Doing the impossible: A note on induction and the experience of randomness

Abstract

The process of induction is formulated as a problem in detecting nonrandomness or pattern against a background of randomness or noise. Experimental approaches that have been taken to evaluate the rationality of human conceptions of randomness are described, and the narrow conceptions of randomness implicit in this experimental literature are contrasted with the broader and less well agreed upon conceptions of randomness in philosophy and mathematics. The relation between induction and the experience of randomness is discussed in terms of signal detection theory. It is argued that an adequate evaluation of human conceptions of randomness must consider the role those conceptions play in inductive inference. (36 ref) (PsycINFO Database Record (c) 2012 APA, all rights reserved)

Full text

Journal
of
Experimental
Psychology:
Learning,
Memory,
and
Cognition
1982,
Vol.
8, No. 6,
626-636
Copyright
1982
by the
American Psychological Association, Inc.
0278-7393/82/0806-0626S00.75
Doing
the
Impossible:
A
Note
on
Induction
and the
Experience
of
Randomness
Lola
L.
Lopes
University
of
Wisconsin-Madison
The
process
of
induction
is
formulated
as a
problem
in
detecting nonrandomness
or
pattern against
a
background
of
randomness
or
noise.
The first
section
of the
article describes
the
experimental approaches
that
have been taken
to
evaluate
the
rationality
of
human conceptions
of
randomness.
The
second section con-
trasts
the
narrow conceptions
of
randomness implicit
in
this experimental lit-
erature with
the
broader
and
less well agreed upon conceptions
of
randomness
in
philosophy
and
mathematics.
The
third
section
discusses
the
relation
between
induction
and the
experience
of
randomness
in
terms
of
signal detection theory.
And
the
fourth
section argues that
an
adequate evaluation
of
human conceptions
of
randomness must consider
the
role those conceptions play
in
inductive
in-
ference.
In
standardized tests
of
reasoning ability,
one
often
finds
questions like this:
What
digit should
go in the
space
at the end of the
series
below?
12233344445555566666?
Almost
certainly, test
makers
and
proficient test
takers
would
agree that
the
answer
is 6.
Another
question
that
might
be
asked
is the
following:
Below
are
three hypotheses concerning
the
source
of the
series above. Rank order
the
hypotheses
from
most
to
least
likely.
a.
The
test maker made
up the
series.
b.
The
series
is
digits
676,512
through 676,531
of
Rand's
One
Million Random Digits.
c.
The
series
is
digits 500,000 through
500,019
of
Rand's
One
Million Random Digits.
Although
the
question
is
unusual, most people
would
rank
the
hypotheses exactly
as
given.
The
series seems almost certainly
to
have been con-
structed
by
human agency,
but if by
some
ex-
traordinary coincidence,
it did
happen
to
occur
in
Rand's
(1955)
table
of
random
digits,
then
it
seems
far
more likely
to
have been
at
some rela-
tively
anonymous
position than exactly
in the
middle
of the
table.
Consider, however, what would happen
if the
This paper
was
facilitated
by a
grant
from
the
Wis-
consin Alumni Research Foundation
and was
presented
in an
earlier version
at the
Bayesian Research Confer-
ence,
Los
Angeles, California, February 1980. Thanks
are
due
Ward Edwards, Hillel
Einhorn,
Dominic Mas-
saro, Gregg Oden,
and
Charles Snowdon
for
their
helpful
criticisms
of an
earlier
draft
of the
manuscript,
and
spe-
cial thanks
are due
Mark
Kac for
allowing
me to
include
his
anecdote
about
the
draft
lottery.
same
two
questions were asked about another
se-
ries:
221215917917683158672
In
this case
it is not at all
clear
how the
series
ought
to be
completed,
nor is it
clear
how the
source hypotheses ought
to be
ordered.
Why do
we
have such
different
intuitions
about
these
two
series?
The
answer
is
obviously that
the first
series
has a
readily discernible pattern whereas
the
sec-
ond
does not.
It is
this pattern that underlies
our
mistaken
intuition that
the first
series
is
less likely
than
the
second
to be
generated
by a
random
pro-
cess.
And it is
also this pattern
that
underlies
and
enables
our
inductive inference
that
the first
series
should
end
with
a 6.
Induction
is how we
discover
for
ourselves what
the
world
is
like.
It
occurs when
we
generalize
past
experience with particular event patterns
to new
and as yet
unobserved
instances
(Harre,
1970).
Scientists
do it; lay
people
do it;
even birds
and
beasts
do it. But the
process
is
mysterious
and
full
of
paradox (cf. Gardner, 1976),
for as
Hume
(1748/1977)
showed long ago, induction cannot
be
justified
on
logical grounds:
No
matter
how
strongly
available evidence
may
seem
to
support
our
current
beliefs
about
the
world,
the
possibility
always
remains that
new
evidence will prove
us
wrong.
Thus,
to do
induction
is
necessarily
to run
the risk of
error,
and
this,
it
will
be
argued, makes
it
difficult
to
evaluate
how
well
the
process
is
being
done.
For if
induction
cannot
be
justified
ratio-
nally,
then what criteria
can be
used
to
judge
whether
it is
being done rationally?
For
human beings, induction
has two
relatively
distinct stages:
the act of
conceiving
a new
idea
or
theory
and the act of
testing
or
justifying
that
626