Baratgin, Jean

Politzer, Guy

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Brief report: What is the chance of winning (or losing) a conditional bet?

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Brief report: What is the chance of winning (or losing) aconditional bet?Guy Politzer, Jean BaratginTo cite this version:Guy Politzer, Jean Baratgin. Brief report: What is the chance of winning (or losing) a condi-tional bet?. BR 1, 2006. <ijn 00130613>HAL Id: ijn 00130613http://jeannicod.ccsd.cnrs.fr/ijn 00130613Submitted on 13 Feb 2007HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestine´e au de´poˆt et a` la diffusion de documentsscientifiques de niveau recherche, publie´s ou non,e´manant des e´tablissements d’enseignement et derecherche franc¸ais ou e´trangers, des laboratoirespublics ou prive´s.May 2006Brief technical reportWhat is the chance of winning (or losing) a conditional bet?Guy Politzer and Jean BaratginInstitut Jean NicodRecent experimental studies have examined lay people's concept of the probability ofconditional sentences (Evans, Handley & Over, 2003; Girotto & Johnson-Laird, 2004;Oberauer & Wilhelm, 2003; Over, Hadjichristidis, Evans, Handley, & Sloman,  2007).This question is important because the answer can help evaluate some claimsregarding the semantics of conditional sentences made (i) by different authors inphilosophical logic (e. g., Stalnaker, 1968; Adams, 1975) and (ii) by the main theoriesof deductive reasoning (Johnson-Laird & Byrne, 2002 for mental models theory; Brain& O'Brien, 1991, and Rips, 1994 for mental rules; Oaksford, Chater & Larkin, 2000for the probabilistic approach). These theoretical approaches will not be developed inthis brief report. They are expounded in Evans & Over (2004).The main result of the experiments just mentioned is that when participantsare asked the probability that a conditional if A then C is true, in the extensional case(e. g., given a pack of cards, if the card is yellow, then it bears a square) there aretwo main answers, namely (i) the conditional probability of the consequent on theantecedent Pr(C/A), and (ii) almost as often, the conjunctive probability Pr(AC); andin the intensional case (e. g., if fertility treatment improves then the world populationwill increase) only the former answer is observed. The probability of the materialconditional, Pr(not-A or B), is hardly ever given.This result supports the theories that predict the conditional probabilityresponse, except for the conjunctive response which requires an explanation.  Thereis one serious problem of comprehension for participants who are asked a questionabout the probability of the truth of a conditional statement. This is because, on theone hand, supporters of the view that conditionals with false antecedents lack a truthvalue (e. g. , Adams, 1975) might claim that the question is meaningless and that,consequently, it is re-interpreted in various ways, one of which would give rise to theconjunctive answer; this would take place especially when, as occurs with materialsdisplaying the logical cases, the AC case is made very salient. On the other hand,there is a possibility that, as claimed by Girotto & Johnson-Laird,  in spite of theprecautions taken by the investigators, the question is re-interpreted as if A, what isthe probability that C is true.In the present investigation, participants were introduced to a betting situation .Methodologically, this has the advantage that one can replace the somehowawkward and artificial request about "the probability that [the conditional sentence] istrue", which may lead to the misundertanding just mentioned, with a request about"the probability that one wins the bet".  In this way, participants should easilyrepresent the probability of the truth of the conditional as the probability of winningthe bet. Also, care was taken to keep processing effort in terms of memory orcalculation as low as possible. Theoretically, conditional bets were introduced by deFinetti (1938). He distinguished bets that one is only able to win or lose,corresponding to bi-valued events, from conditional bets (subordinate in histerminology) corresponding to tri-valued events, whose validity depends onconditions which must be verified. For example, in a race that is to take placetomorrow, the bet on the victory of one of the competitors is of the first type if it isunderstood that  the bet is definitely lost in case this event is not realised, whateverthe reason may be; but it is of the second type if it is agreed that the bet is "null andvoid" in case the race is cancelled or the competitor in question does not participate,etc.Participants and methodParticipants were 102 first year students of psychology or sports sciences at theUniversity of Aix-Marseille-2. They had not received training in probability theory(except for elementary definitions that some of them had learned in high school).They received a questionnaire (in French) which they answered in small groups.There were three questions, the first two presented in counter-balanced order.Instructions and questionsThis drawing represents chips.● ● O   Pierre chooses a chip at random.Marie, who has not seen Pierre's choice, says:I bet that if the chip is square then it is black1) What are the chances that Marie wins her bet? (give a numerical value)2) What are the chances that Marie loses her bet? (give a numerical value)3) Suppose Pierre has drawn a round chip: Do you think that Marie has won or losther bet?ResultsAs the order of the questions had no effect, response freqencies were pooled. Theseare given in the Table below.Betting status of not-A (from question 3)Response f winning losing neither3/4, 1/4 77 19 17 413/7, 4/7 18   3   7   8other   7   0   5   2totals 102  22  29  51The conditional probability answer is Pr(win) = 3/4, Pr(lose) = 1/4. It was given byaround 77% of the participants.The conjunctive response, Pr(win) = 3/7, Pr(lose) = 4/7, was given by around 18%.The few remaining responses show a mixture of conditional probability andconjunction (e. g. , 3/4 and 3/7).DiscussionThe conditional probability response is by far the most frequent, while the conjunctiveresponse is much less frequent than reported by Evans, Handley & Over (2003) andOberauer & Wilhelm (2003) with abstract materials; however, it has not disappeared.It is noteworthy that, given the formulation of question 3 in terms of winning orlosing, the occurrence of the “neither winning nor losing” response was spontaneous.Moreover, it is the most frequent answer (exactly 50%). Had this option been offered,it is likely that it would have appeared even more often.For the theories that predict the conditional probability, this response shouldbe linked with the recognition that in case the chip is not square the bet is void, whilethe conjunctive answer should be linked with the consideration that in that case thebet is lost. There is a corresponding trend in the data but lack of statistical power toconclude with any reliability. Anecdotically, in a pilot group that had an oraldebriefing, there was agreement among the conjunctive respondants that if one is towin any money, the chip would have to be square, so that there are only three casesout of seven where one can get some money: In sum, these participants did notconsider the not-A case (round chips) as making the bet void, rather this case wasviewed as failure to win so that they gave it the same status as the white square.Presumably, in the context of a bet the "voidness" of the not-A case appears moreclearly because people are wary of possible unfulfilled conditions for the bet to hold,whereas in a formal artificial situation such cautiousness does not occur.One could speculate that the betting situation is biased towards winning, sothat the not-A case may be considered as failure to win (typically some money) butcan hardly count as not losing, except if a scenario was designed to render relevantan escape from losing. One may predict the possibility to observe in this situation a1/7, 6/7 answer for winning and losing, respectively, because the void bet is nowconstrued as "not-losing", in a similar way as the "not-winning" construal leads to the3/7, 4/7 answer. In a sense, there is a framing effect in disguise which prevents the1/7, 6/7 solution to occur.ReferencesAdams, E.  (1975). The logic of conditionals. Dordrecht: Reidel.Braine, M. D. S. , & O'Brien, D. P. (1991). A theory of IF: A lexical entry, reasoningprogram, and pragmatic principles. Psychological Review, 98, 182-203.de Finetti, B.  (1938).  La logique de la probabilité. Actes du Congrès International dePhilosophie Scientifique. Sorbonne, 1935.  IV: Induction et probabilité (pp. 31-39). Paris: Hermann.  English translation: The logic of probability (1995).Philosophical studies, 77, 181-190.Evans, J. St. B. T. , & Over, D. E.  (2004).  If.  Oxford University Press.Evans, J. St. B. T. , Handley, S. J. , & Over, D. E. (2003). Conditionals and condtionalprobability. Journal of Experimental Psychology: Learning, Memory, andCognition, 29, 321-335.Girotto, V. , & Johnson-Laird, P. N. (2004). The probability of condirtionals.Psychologia, 47, 207-225.Johnson-Laird, P. N. , & Byrne, R. M. J. (2002). Conditionals: a theory of meaning,pragmatics, and inference. Psychological Review,109, 646-678.Oberauer, K. , & Wilhelm, O. (2003). The meaning(s) of conditionals: Conditionalprobabilities, mental models, and personal utilities. Journal of ExperimentalPsychology: Learning, Memory, and Cognition, 29, 680-693.Oaksford, M. , Chater, N. , & Larkin, J. (2000). Probabilities and polarity biases inconditional inference. Journal of Experimental Psychology:Learning, Memory,and Cognition, 26, 883-899.Over, D. E. , Hadjichristidis, C. , Evans, J. St. B. T. , Handley, S. J. , & Sloman, S. A.(2007). The probability of causal conditionals.  Cognitive Psychology, 54, 62-97.Rips, L. J. (1994). The psychology of proof. Cambridge, Ma: M.I.T. Press.Stalnaker, R. C. (1968). A theory of conditionals. In N. Rescher (Ed.), Studies inlogical theory. American Philosophical Quarterly Monographs. No 2. Oxford:Blackwell.

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Brief report: What is the chance of winning (or losing) a conditional bet?

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