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Journal of Experimental Psychology:
Learning, Memory, and Cognition
1987,
Vol. 13, No.
3,392-MX)
Copyright! 987 by the American Psychological Association, Inc.
0278-7393/87/SOO.7S
Distinguishing Between Random and Nonrandom Events
Lola L. Lopes and Gregg
C.
Oden
University of Wisconsin—Madison
Subjects judged whether binary strings had been generated by a random or a noarandom process.
Half of the strings
were
generated by a Bernoulli process with p ~
.5.
The other half were generated
by
either
a
repetition-biased process
or an
alternation-biased
process.
Subjects
were (a)
not informed
about the nonrandom process, (b) informed about the qualitative nature of the process, or
(c)
given
accurate feedback after each trial about the generating process. The data show that subjects equate
long runs and symmetry with nonrandomness, and high rates of alternation with randomness, mak-
ing them less successful in detecting alternation-biased processes. The data also show that perfor-
mance can
be
improved
by
instructions or feedback.
A
second experiment
using
statistically sophisti-
cated subjects showed that although they perform better than naive subjects, their data are similar
qualitatively. We interpret these results in terms of whether the subject must perform the task in a
null hypothesis mode or
a
maximum likelihood mode.
It is well known that
people are
not naturally good at generat-
ing random sequences (for reviews, see Tune, 1964; Wagenaar,
1972),
Compared with real random sequences,1 human-pro-
duced sequences have too few symmetries and long runs, too
many alternations among events, and too much balancing of
event frequencies over relatively short regions. Although vari-
ous interpretations exist for these results, a case can be made
that the differences stem, at least in part, from a bias to expect
that things that
are
random will
look
random more consistently
than they really do. Thus, taking coin tossing as an example,
the
sequence HTHTTHHT
looks more
random than either HTHH-
HHHH or HTHTHTHT because it has equal numbers of heads
and tails and also because it has no easily discemable pattern.
Support for such a conceptual view can be found in studies of
the judged randomness of sequences (e.g., Falk, 1981; Teigen,
1983;
Wagenaar, 1970).
Outside the laboratory, people have little need to generate
random strings, but they often perform the inductive task of
deciding whether an unexpected event is merely coincidental
(i.e.,
random), or whether it signals a new or hitherto unrecog-
nized regularity in the environment (Lopes, 1982). Consider
Guildenstern in the play Rosencrantz and
Guildensiern
Are
Dead
as he confronts such an event. Eighty-nine times he has
tossed a coin—albeit a different coin each time—and 89 times
it has landed heads. Should Guildenstern accept that this is
merely, as he says, "a spectacular vindication of
the
principle
that each individual coin spun individually is as likely to come
down heads
as
tails and therefore should
cause
no surprise each
This research was supported by Wisconsin Alumni Research Foun-
dation Grant
100691
to Lola
L.
Lopes and
National Science Foundation
Grant BNS80-143I6 to Gregg
C.
Oden.
We
thank Josh Klayman for having introduced us to
Tom
Stoppard's
play, and Michael Waterman and
P.
Revesz for helping us obtain infor-
mation on the Varga teaching experiment.
Correspondence concerning this article should be addressed to Lola
Lopes, Department of Psychology, University of Wisconsin, Madison,
Wisconsin 53706.
individual time it
does"
or is he correct to wonder if this is "in-
dicative of something . . . un-, sub-, or supernatural?" (Stop-
pard, 1967, Act I, Scene 1).
Lopes (1982) has argued that such situations can be usefully
studied in a signal detection framework. An observer is pre-
sented with candidate strings and asked to decide, for each
string, whether it has been generated by a random or a nonran-
dom
process.
A
known proportion of the strings are in fact gen-
erated by a random process, and the rest are generated by a
process that differs in some way from the random process. The
observer may or may not have information about the character
of the nonrandom process. Observers are correct if they judge
a string produced by the nonrandom process (the signal) to be
nonrandom (a hit) or if they judge a string produced by the
random process (the noise) to be random (a correct rejection).
They are incorrect if they judge a string produced by the ran-
dom process to be nonrandom (a false alarm) or a string pro-
duced by the nonrandom process to be random (a miss).
There are two basic approaches to such tasks, the choice of
which depends on how much
the
observer
knows
about
the
non-
random process. The simpler of the two is what we will call
the maximum likelihood (MH) mode. This requires that the
observer have full or partial knowledge of the characteristics of
both the random and the nonrandom
process.
Such an observer
can use this knowledge to decide for each candidate string
whether it
is
closer to what would be expected from the random
process or what would be expected from the nonrandom pro-
cess.
For a sophisticated observer with full statistical knowledge
of the alternative processes, this would amount to
using
a maxi-
1 It
is
not easy to
say
what
a
"real"
random sequence
is.
Although one
might suppose it to be a sequence produced by a real random process,
one cannot know that a real process is random without inferring that
fact from the statistical characteristics of the sequences it
produces.
For
our purposes, it is not necessary to solve this definitional problem. In-
stead,
we
will use the terms random
process
and
randomness
to refer to
the properties expected theoretically of stationary Bernoulli processes.
392
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