A psychophysical analysis of the base rate fallacy

The base rate frequency biasThe 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)

%)8.7(078.)80(.*)99(.)80(.*)01(.

)80(.*)01(.

.).|(*.).(.).|(*.).(

.).|(*.).()|..(

CBPosPCBPCBPosPCBP

CBPosPCBPPosCBP

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: 17.0)20(.*)95(.)80(.*)05(.

)80(.*)05(.)Re(

dOP

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%

¬A A

UncertaintyCertainty Certainty

2€ 1€ 50c 20c 20c 50c 1€ 2€

0 100%

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

)/()/(' XrightOchoicezOrightOchoicezd

Results of a pilot experiment

Dynamic Game

Most Probable Outcome

Bet O X

O 23 15 38

X 17 25 42

40 40 80

d’ 0,5078

Static Game

Most Probable Outcome

Bet O X

O 31 7 38

X 9 33 42

40 40 80

d’ 1,6900

Dynamic Game

Most Probable Outcome

Bet O X

O 8 8 16

X 12 12 24

20 20 40

d’ 0

Static Game

Most Probable Outcome

Bet O X

O 14 5 19

X 6 15 21

20 20 40

d’ 1,1989

In all the trials In the difficult trials

Most Probable Outcome

Bet O X

O 12 1 13

X 2 9 11

14 10 24

d’ 2,3491

Most Probable Outcome

Bet O X

O 9 3 12

X 3 14 17

12 17 29

d’ 1,6034

Most Probable Outcome

Bet O X

O 10 3 13

X 4 10 14

14 13 27

d’ 1,3023

Most Probable Outcome

Bet O X

O 5 7 12

X 5 7 12

10 14 24

d’ 0

Most Probable Outcome

Bet O X

O 11 5 16

X 9 13 22

20 18 38

d’ 0,7151

Most Probable Outcome

Bet O X

O 7 3 10

X 3 5 8

10 8 18

d’ 0,8430

Dynamic Game Static Game

Considering the money

3,5£

3£ 2,5£

2£ 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