E F R A I M F I S C H B E I N , M A R I A SAINATI NELLO AND

M A R I A SCIOLIS M A R I N O

F A C T O R S A F F E C T I N G P R O B A B I L I S T I C J U D G E M E N T S

IN C H I L D R E N A N D A D O L E S C E N T S

ABSTRACT. Six hundred and eighteen pupils, enrolled in elementary and junior-high-school classes (Pisa, Italy) were asked to solve a number of probability problems. The main aim of the investigation has been to obtain a better understanding of the origins and nature of some probabilistic intuitive obstacles. A linguistic factor has been identified: It appears that for many children, the concept of "certain events" is more difficult to comprehend than that of "possible events". It has been found that even adolescents have difficulties in detaching the mathematical structure from the practical embodiment of the stochastic situation. In problems where numbers intervene, the magnitude of the numbers considered has an effect on their probability: bigger numbers are more likely to be obtained than smaller ones. Many children seem to be unable to solve probability questions, because of their inability to consider the rational structure of a hazard situation: "chance" is, by itself, an equalizing factor of probabilities. Positive intuitive capacities have also been identified: some problems referring to compound events are better solved when addressed in a general form than when addressed in a particular way.

I N T R O D U C T I O N

Various studies have been published concerning the developmental aspects of probabilistic thinking, starting especially with the book of Piaget and Inhelder: La Genbse de l'Idbe de Hasard chez l'Enfant (1951) (see for extensive reviews: Fischbein, 1975; Hawkins and Kapadia, 1984; Godino, Batanero and Canizares, 1987; Garfield and Ahlgren, 1988). As for miscon- ceptions in statistical and probabilistic reasoning the main work is that edited by Kahneman, Slovic and Tversky (1982).

A central issue revealed by most of the studies is the relationship between the natural, intuitive approaches individuals' hold with regard to proba- bilistic situations and the formal, mathematically based solutions. Various aspects have already been investigated like the so called heuristics men- tioned by Tversky and Kahneman (availability, representativeness and anchoring), (see Kahneman, Slovic and Tversky, 1982), the negative re- cency effect, (see Fischbein, 1975, pp. 45-48), the notions of certain, possible and impossible events (Fischbein and Gazit, 1984), compound events (Lecoutre and Durand, 1988) etc.

Nevertheless, many aspects still remain obscure either because they have not yet been studied or because of the limited character of the instruments used. The present investigation has been initiated bearing in mind, as its main objective, the preparation of curricula materials for teaching proba-

Educational Studies in Mathematics 22: 523-549, 1991. �9 1991 Kluwer Academic Publishers. Printed in the Netherlands.

524 EFRAIM FISCHBEIN ET AL.

bilities in elementary and junior high school classes in Italy. Our strong belief is that the introduction of a new topic must always be preceded by a systematic psycho-didactical investigation. This is true for mathematics education in general but it is especially true for probabilities. The cultural environment, the ensemble of existing curricula concerning other domains, the socio-economic level of the population, the philosophy behind the didactical methodology etc., may have a certain impact on the children's receptivity for the respective topic. This assertion is particularly true for probabilities.

Just one example, mentioned to one .of us by a colleague. One of his doctoral students came from a country in which 'chance' games were very common. This student was averse to the idea that chance events can be expressed mathematically. In order to formalize a stochastic experience, one has to consider only the mathematical, ideal structure of a chain of events and to eliminate the influence which some concrete irregularities may have. But this student was deeply convinced that, in a chance game, it is impossible to disregard the personal ability of the gambler. This personal ability is the core of the game. The idea of pure chance had no meaning for this student.

The present investigation dealt with a number of concepts which we have considered to be adequate for an introductory course on probabilities. But for the present paper we have selected only those topics and findings which, in our opinion, presented new aspects and which could stimulate new interpretations referring to the intuitive background of probabilistic think- ing. The main topics considered were: Types of events (impossible, possible and certain events); the role of different embodiments of the same mathe- matical structure; compound events. In most of the problems the subjects had to consider the sample space related to a certain event. Therefore, this may be considered the central notion to which we refer in the present paper.

THE METHOD

The Subjects

The subjects were 618 pupils enrolled in six schools in the region of Pisa, Italy. They represented three groups of subjects: 211 subjects in elementary classes (grades 4 and 5), 278 subjects in junior-high school classes (grades 1, 2 and 3 without prior instruction in probabilities) and 130 subjects (grades 1, 2 and 3) with prior instruction in probabilities. We do not have

FACTORS AFFECTING PROBABILISTIC JUDGEMENT 525

a clear image of the nature of this instruction mainly because in Italy there is not an established tradition of teaching probability. Elementary school, grades 4-5, correspond to 9-11 year old pupils and junior high school, grades 1, 2, 3, correspond to 11-14 year old pupils.

The Questionnaires

Two questionnaires (A and B) were used, initially consisting of 14 ques- tions each but in the present paper only the reaction to 6 pairs of questions are analysed. The items in the two questionnaires were parallel, that is they addressed the same type of probability problems but with different embod- iments. Every equestion required the subject to explain his answer. In the present report we have eliminated those aspects which have already been studied elsewhere and which did not offer essentially new insights.

The Procedure

The two questionnaires were administered simultaneously in the usual classroom setting of the subjects. Each subject had to answer the questions of only one questionnaire. A session lasted about 1 hour but there was no time pressure put on the subjects. In the two questionnaires, the order of the questions was randomized so as to balance the effect of order. The symbols A and B refer to the questionnaires. The number attached to each letter in the present text (for instance A1) does not refer to the real order in which the questions were presented but only to their order in the present paper.

RESULTS

The results are exposed and analyzed successively for each pair of parallel questions. The terms "correct" and "incorrect" are sometimes used in the text. We are aware that such a distinction is not an absolute one. Children, as well as adults, possess representations and interpretations which have to be respected in their own right, which have their reasons, and may be adequate in certain situations. But in order to avoid the terms "correct" and "incorrect" we would have to often repeat long explanations which would lengthen the paper and make reading more difficult. "Correct" means simply what is usually accepted in a standard probability text-book.

On the other hand, one has to take into account that, sometimes, children give apparently "correct" answers for wrong reasons. The paper

526 EFRAIM FISCHBEIN ET AL.

copes wi th this difficulty by present ing the ma in types o f jus t i f icat ions the

subjects offer for their answers.

Questions AI and BI: Impossible, Possible and Certain Events

In ques t ion A1, we have cons idered the ro lhng o f a die and the subjects

were asked to indicate whe ther the event o f ob t a in ing a cer ta in n u m b e r is

imposs ib le , poss ib le o r cer tain. The events cons idered were: (a ) A n even

number ; (b) a n u m b e r smal ler t han 7; (c) a n u m b e r b igger than 6; (d) a

n u m b e r b igger than 0; (e) the n u m b e r 5.

In ques t ion B1, one has cons idered the t o m b o l a game with numbers

f rom 1 to 90 and, as above, the subjects had to indicate whe ther the events

o f ob ta in ing a cer ta in n u m b e r was imposs ib le , poss ible o r cer tain. The

number s men t ioned were: (a) A n o d d number ; (b) a n u m b e r smal ler t han

91; (c) 100; (d) a n u m b e r b igger than 0; (e) 31.

L o o k i n g a t the d a t a (see Tables I A and IB) the fo l lowing remarks can be

made : A t bo th age levels the ma jo r i t y o f pupi l s ident i fy adequa te ly possi-

ble, imposs ib le and cer ta in events. The mean ing o f these te rms are the

fo l lowing in our f ramework . "Poss ib le : 0 < P ( E ) < 1 ; " Imposs ib l e " :

P(E) = 0; " C e r t a i n " : P(E) = 1. There is a sl ight i m p r o v e m e n t wi th age at

all i tems. I t seems that , for mos t i tems, ins t ruc t ion had a posi t ive effect, bu t

this effect is sl ight and ra the r inconsis tent . The most surprising result is that

TABLE IA

Possible, impossible and certain events. The symbol (P) stands for: "With previous probability instruction". Percentages of various categories of answers

Elementary (N = 102) Junior high (N = 139) Junior high (P) (N = 65)

Items a b c d e a b c d e a b c d e

The expected answer Ps C IMP C Ps Ps C IMP C Ps Ps C IMP C Ps

No answer 2.0 2,0 2.0 2.0 2.9 0.7 0.7 1.4 1.4 1.4 0 1.5 0 1.5 0

Correct 89.2 62.7 82.3 81.4 86.3 95.0 71.2 84,2 89.9 88,5 96.9 84.6 98,5 92.3 98.5

Incorrect 8.8 35.3 15.7 16.6 11.7 4.3 28.1 14,4 8.7 I0.I 3.1 13.9 1.5 6.2 1.5

Ps - Possible; C - Certain; IMP - Impossible. The terms "correct" and "incorrect" have only a relative meaning. See the text for explana- tion.

FACTORS AFFECTING PROBABILISTIC JUDGEMENT 527

TABLE IB

Possible, impossible and certain events. Percentages of various categories of answers

Elementary (N = 109) Junior high (N = 139) Junior high (P) (N = 65)

Items a b c d e a b c d e a b c d e

The expected answer Ps C IMP C Ps Ps C IMP C Ps Ps C IMP C Ps

No answer 4.6 2.8 2.8 2.8 3.7 1.4 2.2 1.4 2.2 2.9 1.5 3.1 3.1 3.1 3.1

Correct 79.8 55.0 94.4 69.7 70.6 85.6 63.3 93.5 77.7 78.4 92.3 63.1 92.3 76.9 86.1

Incorrect 15.6 42.2 2.8 27.5 25.7 13.0 34.5 5.1 20.1 18.7 6.2 33.8 4.6 20.0 10.8

Ps - Possible; C - Certain; IMP - Impossible.

the category of questions which yielded the lowest rates of adequate answers according to the above definition is that referring to certain events. T o a

ques t ion like: "By rol l ing a die, does one ob t a in a n u m b e r smal ler than 77",

on ly 63% o f j u n i o r high school subjects who got an e l emen ta ry ins t ruc t ion

in probabi l i t i es , answered adequate ly : "The event is ce r ta in" .

W e are used to th ink ing tha t the concep t o f "pos s ib l e " is the mos t sophis-

t ica ted one and that , for younge r chi ldren, it is the logic o f "yes" and " n o "

which is the only one tha t makes sense. I t seems tha t the concep t o f cer t i tude

is much m o r e complex than expected, while tha t o f poss ible develops earher!

O u r exp lana t ion is that , usual ly , one tends to re la te the no t ion o f

" ce r t a in " to tha t o f "un iqueness" . As an effect o f i n t roduc ing the idea o f

mul t ip l ic i ty o f poss ib le results , the no t ion o f possible comes na tu ra l ly in to

mind. But there is a cer ta in nuance which has to be t aken into account .

W h e n asking: "Wi l l one get a n u m b e r b igger than 0?" the percentages o f

answers " the event is ce r ta in" a re much higher (a lways higher than 80

percent) . In this case too, a cer ta in event is assoc ia ted with a mul t ip l ic i ty o f

poss ib le ou tcomes . The on ly exp lana t ion we have found is tha t "b igge r

than 0'" appea r s to the child as a totality - a k ind o f uni ty. One does no t

need to envisage a reper to i re o f possibi l i t ies (which wou ld lead to the idea

o f mult ipl ic i ty) . The " t o t a l i t y " is suggested, direct ly, without any analysis.

Justifications

The fo l lowing symbol i sm is used for ind ica t ing the grade to which the

quo t ed subjects belong: E means " e l e m e n t a r y " classes (4 o r 5). F o r

528 EFRAIM FISCHBEIN ET AL.

instance E4 means Grade 4, elementary school. M means "junior-high school" followed by 1, 2 or 3 indicating the respective grade. MP represents "junior-high-school pupils" who previously received some instruction in probabilities. The justifications given by the children seem to confirm the above explanations:

"Will one obtain a number smaller than 7?"

"I t is possible", says a child, "because it is probable that one may obtain also the number Y' (E4). That is, every number (between one and six) may be obtained. The certain totality is decomposed in the child's mind into a number of possibilities.

The same idea, but expressed differently: "I t is possible, because the numbers begin with 1 and end with 6" (E5), or "I t is possible because with a die one may obtain the 6 or 5 or any other number smaller than 7" (E5).

With regard to the question referring to a number smaller than 91 (the tombola problem): "I t is possible because, among all the numbers, one may obtain 87, 76, 7 4 . . . " (E4).

In fact the whole problem of the concepts "possible, impossible, certain" is psychologically more complex. Many children identify "rare" with impossible. "Will one obtain 31?" (in the tombola game). "I t is impossi- ble", answers the child, "because the probability is very small" (E4). Or: "I t is impossible, because among 90 numbers there is only one 31" (M1). "Is it possible to get 5 when rolling a die? . . . . It is impossible", says the child "because it is only one probability among 6" (M3). "I t is impossible because someone believes that he will win and afterwards it does not come out" (M2). Impossible identified with uncertain: "Will one obtain an even number? . . . . It is impossible because one cannot be sure" (E4).

We have seen, so far, that some of the children tend to identify "certain" with "possible" but the reverse may also occur. "Will one obtain the 31?" "I t is certain because in the tombola there is 31" (E5). "Will one obtain an even number? . . . . It is certain because from 1 to 90 there are even and uneven numbers" (M2). Children also tend to substitute mathematical meanings with subjective expectations: " I f an event is rare, according to my experience, it will not occur", affirm some children. On the other hand, " I f (for instance) 31 may occur then it should occur to somebody, why should it not be myself?." (M3).

The main didactical recommendation, following from the above findings, is that it should not be taken for granted that children understand spontaneously the meaning of the terms "impossible", "possible" and "certain". Children have to be trained to distinguish between rare and

FACTORS AFFECTING PROBABILISTIC JUDGEMENT 529

impossible, and between highly frequent and certain. They have to estimate the likelihood of an event by considering the joint outcomes which com- pose the respective event, and to avoid the spontaneous tendency to consider separately each outcome. Trivial examples like: " I t is possible that it will rain tomorrow" or " I t is certain that the sun will rise tomorrow" are not of much help. The problem is psychologically too complex to be solved didactically by referring to simplistic instances.

Questions A2 and B2: Different Situations, the Same Mathematical (Stochastic) Structure

The subjects were asked to compare the probability of obtaining three times 5 either by rolling one die three times or by rolling, simultaneously, three dice. The aim of this question was to check whether the subjects were able to extract the identical mathematical probabilistic structure from the two different embodiments (see the results in Table IIA).

Two main types of "unequal probabilities" are mentioned in Table IIA. Type I: The subjects consider that, by successively throwing the die, they have a higher chance to obtain the expected result. Type II: The subjects consider that by throwing three dice simultaneously they will have a higher chance to obtain the expected results. The percentages of the two types are calculated only by referring to the total number of "unequal" answers. The same explanation holds also for Table IIB (tossing three coins successively or simultaneously).

As one can see, only half of the younger subjects consider that the two procedures lead to the same probability. There is an improvement with age

Throwing 3 dice simultaneously TABLE IIA

or successively. (Percentages) For the two types of unequal probabilities see the text.

Elementary Junior high Junior high (N = 102) (N = 139) (P) (N = 65)

No answer 11.8 16.6 16.9 Probabilities are equal 50.0 62.6 70.8 Probabilities are different 38.2 20.9 12.3

Type I "unequal" 59.0 65.5 50.0 Type II "unequal" 33.3 20.7 37.5

530 E F R A I M F I S C H B E I N ET AL.

TABLE IIB

Tossing 3 dice simultaneously or successively. (Percentages) For the two types of unequal probabilities see the text.

Elementary Junior high Junior high (N = 109) (N = 139) (P) (N = 65)

No answer 7.4 12.2 3.1 Equal probabilities 55.0 57.6 76.9 Different probabilities 37.6 30.2 20.0

Type I unequal 58.5 66.7 69.2 Type II unequal 39.0 23.8 30.8

(about 60 percent of correct answers in junior high school) and with instruction (about 70 percent of correct answers in children who received some instruction on probabilities). The answers claiming that the chances are different are distributed in an unequal manner. About twice as many answers consist in affirming that those who throw a die three times successively have a higher chance of obtaining the sequence 5, 5, 5.

A similar question (questionnaire B) referred to the tossing of a coin: The probability of obtaining three times "head" (H, H, H) either by tossing a coin three times or by tossing three coins simultaneously. The distribution of the answers (see Table IIB) is about the same as with the dice version. Here, too, the predominant "unequal" answers express the idea that the expected outcome (H, H, H) has a higher chance when tossing one coin three times successively.

The subjects gave different explanations for their preference for succes- sive trials. "By roiling one die at a time, one may use the same type of rolling" (E4). "This way (tossing one coin at a time) the coin always falls on the same side" (E5). "Because the coins do not knock against each other and therefore do not follow diverse paths" (M1). As a matter of fact, the explanations express the same essential idea: by throwing a die (or a coin) one at a time, the process can be better controlled and a desired regularity can be better achieved.

The opposite solution (less frequent) also has its justifications: It is better to throw the coins (or the dice) together in order to obtain the same outcomes: "Because the same force is imparted" (E5). "One can launch in the same way" (M1).

The general idea is then, that the outcomes can be controlled by the individual. The mathematical, probabilistic structure has not yet been

FACTORS AFFECTING PROBABILISTIC JUDGEMENT 531

detached from the concrete circumstances and considered in its abstract generality.

Let us mention some examples of justifications for answers mentioning the equality of odds:

�9 "3 (coins) x one time ( "una volta") = 3 (times) one coin" (M3). �9 "I t is not important whether I toss the coins in different moments

or in the same moment, in both cases there is the same probability" (MP3).

It is important to use the type of questions mentioned here in the instruction process. From our results one may conclude that there are many children, even in junior high school classes, who do not detect the identical mathematical structure in practically different situations. What the child does not understand is that the situations have to be considered as mere embodiments for ideal experiments, that mathematics deals with ideal, abstract operations and entities. To teach the child that probability is a branch of mathematics and, consequently, it has to do with abstract ideal, formally defined objects - like geometry or arithmetic - is one of the main tasks of the instruction process. Comparing different embodiments of identical mathematical structures is an effective procedure for attaining that aim, not only formally, but also intuitively. The affirmation that mathemat- ics deals with abstract, ideal, formally defined objects does not imply that mathematics deals with objective, eternally fixed entities. Mathematics has an historical character and mathematical concepts are, in fact, psychologi- cally conditioned. Definitions and interpretations change. But it is generally accepted today that every mathematical theory is based on a system of axioms explicitly formulated. Axioms may be replaced with the condition that they will not lead to contradictions. As mentioned above, the terms "correct" and "incorrect" used sometimes in the present text have only a relative character. As a matter of fact, we see the process of learning and understanding as a reciprocal process of communication between a referent situation and the mathematical structure. "Any theory", says Feller, "nec- essarily involves idealization, and our first idealization concerns the possi- ble outcomes of an 'experiment' or 'observation'. If we want to construct a model, we must at the outset reach a decision about what constitutes a possible outcome of the idealized experiment" (Feller, 1960, p. 8). For example: "When a coin is tossed, it does not necessarily fall head, or tail: it can roll away or stand on its edge. Nevertheless, we shall agree to regard 'head' and 'tail' as the only possible outcomes of the experiment" (Feller, op. cit., p. 7).

532 EFRAIM FISCHBEIN ET AL.

The child who learns about probabilities has to get used to this type of idealization process, governed by formal conventions as it happens in other branches of mathematics, and first of all in geometry. The terms "correct" or "incorrect" used sometimes in the present text have their meaning established with regard to such commonly accepted conventions.

Compound Events (I)

According to previous findings, the concept of compound events seems to raise some difficulties (see Fischbein, Barbat and Minzat, 1971; Fischbein and Gazit, 1984; Lecoutre and Durand, 1988). We decided to consider again this problem systematically, raising dice and coin questions. So far, the main concern had been with the role of order in generating various outcomes accounting for the same event. We decided to consider again these types of problems but also to extend the range of the questions asked in the following ways: (a) We studied compound events which may be obtained by summing up different pairs of numbers; (b) We studied compound events including outcomes in a general form (for instance, the probability of obtaining a couple of equal numbers by throwing two dice vs. a couple in which the numbers are different) without specifying the numbers; (c) We studied the justifications offered by the subjects for their answers and thus we obtained a much deeper insight into the subject's internal models.

Questions A3 and B3

"Let us consider the rolling of two dice. Is it more likely to obtain 5 with one die and 6 with the other, or 6 with both dice? Or is the probability the same in both cases?"

"When tossing two coins which result is more likely: to get 'head' with one coin and 'tail' with the other, or to get 'head' with each of the two coins; or is the probability the same for both results?"

One might assume that some children have considered the pairs as ordered pairs and then the correct answer would be that the couples (5, 6) and (6, 6) are equiprobable. This is also true for the couples (H, T) and (H, H). But this was never the case. Analyzing the justifications, it became clear that no order was considered and that the very frequent "equiprob- able" type of answer was mostly justified by the effect of chance or by considering separately the two elements 5 and 6 or H and T.

F A C T O R S A F F E C T I N G P R O B A B I L I S T I C J U D G E M E N T 533

The first of the two problems has also been used by Lecoutre and Durand (1988). They discovered that most of the subjects answered, incorrectly, that the two events have the same probability and that this bias is very resistant. Various attempts to change it did not have a significant effect (for instance, coloring the two dice differently). Lecoutre and Durand identified a number of "models" used by the subjects to justify their answers. In the case of the answer "the two probabilities are different" Lecoutre has found justifications like: "I t is more seldom to obtain twice the same result", but also a correct combinational type of justification ("Because there are more possibilities with the other combinations than with the doubles") (Lecoutre and Durand, op. cit., p. 364).

With regard to the answers of the type "the same probability" Lecoutre and Durand also mention some patterns of justifications: "I t is a hazard guess in both cases", "The dice are thrown at the same time" or "The two results (5 or 6), have the same chance" (ibid.). Let us now come to our findings.

TABLE IIIA

Comparison between P(6, 6) vs. P(5, 6 and 6, 5). (Percentages) For the two types of miscon- ceptions see the text.)

Elementary Junior high Junior high (P)

No answer 14.7 18.7 12.3 P(6, 6) # P(5, 6 and 6, 5) 22.6 18.7 9.2 Misconceptions 62.7 62.6 78.5

Type I errors 73.4 89.6 92.2 Type II errors 12.5 4.6 5.9

TABLE IIIB

Comparison between P(H, H) vs. P(H, T and T, H). (Percentages) For the two types of misconceptions see the text.

Elementary Junior high Junior high (P)

No answer 13.8 16.5 20.0 P(H, H) # P(H, T and T,H) 21.1 10.I 18.5 Misconceptions 65.1 73.4 61.5

Type I errors 63.3 73.4 87.5 Type II errors 5.6 2.9 2.5

Two types of misconceptions are mentioned in Tables I l iA and IIIB.

534 EFRAIM FISCHBEIN ET AL.

In Table IIIA - Type I means: same probability for obtaining the couple (6, 6) or the couple (5, 6). Type II: The couple (6, 6) is more likely. For Table IIIB - Type I: The couples HT and H H are equiprobable. Type II: It is more likely to get the couple HH. The percentages of the two types of misconceptions are calculated on the basis of the total number of errors.

Inspecting Tables IIIA and IIIB one can clearly see that only a small proportion of answers indicate that the probabilities are different. We deal here with a well-known misconception. The two outcomes (either 6, 6 and 5, 6 or TH and HH) are considered equivalent at all age levels by most of the subjects without, as we said above, implying a given order. There is also no improvement with instruction. On the contrary, there are less correct answers in older children who received a certain instruc- tion than in younger children who did not receive any instruction in probabilities!

Our first explanation of this finding was that there is no natural intuition for evaluating the probability of a compound event. As a matter of fact, things seem to be much more complex, as one will see later on when considering other types of compound events (which reveal the existence of a natural intuition in this respect!).

With regard to the presently discussed problem, one deals with a special type of compound event. Lecoutre and Durand refer to the term "rchange- abilitr" considered to be more fundamental than the notion of indepen- dence and which may explain the above bias. It seems that, naturally, the various possible orders of a set of elementary results are not counted separately (for instance HT and TH or 5-6 and 6-5) when defining the magnitude of the sample space. It is this special type of compound event based on order which is intuitively deficient.

Analyzing the nature of the misconceptions, one can see that almost all of these answers affirm that the two outcomes have the same probability. The idea that the probability of the couple 5-6 (the order being indifferent) is twice that of 6-6 can be reached only by getting some representation of the corresponding sample space. The basic justification offered by the children is that, in both cases, one deals with chance events. "The probabil- ity is the same because one may obtain 6-6 and 6-5 or none of these results" (E4). "In my opinion the probability is the same because the obtained number is a surprise" (E4). "The probability is the same because there is the same number of 6 and of 5 in both dice" (M2). "The same, because one cannot know which faces one will obtain with both dice"

F A C T O R S A F F E C T I N G P R O B A B I L I S T I C J U D G E M E N T 535

(M1). "The probability is the same, because one cannot determine some- thing which depends only on the motion of a small object like a die, which is thrown by each person in a different manner" (M3).

It is interesting to consider also the justifications given by junior high school pupils who received a certain instruction in probabilities. By consid- ering these justifications, one can learn something about why these students gave lower percentages of correct answers than younger ones without instruction.

"Each die is independent from the other. The probability that with one die, one will obtain a certain number is 1/6 and it is the same probability that one will obtain the same number with the other" (MP3).

It is clear that this student has been taught probabilities. He used the basic notions he has been taught: the concept of independence and the fact that the six faces are equiprobable. Combining these two ideas and considering the two possible outcomes 5 and 6 separately (simple events) one comes to the conclusion that the couples, 5-6 and 6-6 have the same probability!

Two main ideas are then used to justify the equality of the probabilities of getting 6-6 and that of getting 5 and 6: (a) The more primitive idea that both events are the effect of chance and therefore there is no reason to expect one more than the other; and (b) The more sophisticated idea that 5 and 6 are equiprobable and therefore every event, representing a binary combination of them, has the same probability.

On the other hand, most of the subjects who affirm that the couple (6, 6) or the couple (H, H) are less probable do so because they consider that identical results appear less often than different results:

"The 'probability' of obtaining (6, 6) is small. It is more likely to obtain a 6 and a 5" (E2).

"I t is more likely to obtain a 5 and a 6 because the couples of equal numbers represent a minority" (E3).

One has also to mention that some of the 13 to 14 year old pupils, who received some instruction in probabilities, were able to find the correct answer and to justify it correctly:

"I t is more probable to obtain a 5 and a 6 because there are two poss- ibilities (5, 6) and (6, 5) while (6, 6) represents only one possibility" (MP3).

"I t is more probable to obtain an H and a T because one may obtain with one coin H and with the other T and vice versa" (MP3).

536 EFRAIM FISCHBEIN ET AL.

These f indings c o r r o b o r a t e genera l ly those o f Lecou t re and D u r a n d

(1988). They show also tha t one gets the same pa t t e rn o f results when using

coins ins tead o f dice, which po in ts to the genera l i ty o f the findings.

The fo l lowing two ques t ions A4 and B4 cons ider the same type o f

p r o b l e m s as A3 and B3 bu t in a genera l form. We h o p e d that , this way, we

could get new insights in to this in teres t ing p rob lem.

Questions A4 and B4

"One rol ls two dice. Which is more p robab le : to ob ta in the same n u m b e r

wi th bo th dice, or different number s?"

"One tosses two coins. W h i c h is more likely: to ob t a in the same face with

bo th coins o r different faces? Or is the p robab i l i t y the same?"

TABLE IVA

Two dice. The probability to get the same number compared with the probability to get different numbers. (Percentages). See the text for the two types of misconceptions

Elementary Junior high Junior high (P)

No answer 9.8 14.4 18.5 Correct answers 34.3 43.2 38.4 (Probabilities are different) Misconceptions 55.9 42.4 43.1

Type I 71.9 98.3 100.0 Type II 10.5 1.7 0.0

TABLE IVB

Two coins. The probability to get the same faces compared with the probability to get different faces. (Percentages). See the text for the two types of misconceptions

Elementary Junior high Junior high (P)

No answer 17.4 23.0 23.1 Correct answers 50.5 60.5 58.4 (The same probability) Misconceptions 32.1 16.5 18.5

Type I 65.7 65.2 75.0 Type II 22.9 21.7 8.3

The results are expressed in Tables I V A and IVB. Exp lana t ions for the

two ma in types o f misconcept ions :

FACTORS AFFECTING PROBABILISTIC JUDGEMENT 537

Question A4 (two dice). Type I: The probability of obtaining two equal numbers and the probability of obtaining two different numbers with two dice are the same. Type II: There is a greater likelihood of obtaining equal numbers than different numbers.

Question B4 (two coins). Type 1: There is a greater likelihood of obtaining different faces. Type II: There is a greater likelihood of obtaining the same faces.

These two questions raise the same problems as A3 and B3 but in a generalized form. We could have expected the same type of results as those obtained with the previous questions. As a matter of fact, the results are different. At all age levels and for both problems (dice and coins) the percentages of correct answers are visibly higher for the generalized form of the questions than for the specific ones.

This is a fundamentally new finding. First, let us remark that the higher percentage of correct answers for the generalized form of questions is not related to the equality or non-equality of probabilities.

For the specific questions (A3 and B3), the correct answer is that the probabilities are different for (6-6) versus (5-6) and for (H-H) versus (H-T). For the generalized questions the correct answer for the coin problem is that the probabilities are equal while for the dice problem the probability of getting the same number is smaller than that of getting different numbers.

The only plausible explanation is that many of the subjects (those who answer correctly) possess the inuitive capacity to evaluate globally the magnitude of the sample space and of its structure. But in order to elicit this capacity, the question has to be asked in a generalized form. When the question is not asked in the generalized form, the answers are based on other interpretations (the primary intuition of chance or the additive combination of two independent outcomes) which both lead to incorrect answers. It is then interesting to see how the subjects explain their correct answers.

A4 (dice). Some of the subjects refer to their previous experience.

"I t is more likely to obtain different numbers because I have always obtained such a result" (E4).

"It is more likely to obtain different numbers because one seldom obtains the same numbers" (E5).

Quantitative Additive Evaluations

"I t is more likely to obtain different numbers because from 6 possibilities 5 are different" (M2).

538 EFRAIM FISCHBEIN ET AL.

"I t is more likely to obtain different numbers because among 12 it is more difficult to obtain 2 equal numbers" (M2).

Multiplicative Evaluations

"There is more likelihood of obtaining different numbers because there are more combinations. For equal numbers, one may obtain (1, 1) ( 2 , 2 ) . . . ( 6 , 6 ) while for different numbers one may have (1,2) (1, 3 ) . . . (1, 6) (2, 1)(2, 3 ) . . . (2 , 6 ) . . . " (M2).

"I t is more likely to obtain different numbers because, in order to get equal numbers, we have 6 possibilities but for obtaining two different numbers one has 30 possibilities" (MP3).

"The probability is higher for obtaining different numbers: the probability of obtaining equal numbers is 1/6 while the probability of obtaining different numbers is 5/6" (MP3).

It is important to emphasize that all the subjects mentioned above (and many others whom we did not quote) have answered to the specific question (6-6 versus 5-6) that the probability is the same. As we have already mentioned, it is the generalized form of the question which arouses either intuitively, globally or as an effect of instruction (analytically) a representation of the sample space (complete or incomplete).

Let us quote some justifications of correct answers for the coin problem. In this case, the correct answer is that the two probabilities (the same faces versus different faces) are equal (while for obtaining the couple H & T the chances are the double compared with those of obtaining H with both coins). As we have seen, to this question one has obtained relatively high percent- ages of correct answers, as compared with the specific question. Nevertheless most of the subjects did not offer explanations in which any representation of the corresponding sample space is obvious.

"The probability is the same because the results cannot be predicted" (M2).

"The probability is the same because one may obtain either the same or different faces" (E5).

In very few cases, subjects who possessed a certain instruction in probabilities were able to describe the corresponding sample space:

"The probability is the same because one may obtain HT, HH, TH, TT" (MP3). The same pupil gave the erroneous answer: "The same probability", also to the corresponding specific question. It is the generality of the

FAC TORS A F F E C T I N G PROBABILISTIC J U D G E M E N T 539

question which arouses the representation of the corresponding sample space.

Another example of a correct answer based on the representation of the sample space: "In my opinion, one may get either head and tail or tail and head or the same faces for both the head and the tail" (MP2). In this case, the child answered also "the same probability" (incorrectly) to the specific question.

Summarizing the results obtained with questions A4 and B4 one may contend that:

(a) Some of the subjects possess the capacity to evaluate intuitively the magnitude of the sample space corresponding to a stochastic experi- ment.

(b) This intuitive capacity improves spontaneously with age. (c) This capacity is aroused by generalized forms of questions (which

evoke the idea of multitude). (d) The proportion of correct evaluations of sample spaces (when comparing

probabilities) is greater if the sample space is richer (better results were obtained with the dice than with the coins in the A4 and B4 problems).

A didactical recommendation is to analyze, with the students, the probabil- ities in the two different forms: the particular and the generalized form of the questions. This, in our opinion may represent an efficient way for improving the corresponding intuitive background for the notion of com- pound event.

Compound Events ( H)

Two other pairs of questions of a different type are also devoted to the concept of compound event. In these questions, sums of couples of numbers obtained by throwing two dice are considered. One hypothesis was that the magnitude of the sum considered may play a role in determin- ing the probability estimation. More specifically, we supposed, initially, that, intuitively, the naive subject would bet on greater sums (bigger numbers) no matter the magnitude of the corresponding sample space. This was based on the more general assumption that in naive subjects the notion of sample space does not develop itself spontaneously. As we have already seen, this assumption has been contradicted by some of our findings.

The two pairs of questions whose analysis follows explore the same problem with different means.

540 EFRAIM FISCHBEIN ET AL.

Questions A5 and B5

Question AS: " C o n s i d e r i n g the sum o f the po in t s o b t a i n e d when roi l ing a

pa i r o f dice, will you be t on 3 o r on 6? W h y ? "

Question B5: "Cons ide r i ng the sum o f the po in ts ob t a ined when rol l ing a

pa i r o f dice, wou ld you bet on 7 o r on 10? W h y ? "

The two ques t ions seem to be abso lu te ly similar , bu t there is an impor -

tan t difference. In the first case, the n u m b e r wi th a h igher p robab i l i t y is:

" 6 " ( t ha t is the b igger n u m b e r in the pa i r 3 and 6). In the second case, the

chosen n u m b e r should be "7" , tha t is the smal ler o f the two numbers (7

and 10).

As an effect, the resul ts a re s t r ik ingly different. As one can see f rom

Tab les V A and VB, in the case o f ques t ion A5, 6 0 - 7 0 % o f the answers

were correc t whils t in the case o f ques t ion B5, we got respect ively 19.3%,

34.5% and 46.2% correct choices (see Tables V A and VB),

TABLE VA

Two dice. Comparison between the probabilities to get the sums 3 and 6. (Percentages). For the two types of errors, see the text

Elementary Junior high Junior high (P)

No answer 7.8 14.4 10.8 The correct choice (6) 62.8 61.1 73.8 Incorrect answers 29.4 24.5 15.4

Type I errors 23.3 26.5 80.0 Type II errors 43.3 50.0 10.0

TABLE VB

Two dice. Comparison between the probabilities to get the sums 7 and 10. (Percentages). For the two types of errors see the text

Elementary Junior high Junior high (P)

No answer 11.9 19.4 12.3 The correct choice (7) 19.3 34.5 46.2 Incorrect answers 68.8 46.1 41.5

Type I errors 22.7 28.1 33.3 Type II errors 56.0 57.8 63.0

FACTORS AFFECTING PROBABILISTIC JUDGEMENT 541

We consider in the two Tables two different types of misconceptions.

Question A5 - Type I: The probability of obtaining the sum 3 and that of obtaining the sum 6 are equal. Type II error: it is more likely to get the sum 3.

Question B5 - Type I: The probability of obtaining the sum 7 and that of obtaining 10 are equal. Type II: it is more likely to obtain the sum l0 than to obtain the sum 7.

There seems to be only one explanation of these results: Most of the subjects chose the bigger number as the favourite one, no matter the sample space. In question AS, the bigger number 6 is really more likely to win and so their choice is apparently correct (though, in fact, based on a wrong reason). In question B5, it is the smaller number, 7, which presents the higher probability and, therefore, a relatively low percentage of the subjects made the correct choice. In this case, there is a clear progress as an effect of age and instruction. The justifications confirm, globally, the above explanation but the picture one gets is, in fact, more complex.

Some of the children claim that they will bet on 6 simply because it is bigger.

"The 6, because it is a bigger number and two dice may give 3 + 3, 1 + 5, 2 + 4" (ES). "The 6 because it is bigger than 3" (M1). "On 6 because it is bigger than all the numbers" (MP3).

Finally, some of the children try to justify their preference for 6, indicating some or all the possible outcomes related to 6.

"I will bet 6 because, in the case of 3, one wins only with 1 and 2, while in the case of 6 one wins with l and 5, 4 and 2, 3 and 3" (MP3).

This child has certainly an intuitive idea of the role of the magnitude of the sample space, but he considers only a part of it (that is, without considering the order as a factor of differentiating outcomes).

There are also a few cases of children who were able to indicate all the possible outcomes which constitute the expected event.

"(I bet) on 6 because it may be obtained from more numbers, that is: 1 and 5, 4 and 2 . . . " (and the child mentions all the possibilities) (ES).

Considering their justifications, one may conclude that many children have an intuitive idea of the relationship between the likelihood of an event and the magnitude of the corresponding sample space. Even when they choose the bigger number incorrectly (10 from the pair 7-10) they still support their option by pointing to a bigger number of possible combina- tions. For instance:

542 E F R A I M F I S C H B E I N ET AL.

"The 10, because in order to get 7 one has only 4 and 3, and 3 and 4 while for 10 one has 5 and 5, 6 and 4, 4 and 6" (MP3).

One may then distinguish the following levels:

(a) The child indicates the bigger number without any reference to the constitutive outcomes.

(b) The child indicates only some of the possible outcomes, which leads him to an erroneous evaluation.

(c) The child decribes all the possible pairs but does not understand that each pair of different numbers has to be considered twice, that is considering the order too.

(d) The child is able (even spontaneously) to imagine all the possible outcomes, and consequently, to conclude correctly.

Some of the answers point to the equality of the possibilities. In this case, the child refers only to the general ideas of chance and good luck.

"It is the same to bet either on 10 or 7: The possibility is the same, it is enough to have good luck" (E5).

An interesting observation: Considering only the inadequate reactions, one finds that the percentage of answers indicating equal chances for both numbers, increases with age and instruction for both items!

The main finding of this part of the research is that most of the subjects seem to possess an intuitive idea of the realtionship between probability and the size of a corresponding sample space. The non-adequate options were generated mainly by a lack of technical knowledge on how to build this sample space. In addition, there is no natural intuitive understanding of the fact that, in constructing the sample space, one has to consider also the order of the elementary results which constitute the outcomes. Only a few subjects seem to have intuitively this understanding.

Questions A6 and B6

The questions A6 and B6 are of the same type as A5 and B5.

Question A 6. Luca and Paolo play with a pair of dice. If the sum of the points is 3, Luca is the winner. If the sum of the points is 11, Paolo is the winner. Which of the following answers seems to you to be the correct one7 Why?

Luca is the favourite Paolo is the favourite Luca and Paolo have the same chance

FAC TORS A F F E C T I N G PROBABILISTIC J U D G E M E N T 543

Question B6 is similar but the numbers considered are 2 (Paolo) and 12 (Piero). In both problems, the probability is the same for each number in the pair. Therefore, the only factor which apparently may affect the choice is the magnitude of the numbers in each pair. This way, we hoped to get more information concerning the role of the magnitude of the expected number when assessing its probability.

The results are presented in Tables VIA and VIB. Types of misconceptions:

Question A6 - Type I: The probability of obtaining the sum 3 is higher than the probability of obtaining the sum 11. Type II: The probability of obtaining the sum 11 is higher.

Question B6 - Type I: The probability of obtaining the sum 2 is higher. Type II: The probability of obtaining the sum 12 is higher.

The proportion of those who do not answer question A6 at all increases with age and instruction. The proportion of those who answer correctly

TABLE VIA

Two dice. Comparison between the probabilities to get the sums 3 and 11. (Percentages). For the two types of errors see the text

Elementary Junior high Junior high (P)

No answer 2.9 13.7 26.1 Correct answers 30.4 51.8 38.5 Incorrect answers 66.7 34.5 35.4

Type I 19.1 12.5 4.3 Type II 77.9 83.3 95.7

TABLE VIB

Two dice. Comparison between the probabilities to get the sums 2 and 12. (Percentages). For the two types of errors see the text

Elementary Junior high Junior high (P)

No answer 0.9 3.6 4.6 Correct answers 50.5 58.3 67.7 Incorrect answers 48.6 38.1 27.7

Type I 18.9 26.4 5.6 Type II 81.1 73.6 94.4

544 EFRAIM FISCHBEIN ET AL.

(the chances are equal) increases with age but not as an effect of instruc- tion. The younger subjects and those who possess some experience with probabilities gave less than 50% correct answers.

With question B6, the situation is different. Very few subjects do not answer at all. The percentage of correct answers increases with age, and in all three groups more than half of the subjects answered correctly.

Considering the results obtained with both questions, it is obvious that the magnitude of the numbers is a decisive factor of the choice. Two questions have to be answered: (a) Why do the subjects prefer the bigger number? (b) Why is the second problem easier?

Justifications for A 6 and B6

A 6 - Answers Mentioning Equal Probabilities (correct)

Some of the subjects refer to the pairs of numbers the sum of which is 3 and 11 respectively.

"The chances are equal because 11 is obtained from 5 and 6 and 3 is obtained from 1 and 2" (E5).

"They have the same chances: Luca must obtain (2,1) and Paolo (5,6)" (MP3).

Almost all of these subjects who consider correctly the numbers 1 and 2 and 5 and 6 respectively do not see that each pair has to be considered twice. Again, there is no natural intuition supporting the idea that each pair is in fact a compound event. But the range of explanations is richer:

"Luca and Paolo have the same chance because one does not know which number one will get" (E5).

"The chance is the same because 3 is too low and 11 is too high" (M3).

A 6 - Answers Referring to Different Probabilities (incorrect)

"Paolo has a higher chance because he has the bigger number" (E4).

"Paolo is advantaged because there are 5 possibilities to obtain 11, that is 8 + 3, 10 + 1, 6 + 5, 9 + 2, 7 + 4 while Luca has only one possibility, that is, 2 + 1" (E5).

"Paolo is advantaged because with two dice one obtains almost always numbers bigger than 3" (M1).

FACTORS AFFECTING PROBABILISTIC JUDGEMENT 545

"Paolo is advantaged because 3 may be obtained only with 1 + 2 (which will not be obtained very often), while 11 may be obtained with many numbers" (M2).

"Paolo is advantaged because he has the higher number" (M3).

"Paolo is advantaged: in fact, for Paolo 1 and 10, 2 and 9, 3 and 8, 7 and 4, 6 and 5, 5 and 6, 4 and 7, 3 and 8, 2 and 9, 1 and 10; for Luca 2, 1 and 1, 2" (MP3).

"Paolo is the more advantaged because he has more points" (MP3).

"Paolo is more advantaged because with dice it is more likely to obtain big numbers" (MP3).

There are two main types of justifications for the choice of the bigger number: (a) Some of the subjects simply chose the bigger number because "it is bigger"; (b) Others try to identify the pairs which would yield 11 but forget the limitation imposed by the conditions of the game (the 6 being the biggest possible number). We suppose that even those subjects who, without explaining, chose the number 11 as being more likely, have in fact in mind, tacitly, the multiplicity of possible combinations (but also forget- ting the specific limitations of the dice game).

Q u e s t i o n B 6 - C o r r e c t A n s w e r s

Although the two questions A6 and B6 are apparently similar, the results obtained are different: higher percentages of correct answers ("the proba- bilities are equal") have been obtained with question B6 as compared with question A6. A plausible explanation seems to be the following. In question B6, the two numbers considered are 2 and 12, which may be obtained only in one way (1 + 1 and 6 + 6). This makes easier the task of producing (tacitly or explicitly) the corresponding sample spaces (what Tversky and Kahneman call "the heuristic of availability" (1973)). Let us consider some examples referring to correct answers:

"The probability is the same because with the two dice, one may obtain 12(6+6) and 2(1 + 1)" (E5).

"The same probability, because by rolling two dice, one may get 2 as one may get 12" (E4).

"The 12 may be obtained only with 6, 6 and the 2 with 1, 1. Therefore, from the 36 possible combinations, each one of the two numbers has only one chance" (MP3).

546 EFRAIM FISCHBEIN ET AL.

Question B6 - Erroneous Answers

The main justification for the incorrect choice of 12 is based on the idea (as it was for question A6) that 12 may be obtained as a result of more possible combinations than 2. That is, in this case too, the subjects forget the constraints imposed by the dice game.

"Piero has the bigger chance because 2 may be obtained only with two equal numbers while 12 may be obtained also with different numbers" (M2).

"Piero has the advantage because 2 is obtained only with (1, 1) while 12 has more possibilities to be obtained: (6, 6) (8, 4), (7, 5), (3, 9)" (MP3).

There were also a few subjects who forgot completely the conditions of the game and simply considered that the higher number always wins.

"Piero has the advantage because in a dice game if one has 10 more points you are absolutely sure to win" (M2).

The results obtained with question A6 and B6 support the conclusion previously drawn from the reactions to questions A5 and B5: Children develop a natural, intuitive tendency to evaluate the probability o f a compound event on the basis of the corresponding sample space. This is a fundamental finding to be considered both from the psychological and the didactical point of view.

The following obstacles interfere, conflicting with the correct evaluation:

(a) As already mentioned, there is no intuitive, natural support for count- ing separately as distinct outcomes, the same groups of results in different sequences (for instance 5, 6 and 6, 5).

(b) The subjects tend to forget, sometimes, the specific conditions of the stochastic experience and the sample space is constructed without considering the respective limitations (for instance in a dice game numbers like 7, 8 etc. are also considered).

(c) Many subjects do not possess a systematic technique for producing all the possible outcomes related to an event.

(d) Availability seems to be an important factor in the intuitive evaluation of the magnitude of a sample space. Subjects have a better chance to compare correctly the possibilities of getting (by adding two numbers) 2 or 12 than to get 3 or 11 (in a dice game).

(e) In some subjects, the general idea of chance is still sufficiently active for hiding the specific condition of the problem and leading to the idea of equal chances ("the probabilities of the two events are the same because both are chance events").

FACTORS AFFECTING PROBABILISTIC JUDGEMENT 547

But beyond the effect of these factors - except the last one - one may postulate the existence, in naive subjects, of an intuitive capacity to evaluate the magnitude of the sample space corresponding to a certain stochastic experience. This finding corroborates those related to formerly analyzed questions (in which the subjects were asked to compare the probabilities of obtaining identical versus different results when rolling a couple of dice or tossing two coins).

SUMMARY AND CONCLUSIONS

The psychological aspects of the concept of probability seem to be much more complex, in many respects, than it is usually considered. If one intends to develop by instruction a strong, correct, coherent, formal and intuitive background for probabilistic reasoning, one has to cope with a large variety of misunderstandings, misconceptions, biases and emotional tendencies. Such distortions may be caused by linguistic difficulties, by a lack of logical abilities, by the difficulty to extract the mathematical structure from the practical embodiment, by the difficulty to accept that chance events may be analyzed from a deterministic-predictive point of view.

I. The subjects do not have in mind a clear definition of the terms "possible", "impossible" and "certain". The term "certain", especially, entails a certain difficulty when it is related to a compound event (for instance, the probability of obtaining a number smaller than 7 when rolling a die). Some subjects confuse "rare" with "impossible".

2. The notions of compound and simple events also raise interesting psychological and didactical problems. It is a topic which deserves more attention, considering its intuitive complexity and the variety of situa- tions which it may generate. The following situations have been iden- tified:

There is no natural understanding of the fact that, in a sample space, possible outcomes should be distinguished and counted separately if the order of their elementary components is different. Our intuition is surprised by this discovery. Should we conclude that, generally, the idea of a compound event and the evaluation of its probability lack any intuitive background? The answer is negative. With other types of problems many subjects seem to display a certain, natural feeling of the role of the magnitude of the corresponding sample space in determining the probability of a compound event.

548 EFRAIM FISCHBEIN ET AL.

There are, in our investigation, two types of problems which support this conclusion. The first case is that in which couples of results are considered in a general form: "By rolling two dice is it more likely to get two different numbers or the same numbers?" A similar question has been asked referring to coins.

In this general form, the questions yielded much more correct an- swers. In the case of coins most of the subjects answered correctly at both age levels. The justifications usually refer to the corresponding sample space (though, very often, incompletely described).

This finding indicates clearly that, in certain circumstances, the subjects are able to evaluate spontaneously the probability of a com- pound event. This ability is strengthened and really put into profit when the question is addressed in a general form.

The second type of questions used by us which reveals the natural capacity of the subjects to cope with compound events is that in which the subjects are asked to compare the probabilities of getting certain numbers obtained by addition when rolling two dice. Generally speaking, many subjects seem to be able to relate spontaneously the estimations of probabilities to the magnitude of sample spaces though, sometimes, the description of these sample spaces is incorrect. The magnitude of the numbers the probabilities of which one has to compare represents a major factor in the subjects' decisions. The fact that a number is bigger is frequently associated with a higher probability without any other justification than the magnitude of the number.

3. The capacity to abstract the identical mathematical (probabilistic) structure from different experimental embodiments: The probability of obtaining (5, 5, 5) by throwing a die three times or by throwing simul- taneously three dice; the probability of obtaining (H, H, H) by tossing a coin three times or by tossing three coins simultaneously.

The majority of subjects answered both questions correctly: the probabilities are equal. There is an increase with age and instruction of the proportion of correct answers. Most of the errors consisted in affirming that by performing the operation three times successively, one has a better chance to get the expected event. The reason: this way the effect can be better controlled by the subject.

4. An important factor influencing some of the subjects' reactions (even in adolescents) remains the capacity to synthesize the necessary and the random in the concept of probability (see Fischbein, 1975). The idea that an outcome of a stochastic experiment depends only on chance -

FACTORS A F F E C T I N G PROBABILISTIC J U D G E M E N T 549

no matter what the given conditions - renders then any prediction baseless. This, in our opinion, is a very important finding and it should be treated with much care by teachers and didacticians.

REFERENCES

Feller, W.: 1960, Introduction to Probability Theory and its Applications, Vol. 1, John Wiley, New York.

Fischbein, E.: 1975, The Intuitive Sources of Probabilistic Thinking in Children, Reidel, Dordrecht.

Fischbein, E., Barbat, I. and Minzat, I.: 1975, 'Primary and secondary intuitions in the introduction of probability', in Fischbein, E. (ed.), The Intuitive Sources of Probabilistic Thinking in Children, Appendix 1, pp. 139-155, Reidel, Dordrecht.

Fischbein, E. and Gazit, A.: 1984, 'Does the teaching of probability improve probabilistic intuitions?', Educational Studies in Mathematics 15(1), 1-24.

Garfield, J. and Ahlgren, A.: 1988, 'Difficulties in learning basic concepts in probability and statistics: implications for research', Journal for Research in Mathematics Education 19(1), 44-63.

Godino, J. D., Batanero, C. and Canizares, J.: 1987, Azar y Probabilidad, Sintezis, Madrid. Hawkins, S. A. and Kapadia, R.: 1984, 'Children's conceptions of probability - a psycholog-

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Heuristics and Biases, Cambridge University Press, Cambridge. Lecoutre, M. and Durand, J.: 1988, 'Jugements probabilistes et mod61es cognitifs: Etude d'une

situation aleatoire', Educational Studies in Mathematics 19(3), 357-368. Piaget, J. and Inhelder, B.: 1951, La Gen~se de l'Id~e de Hasard chez L'Enfant, PUF, Paris. Tversky, A. and Kahneman, D.: 1973, 'Availability: A heuristic for judging frequency and

probability', Cognitive Psychology No. 5, 207-232.

Tel-Aviv University, School of Education, Tel-Aviv, Israel

EFRAIM FISCHBEIN

Didactical Seminary,

Department o f Mathematics ,

University of Pisa, Italy

MARIA SAINATI NELLO and MARIA SCIOLIS MARINO