Psy/Orf 322: PROBABILISTIC THINKING

• We think about probabilities, because actions depend on what is likely to happen.

• Theorists disagree about how we make these inferences.

• Probability calculus: self-evident rules, e.g. extensional notion that prob(event) = sum of probabilities of different ways in which it can occur.

• Do people reason according to probability calculus?

“Someone with only the most modest knowledge of probability mathematics could have won himself the whole of Gaul in a week.” -- Ian Hacking (1975)


SOME SIMPLE JUDGMENTS

1. In the U.S. which is most probable: death in automobile accident, by stroke, or by stomach cancer?

2. What’s the probability of a civil war in Iraq?

3. In a box, there is a red marble or a green marble, or both. What’s the probability that there is both the red and the green marble in the box?


THE MEANING OF PROBABILITY

• What do such assertions mean:

probability (civil war in Iraq) = 0.6?

In what circumstances would it be true? or false?

• Philosophers argue seriously about

interpretation of probabilities:

subjective belief

(assertion above is sensible)

limit on a relative frequency

(assertion above is meaningless)

partial logical entailment (?)


‘Naive’ performance

• How do you infer probability of: death in auto accident, by stroke, or by stomach cancer?

Correct answer:

stroke > stomach cancer > auto accident

Method: use available evidence, e.g. frequency in media (use of heuristics studied by Tversky and Kahneman).

• How do you infer probability of: red & green marble in box?

Method: rules or models.


MENTAL MODELS & PROBABILITIES

Three assumptions:

1. Truth: people use what’s true, not false [last lecture].

2. Equiprobability: if no information to the contrary, each model represents an equiprobable alternative.

Cf. Laplace’s ‘indifference’ over events by which he proved that the odds that the sun will rise to-morrow are 1,826,214 to 1.

3. Proportionality: p(event) = proportion of models in which it occurs.


A problem

If one of the following assertions is true then so is the other:

A green if and only if a blue.

There is a green.

Which is more likely to be in the box, green or blue?

90% say: equiprobable.

It’s an illusion!

Both assertions true: G B

Both false: not-G B

\ Blue more probable than green.

• Moral: people use models, not (valid) formal rules from probability calculus.


A problem

• Phil has two children. One is a girl. What’s the probability that the other is a girl?

Most people say: approx 1/2

Conditional probability:

prob(A/B)

i.e. probability of A, given that B is the case.

Correct answer:

1st born 2nd born

girl girl

girl boy

boy girl

boy boy

\ prob(other is girl/one girl) = 1/3

• Why do people go wrong?

FIRST ERROR IN REASONING ABOUT CONDITIONAL PROBABILITIES

• Failure to detect that question is about conditional probability, as opposed to simple probability. Hence, inappropriate models for problem:

girl

boy

• prob(A) = 1/2, and prob(B) = 1/2.

What is probability of A and B?

Answer depends on p(A/B):

prob(A & B) = p(A)p(B/A)

or equivalently = p(B)p(A/B)

Because p(A)p(B/A) = p(B)p(A/B), we have:

p(B/A) = p(B)p(A/B) [Bayes’s theorem]

p(A)


A PROBLEM

The suspect’s DNA matches the crime sample. The probability of a DNA match is 1 in a million if the suspect is not guilty. Is the suspect likely to be guilty?

Why do people go wrong?

p(DNA matches/not guilty) = 1 in a million

They build models with frequencies:

Frequencies

¬ Guilty DNA matches 1

. . . 999,999

and flesh them out:

Frequencies

¬ Guilty DNA matches 1

Guilty DNA matches 999,999

Suppose the PARTITION is: Frequencies

¬ Guilty DNA matches 1

¬ Guilty ¬ DNA matches 999,999

Guilty DNA matches 9

Guilty ¬ DNA matches 0

p(DNA matches/not guilty) = 1 in million

BUT: p(not guilty/DNA matches) = 1 in 10

• SECOND ERROR: hard to hold all models in mind.


BAYESIAN INFERENCE

• One bag contains 70 red and 30 blue chips; another bag contains 30 red and 70 blue chips. One bag chosen at random. From it, 12 chips are selected at random with replacement. Result: 8 red and 4 blue chips.

What’s prob that ‘70 red’ bag was chosen?

People’s estimates average around: 0.8

Bayes’s theorem: 0.967

Moral: people don’t use Bayes’s theorem to infer posterior probabilities.

• But, people can infer posterior probabilities. How do they do it?

CONCLUSIONS

• Naïve individuals have some ability to reason about probabilities.

• It appears to be based, not on the probability calculus, but on mental models.

• Frequencies make the arithmetic easier; but no evidence for an innate module for reasoning about frequencies.