A psychophysical analysis of the base rate fallacy

The base rate frequency bias The mammography problem The probability of breast cancer is 1% for a woman at age forty who participates in routine screening. If a woman has breast cancer, the probability is 80% that she will get a positive mammography. If a woman does not have breast cancer, the probability is 9.6% that she will also get a positive mammography. A woman in this age group had a positive mammography in a routine screening. What is the probability that she actually has breast cancer? (Gigerenzer and Hoffrage, 1995) The information we have The “a priori” probability of a breast cancer: P(B.C.) The conditional probabilities to be positive with and without B.C.: P(Pos|B.C.), P(Pos|¬B.C.) The correct solution of the problem The most frequent subjective estimations The 95% of the subjects give an estimation of the probability for the woman to have a breast cancer between 70% and 80% (Eddy, 1982)

The dual process hypothesis Strategic processing –Mental representations –Examples of theories: Human problem solving (Newell & Simon, 1972) Mental models (Johnson-Laird, 1989) –Features: Effortful processing Goal-directed Automatic processing –Features: No attentional control required Obliged course Stimulus dependent –Examples of phenomena Conditioned reflexes Implicit memory

A bayesian game with cards There is a pack of cards, every card has a symbol on a side and a colour on the other side. The symbol could be an O or an X. The colour could be Red or Blue. The probability that a card has an O on one of its side is 5%. If a card has an O on a side the probability is 80% that it is Red on the other side. If a card has an X on a side the probability is 20% that it is Red on the other side. A card from the pack is Red. Which is the probability it has an O on the other side? The outcomes: O or X The data: Red or Blue The information we have: P( O), P( Red| O), P( Red| X) The information we want: P( O| Red) The solution:

The design of the experiment The independent variables The type of presentation of the information: dynamic or static, varying through the treatments The real probability of the outcome given by the Bayes theorem and varying from trial to trial The difficulty: the incidence of the a priori probability on the real probability, varying through the trials with the same real probability The dependent variable The subjective probability: the choice for an outcome and the amount of money of the bet The hypothesis In the second treatment the subjective probability fits with the real probability better than in the first treatment The goodness of fit measure: The d’ : the difference between the z points of the proportions of choices for the outcome when its probability is higher and lower than 50%

The subjective probability The external validity problem –Should we trust the statements of the subjects? –Do the subjects understand the numbers? The subjective probability –The probability is the relative willingness to invest in the occurrence of a future event (De Finetti, 1931) 50%30%10%70%90% ¬AA UncertaintyCertainty 2€1€50c20c 50c1€2€ 0100% Choice:

The meaning of the d’ 0 10% 30% 50% 70% 90% 100% d’ Wrong bets for O Correct bets for O Wrong bets for X Correct bets for X

Results of a pilot experiment Dynamic Game Most Probable Outcome BetOX O X d’0,5078 Static Game Most Probable Outcome BetOX O31738 X d’1,6900 Dynamic Game Most Probable Outcome BetOX O8816 X d’0 Static Game Most Probable Outcome BetOX O14519 X d’1,1989 In all the trialsIn the difficult trials

Most Probable Outcome BetOX O12113 X d’2,3491 Most Probable Outcome BetOX O9312 X d’1,6034 Most Probable Outcome BetOX O10313 X d’1,3023 Most Probable Outcome BetOX O5712 X d’0 Most Probable Outcome BetOX O11516 X d’0,7151 Most Probable Outcome BetOX O7310 X d’0,8430 Dynamic GameStatic Game Considering the money 3,5£ 3£ 2,5£ 2£ 3£ 2,5£ 3,5£

Conclusions The base rate fallacy in these games is not so pervasive as in the written problems It seems to depend more by the amount of ambiguity of the information than by the kind of cognitive process the subjects use There is a clear correlation between the amount of money the subjects invest and their ability to guess the most probable outcome