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
::
::
::
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”
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