Understanding Inductive Reasoning and Bayes' Theorem: Probability and Judgment, Slides of Brain and Cognitive Science

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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.

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 likelihood of being robbed ~H likelihood of no robbery E|H likelihood of door being left ajar during a robbery E|~H likelihood of door ajar without robbery

P E H P H P E H P H

P E H P H

P H E

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.

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.

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 3 rd^ 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.

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

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.