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
probability calculus?
according
to
“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
Both false:
not-G
B
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
girl
girl
boy
boy
2nd born
girl
boy
girl
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:
¬ Guilty
Frequencies
1
999,999
DNA matches
. . .
and flesh them out:
¬ Guilty
Guilty
DNA matches
DNA matches
Suppose the PARTITION is:
¬
Guilty
DNA matches
¬
Guilty
¬
DNA matches
Guilty
DNA matches
Guilty
¬
DNA matches
p(DNA matches/not guilty)
BUT: p(not guilty/DNA matches)
=
=
Frequencies
1
999,999
Frequencies
1
999,999
9
0
1 in million
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
probabilities. How do they do it?
posterior
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.