Abstract: Many daily life events, from lotteries to coincidental encounters, occur partly or entirely randomly or “by chance.” Six experiments, in two different languages, explored how perceptions of randomness are related to the perceived probability of the same events—specifically, whether low-probability events were viewed as more random than similar events that were judged (rightly or wrongly) to be more likely. The experiments suggest that low-probability outcomes of stochastic events are indeed considered as being more random than medium and highly likely outcomes, even when all are produced by a “blind” (hence random) process. Degree of randomness involved in catching a bus was inversely related to the subjective probability estimates of the same event, both for correct and incorrect estimates. Unlikely coincidences were perceived to be more random than the same events presented in a more likely frame. The outcome of a match between two soccer teams was deemed to be more random when the weaker team wins than when the stronger team wins. Only extremely deviant outcomes—for instance, a top student who fails on two successive exams—made some people reject the randomness account, presumably believing that such extreme events must have a causal explanation. We conclude that people generally associate randomness with low-probability events, indicating outcomes that “cannot be predicted.”
Check also Are random events expected to be small? Karl Halvor Teigen, Alf Børre Kanten. Psychological Research, September 30 2019. https://www.bipartisanalliance.com/2019/09/randomness-and-related-concepts-events.html
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General discussion
The present set of studies constitutes the first attempt to examine empirically how perceived randomness of singular events
is related to their judged probability. We reviewed arguments
for different, seemingly plausible accounts for such a relationship, but subsequent experiments offered most consistent support for the low-probability account: Low-probability events
will be perceived as more random than comparable outcomes
that are estimated (rightly or wrongly) to have a higher probability of occurrence. This supports the more informal observations by Shanahan and Porfeli (2006) and Jolfaee et al. (2014),
and is congruent with the fact that real-life stories about random happenings, particularly coincidences, are typically illustrated by descriptions of low-probability events (Bandura,
1982; Johansen & Osman, 2015).
The results of the first four experiments all supported the
low-probability hypothesis rather than a process account, endorsed by scientists who think that randomness “objectively”
depends upon the generating mechanism (Fitelson &
Osherson, 2015; Lecoutre et al., 2006; Nickerson, 2002).
Experiment 5, which compared probability estimates and perceived randomness of selected soccer results, indicated a potential boundary condition for this rule. Results that were considered highly likely were again perceived as being less random than those that were not considered so likely, whereas
exceptionally extreme unlikely results turned out to be ambiguous, as participants were split into those who thought that
such results indicated a very high degree of randomness and
those who thought that such results could not be due to
chance. This split was replicated in Experiment 6, where very
deviant exam grades were viewed as being either more random (because they were unusual) or less random (because
there had to be a reason). However, for low-probability events
to appear nonrandom a nonrandom explanation has to be
available. Thus, unexpected grades (Experiment 6a) are easier
Table 4 Mean probability estimates and randomness scores (with mean
absolute deviations [MAD]) for poor exam grades and lottery wins,
Experiments 6a and 6b
Scenario Probability (1–7) Randomness (1–7)
Mean score Mean score MAD
Exam scenario (6a)
Two poorer grades (Ann) 3.34 a 3.16 a 1.36 a
Two failed grades (Carol) 1.93 b 3.24 a 2.02 b
Lottery scenario (6b) Objective probability
One win 1/20 6.29 b 0.88 c
Two wins 1/400 6.11 b 1.06 d
Note. Numbers with different subscripts in each column are significantly
different from each other (p < .001 for Experiment 6a and p < .05 for
Experiment 6b)
perceived as nonrandom than unexpected lottery results
(Experiment 6b).
Comparing our results with randomness judgments performed within the binary sequence paradigm, a striking contrast can be observed. In these studies patterns were judged
less random when they were perceived as unlikely outputs
from a series of tosses by a coin. In Falk and Konold’s
(1997) studies, ratings of “apparent randomness” were in fact
obtained by asking participants about the likelihood of achieving particular sequences randomly, and then concluding that
unlikely outcomes meant not random. Our studies indicate the
opposite—namely, that for singular events in daily life,
unlikely indicates more random. These differences suggest
that studies of binary sequences have limitations as a universal
model of the perception of randomness (see also Matthews,
2013, on the evaluation of streaks of different kinds).
These evaluations are not as incompatible as they may
seem. Judgments of randomness in sequences assume the existence of a random generator producing inchoate strings,
where order is “surprising” (Feldman, 2004) and anomalous.
Participants in these tasks make judgments based on an assumption of disorder, where observations of irregularity are
expected and, in a way, considered as default. Outside of this
rather artificial universe, however, people will look for, and
expect, some degree of predictability and order. They will find
irregular events to be the exception rather than the rule, and
only draw conclusions about randomness in the “unlikely”
case of aberrant events that cannot be predicted. By this epistemic attitude they manage to preserve a model of the world
as basically orderly and explainable.
In the present studies, we have compared participants’ solutions to a task of predicting outcomes (probability estimates)
with a task of postdicting hypotheses (randomness ratings). In
terms of conditional probabilities, this changed their task from
considering p(data | H) to expressing their opinions about the
inverse relationship p(H | Data). To do so in a meaningful way,
one needs to have an idea not only about p(data | H), but also
about the prior probability of potential alternative hypotheses,
and how compatible the actual outcomes are with both alternative hypotheses, as required by Bayes’s theorem. It is reasonable to assume that a search for alternative hypotheses will
emerge when p(data | H) is quite low, unless the process is so
well described that the role of other contributing causes can be
controlled for. To illustrate, the winner of the lottery in
Experiment 1a had only 10% chance, yet the fact that he
won could not rule out randomness, as the process of a blind
draw did not allow the winning to be explained in any other
way. This was replicated in Experiments 1b and 6b with even
lower probabilities. In contrast, the factors responsible for the
outcome of a soccer match include situational determinants
and skills, making it easier to produce a narrative that highlights factors different from randomness. So, when an extraordinary and unlikely 5-0 outcome happens, some participants
decided to look for potential reasons that made this result
appear more plausible, and hence less random than before.
These apparently divergent judgments do not invalidate the
link between low probability and high perceived randomness,
as they may simply stem from an attempt to think that extraordinary outcomes must be due to an overriding cause, making
them more likely (and less random) than originally assumed.
A similar process can be observed when exceptional coincidences are “explained” by recourse to magical or supernatural
forces. They then become more likely (and not random) by
means of an unlikely (magical) theory of coincidences
(Griffiths & Tenenbaum, 2007).
We do not claim that all low-probability events will be
regarded as random, and certainly not that probability is the
only determinant for attributing an outcome to randomness, or
chance. Other potential determinants of randomness could be
examined by manipulating other variables—for instance,
causal factors, skill, intentionality, and effort. Our results indicate that low probabilities constitute one (supposedly important) facet in the perception of randomness, though we do not
claim that it is the only one.
Why should low p events be regarded as more random? Low
p events need not be uncaused or unintentional, and even quite
infrequent happenings (like Halley’s comet appearing once in
75 years) can form a pattern. We propose that the answer may
reside in two related characteristics of subjective randomness.
People consider an event as random if it appears as
disconnected from the general flow of events, like an unmotivated cough in the middle of a sentence or an unexpected computer crash before the document is saved. In these cases, the
“random” events cannot be conceived as causally related to the
main story line. It is also in the nature of such events to be
unforeseeable. They are deviant and form exceptions to the rule.
By this logic, it was less foreseeable (and more random) that
Karl in Experiment 1 drew a winning marble from his 10% urn
than John did, as John’s urn contained 90% winning marbles.
Indeed, unpredictability has been suggested as an optional
definition of randomness—for instance, as independence (zero autocorrelations) between the parts of a random sequence,
where one part cannot be predicted from preceding parts
(Neuringer, 1986; Nickerson, 2002). Also, in the case of blind
draws from an urn with unequal frequency of different colors,
all marbles (but not all colors) have the same probability of
being chosen, implying unpredictability on the level of individual marbles (but not on the level of colors). Our participants appear to go one step further and reason that degree of
randomness depends on outcome features: a drawing of the
dominant color is less random, by being more expected than
other colors.
The present studies did not instruct participants as to what
should be meant by a random or a nonrandom outcome. This
was done deliberately, in order to avoid directing our respondents towards a specific interpretation of the term. Like
probability, randomness is a polysemous concept (Hertwig &
Gigerenzer, 1999), having multiple, related meanings.
Accordingly, we do not claim that this term was used by all
participants in exactly the same way. Notwithstanding, the
results showed a remarkable consistency of low-probability
events being rated as being more random across a wide range
of situations with speakers of two different languages.
The idea that random events are rare can have several important implications. One is exaggerated beliefs in foreseeability and control, as indicated by research on the hindsight bias
(Roese & Vohs, 2012). It might reinforce a preference for
intentional rather than accidental explanations of behaviour
as demonstrated by studies of the “intentionality bias”
(Reich, Kupor & Smith, 2018; Rosset, 2008). It may lead
historians, social scientists, and psychologists to underestimate the role of randomness in shaping individual and collective history (Krantz, 1998; Sunstein, 2015), and make them
look for patterns, plans, and explanations behind phenomena
that cannot be adequately attributed to single causes or to a
purposeful design.
An association between randomness and low probability
may suggest that random events are in themselves
insignificant and do not, as a rule, give rise to important
changes. As John Stuart Mill (1856) observed, people tend to
think that effects share important characteristics with their
causes; large effects are supposed to have large causes, and evil
effects are supposed to flow from evil forces (Nisbett & Ross,
1980). By a similar logic, one may believe that random and
unlikely events can be taken lightly, if they have only in their
power to produce slight and insignificant consequences. What
people think about the “magnitude” of random events and their
potential causal power has been recently been examined by
Teigen and Kanten (2019). Their perceived low probability
suggests that they can be easily ignored or discounted as “exceptions.” Correspondingly, one rarely stops to consider whether frequent or more noteworthy events could be due to chance.
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