the correct posterior odds are 8 to 1 for the 4:1 sampleand 16 to 1 for the 12:8 sample

assuming equal prior probabilities.However

most people feel that the first sample provides much strongerevidence for the hypothesis that the urn is predominantly red

because theproportion of red balls is larger in the first than in the second sample. Hereagain

intuitive judgments are dominated by the sample proportion and areessentially unaffected by the size of the sample

which plays a crucial rolein the determination of the actual posterior odds.5 In addition

intuitiveestimates of posterior odds are far less extreme than the correct values.The underestimation of the impact of evidence has been observedrepeatedly in problems of this type.6 It has been labeled âconservatism.âMisconceptions of chance. People expect that a sequence of eventsgenerated by a random process will represent the essential characteristicsof that process even when the sequence is short. In considering tosses ofa coin for heads or tails

for example

people regard the sequence H-T-H-T-T-H to be more likely than the sequence H-H-H-T- [enc. IT-T

which doesnot appear random

and also more likely than the sequence H-H-H-H-T-H

which does not represent the fairness of the coin.7 Thus

people expectthat the essential characteristics of the process will be represented

notonly globally in the entire sequence

but also locally in each of its parts. Alocally representative sequence

however

deviates systematically fromchance expectation: it contains too many alternations and too few runs.Another consequence of the belief in local representativeness is the well-known gamblerâs fallacy. After observing a long run of red on the roulettewheel

for example

most people erroneously believe that black is now due

presumably because the occurrence of black will result in a morerepresentative sequence than the occurrence of an additional red. Chanceis commonly viewed as a self-correcting process in which a deviation inone direction induces a deviation in the opposite direction to restore theequilibrium. In fact

deviations are not âcorrectedâ as a chance processunfolds

they are merely diluted.Misconceptions of chance are not limited to naive subjects. A study ofthe statistical intuitions of experienced research psychologists8 revealed alingering belief in what may be called the âlaw of small numbers

â accordingto which even small samples are highly representative of the populationsfrom which they are drawn. The responses of these investigators reflectedthe expectation that a valid hypothesis about a population will be