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Misconceptions and Fallacies Concerning Probability Assessments

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Law of Averages Toss coin many times. What happens as number of tosses increases? Law of Averages Says: n The percentage of heads should become very close to 50%.

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John Kerrich’s Coin-tossing Experiment Number of Tosses Proportion Heads

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Law of Averages IN GENERAL Law of Averages says: Averages and proportions vary less from the “expected” as sample size increases

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Two Misconceptions

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1. Question: A coin is tossed either 2 times or 100 times. You win $2 if # of heads = # of tails. Which has a better chance of winning ? n 2 times OR n 100 times

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In fact: As # of tosses increases, the chance of “exactly 50% Heads” decreases

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John Kerrich’s Coin-tossing Experiment (Text, p.274) Number of Number of Difference from Tosses Heads Expected 10 4 -1 20 10 0 30 17 2 40 21 1 50 25 0 8,000 4,034 34 9,000 4,538 38 10,000 5,067 67 :::: :::: ::::

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John Kerrich’s Coin-tossing Experiment : Number of Tosses # Heads - Half the # of Tosses

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# of Tosses Probability of between 49.9% and 50.1% HEADS 100.08 10,000.17 1,000,000.95 So - as the number of tosses increases, the chance that the % of HEADS is close to 50% INCREASES

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Exercise 4, page 277 (a) there are more than 60% Heads. In each of the following situations, you will toss a coin 10 times or 100 times. Which is better if you win whenever: (b) there are more than 40% Heads. (c) there are between 40% and 60% Heads. (d) there are exactly 50% Heads. (c’) there are between 60% and 80% Heads.

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2. How Does the Law of Averages Work? n By “compensation” or “adjustment” ? NO! Kerrich example: 130 instances of “HHHH” next toss: 69 H’s 61 T’s (no adjustment) –gambler’s fallacy !

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2. How Does the Law of Averages Work? Actually, the Law of Averages works by swamping. –isolated discrepancies become unimportant as number of tosses increases

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1. The Availability Heuristic -- probability assessment is based on instances that you can remember Example: deaths by homicide or from a stroke -- Which happen more often? (strokes cause about 11 times as many deaths as homicides) Distortion of Subjective Probabilities

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2. Representative Heuristic -- assignment of higher probabilities than are realistic based on how one imagines things will happen Example: Bank Teller Linda is 31 yrs old, single, and outspoken. As a student she was involved with issues of discrimination and social justice, and she also participated in the anti-nuclear demonstrations. Which is more likely? (a) Linda is a bank teller (b) Linda is a bank teller who is active in the feminist movement.

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Conjunction Fallacy -- when chance of 2 or more events both occurring is given a higher chance than the individual events.

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3. Anchoring –risk perception can be distorted by providing an “anchor” or reference point Example: chance of nuclear war Questionnaire: (a) Do you think the chance of a nuclear war is higher or lower than 1% (b) Do you think the chance of a nuclear war is higher or lower than 90% Problem: conservatism in probability revision

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4. Forgotten Base Rates –ignoring information concerning likelihood of an event Example: Physicians and Rare Diseases Situation: -- a patient may have a rare disease -- diagnostic test is positive -- what is chance the patient has the disease? Doctors often over-estimate the chance that the patient has the disease - in one such study the doctors’ estimates were 10 times too high

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5. Overconfidence/Optimism –psychologists have found that people tend to have subjective probabilities about themselves that are unrealistically optimistic Example: “It’ll never happen to me” -- leads to foolish risk taking

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6. Gambler’s Fallacy -- “I’m due”

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