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Cognitive Processes PSY 334 Chapter 11 – Judgment and Decision-Making Inductive Reasoning Processes for coming to conclusions that are probable rather than certain. As with deductive reasoning, people’s judgments do not agree with prescriptive norms. Baye’s theorem – describes how people should reason inductively. Does not describe how they actually reason. Baye’s Theorem Prior probability – probability a hypothesis is true before considering the evidence. Conditional probability – probability the evidence is true if the hypothesis is true. Posterior probability – the probability a hypothesis is true after considering the evidence. Baye’s theorem calculates posterior probability. Burglar Example Numerator – likelihood the evidence (door ajar) indicates a robbery. Denominator – likelihood evidence indicates a robbery plus likelihood it does not indicate a robbery. Result – likelihood a robbery has occurred. Baye’s Theorem H ~H E|H likelihood likelihood likelihood robbery E|~H likelihood of being robbed of no robbery of door being left ajar during a of door ajar without robbery P( E | H ) P( H ) P( H | E ) P( E | H ) P( H ) P( E |~ H ) P(~ H ) Baye’s Theorem P(H) = .001 P(~H) = .999 P(E|H) = .8 P(E|~H) = .01 from police statistics this is 1.0 - .001 Base rate P( E | H ) P( H ) P( H | E ) P( E | H ) P( H ) P( E |~ H ) P(~ H ) (.8)(.001) P( H | E ) .074 (.8)(.001) (.01)(.999) Base Rate Neglect People tend to ignore prior probabilities. Kahneman & Tversky: 70 engineers, 30 lawyers vs 30 engineers, 70 lawyers No change in .90 estimate for “Jack”. Effect occurs regardless of the content of the evidence: Estimate of .5 regardless of mix for “Dick” Cancer Test Example A particular cancer will produce a positive test result 95% of time. If a person does not have cancer this gives a 5% false positive rate. Is the chance of having cancer 95%? People fail to consider the base rate for having that cancer: 1 in 10,000. Cancer Example Base rate P(H) = .0001 P(~H) = .9999 P(E|H) = .95 P(E|~H) = .05 likelihood of having cancer likelihood of not having it testing positive with cancer testing positive without cancer P( E | H ) P( H ) P( H | E ) P( E | H ) P( H ) P( E |~ H ) P(~ H ) (.95)(.0001) P( H | E ) .0019 (.95)(.0001) (.05)(.9999) Conservatism People also underestimate probabilities when there is accumulating evidence. Two bags of chips: 70 blue, 30 red 30 blue, 70 red Subject must identify the bag based on the chips drawn. People underestimate likelihood of it being bag 2 with each red chip drawn. Probability Matching People show implicit understanding of Baye’s theorem in their behavior, if not in their conscious estimates. Gluck & Bower – disease diagnoses: Actual assignment matched underlying probabilities. People overestimated frequency of the rare disease when making conscious estimates. Frequencies vs Probabilities People reason better if events are described in terms of frequencies instead of probabilities. Gigerenzer & Hoffrage – breast cancer description: 50% gave correct answer when stated as frequencies, <20% when stated as probabilities. People improve with experience. Judgments of Probability People can be biased in their estimates when they depend upon memory. Tversky & Kahneman – differential availability of examples. Proportion of words beginning with k vs words with k in 3rd position (3 x as many). Sequences of coin tosses – HTHTTH just as likely as HHHHHH. Gambler’s Fallacy The idea that over a period of time things will even out. Fallacy -- If something has not occurred in a while, then it is more likely due to the “law of averages.” People lose more because they expect their luck to turn after a string of losses. Dice do not know or care what happened before. Chance, Luck & Superstition We tend to see more structure than may exist: Avoidance of chance as an explanation Conspiracy theories Illusory correlation – distinctive pairings are more accessible to memory. Results of studies are expressed as probabilities. The “person who” is frequently more convincing than a statistical result. Decision Making Choices made based on estimates of probability. Described as “gambles.” Which would you choose? $400 with a 100% certainty $1000 with a 50% certainty Utility Theory Prescriptive norm – people should choose the gamble with the highest expected value. Expected value = value x probability. Which would you choose? A -- $8 with a 1/3 probability B -- $3 with a 5/6 probability Most subjects choose B Subjective Utility The utility function is not linear but curved. It takes more than a doubling of a bet to double its utility ($8 not $6 is double $3). The function is steeper in the loss region than in gains: A – Gain or lose $10 with .5 probability B -- Lose nothing with certainty People pick B Framing Effects Behavior depends on where you are on the subjective utility curve. A $5 discount means more when it is a higher percentage of the price. $15 vs $10 is worth more than $125 vs $120. People prefer bets that describe saving vs losing, even when the probabilities are the same.