Misconceptions and Fallacies Concerning Probability Assessments.
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Transcript of Misconceptions and Fallacies Concerning Probability Assessments.
Misconceptions and
Fallacies Concerning
Probability
Assessments
Law of Averages Toss coin many times. What happens
as number of tosses increases?
Law of Averages Says: The percentage of heads should
become very close to 50%.
John Kerrich’s Coin-tossing Experiment
0.4
0.5
0.6
10 100 1000 10000
Number of Tosses
Proportion Heads
Law of Averages
IN GENERAL
Law of Averages says:
Averages and proportions vary less from the “expected” as sample size increases
Two Misconceptions
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?
2 times
OR 100 times
In fact:
As # of tosses increases, the chance of “exactly 50% Heads” decreases
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:
-10
10
30
50
70
10 100 1000 10000
Number of Tosses
# Heads - Half the # of Tosses
# ofTosses
Probability of between49.9% 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
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.
2. How Does the Law of Averages Work?
By “compensation” or “adjustment” ? NO!
Kerrich example: 130 instances of “HHHH” next toss:
69 H’s
61 T’s
(no adjustment)
– gambler’s fallacy
!
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
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
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 ofdiscrimination and social justice, and she alsoparticipated 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.
Conjunction Fallacy
-- when chance of 2 or more events both occurring is given a higher chance than the individual events.
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
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
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
6. Gambler’s Fallacy
-- “I’m due”