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”