and only one chip of a different color. Therefore, it istempting to conclude that it is more likely to get two twin-chips than two chips of different colors. Consider anothersituation where one throws two regular dice whose faces

are numbered from 1 to 6. Which event is more likely: get-ting 6–6 or getting 6–5? You might focus on the possibilitythat one die lands on 6. Accordingly, you might infer thatthe other die has the same chances of landing on 5 or 6,and conclude that the chances of getting 6–6 are the sameas those of getting 6–5. Indeed, Fischbein and Schnarch(1997) found that children and adolescents whose agesranged from 10 to 17 years, as well as college studentswho were prospective teachers specializing in mathemat-ics, judged that these events were equiprobable. This judg-ment, however, is incorrect. There is only one possibility toobtain two 6s, but two possibilities to obtain a 6 and a 5,given that each of the two dice can land on 6 or 5. The sameerror can be found even in the work of eminent mathema-ticians, such as Leibniz. In a letter dated 1714, he wrote:‘‘. . . with two dice, it is as doable to throw a twelve as tothrow an eleven for each can only be done in one way’’(quoted by Dudley Sylla, 1998, p. 50). In sum, in somecases, individuals produce incorrect judgments becausethey disregard possibilities necessary for a proper exten-sional treatment. Despite its limits, however, such a treat-ment remains a basic and early acquired way to reasonabout uncertain events.

In conclusion, our results make three contributions tothe understanding of naive probabilistic reasoning. First,they show that children’s probabilistic competence is largerthan what previously thought. Since the Sixties, evidenceexisted that children have correct intuitions about theprobability of events characterized by a simple property(e.g., Davies, 1965; Goldberg, 1966; Yost, Siegel, & Andrews,1962). Recent studies have shown that they also have a cor-rect intuition of the basic principles that link prior and pos-terior probability (Girotto & Gonzalez, 2008). The currentresults show that children are even able to make correctpredictions about combinations of properties. The abilityto make such predictions appears to emerge at about sixyears, whereas the ability to predict events defined by asimple property appears to emerge at about four years(e.g., Goldberg, 1966; Teglas et al., 2007, Studies 3 and 4).However, contrary to Fischbein’s claim (1975), the abilityto predict combinations of properties does not depend onformal instruction, given that children exhibit it beforereceiving any training in probability and combinatorics.Historical evidence further corroborates such a conclusion.Before the emergence of a norm for measuring chance inthe seventeenth century, medieval authors understood thatthe expectation of a given sum of points in a throw of dicedepended on the number of ways in which this sum couldoccur (e.g., Franklin, 2001; Girotto & Gonzalez, 2005). Sec-ond, contrary to the thesis that the human mind can onlymake predictions on the basis of previously encounteredevents (e.g., Cosmides & Tooby, 1996), we showed that chil-dren make suitable predictions that do not depend on pre-vious experience: For example, they predict whether tworandomly chosen pieces of a triangle will be contiguouswithout a previous observation of the occurrence of theseoutcomes. A mental sampling of the possibilities thatinstantiate and do not instantiate the relation suffices tomake a correct judgment. Finally, our results extend theextensional theory of probabilistic reasoning by showingthat naive individuals do not need to enumerate the various

M. Gonzalez, V. Girotto / Cognition 120 (2011) 372–379 379

possibilities in order to infer which is the more likely out-come. In many situations, their ability to compare possibil-ities in an approximate way leads them to make suitablepredictions and optimal choices.

Acknowledgements

We thank two anonymous referees for their comments,as well as Ilaria Camozzo, Paolo Cappelletto and StefaniaPighin for their help in data collection, and the childrenand the schools for their collaboration. Portions of this re-search were presented at the Workshop Probabilistic Mod-els of Cognitive Development, Banff International ResearchStation, Canada, May 2009.

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