No time for waiting: Statistical structure reflects subjective complexity (original) (raw)

Why are people bad at detecting randomness? A statistical argument

Journal of Experimental Psychology: Learning, Memory & Cognition.

Errors in detecting randomness are often explained in terms of biases and misconceptions. We propose and provide evidence for an account that characterizes the contribution of the inherent statistical difficulty of the task. Our account is based on a Bayesian statistical analysis, focusing on the fact that a random process is a special case of systematic processes, meaning that the hypothesis of randomness is nested within the hypothesis of systematicity. This analysis shows that randomly generated outcomes are still reasonably likely to have come from a systematic process, and are thus only weakly diagnostic of a random process. We tested this account through three experiments. Experiments 1 and 2 showed that the low accuracy in judging whether a sequence of coin flips is random (or biased towards heads or tails) is due to the weak evidence provided by random sequences. While randomness judgments were less accurate than judgments involving non-nested hypotheses in the same task domain, this difference disappeared once the strength of the available evidence was equated. Experiment 3 extended this finding to assessing whether a sequence was random or exhibited sequential dependence, showing that the distribution of statistical evidence has an effect that complements known misconceptions.

Why Streaks Are Special: The Time of Patterns

People seek for patterns and pay particular attention to streaks even when they are generated by a random process. The present paper examines statistics of pattern time in sequences generated by Bernoulli trials. We demonstrate that streak patterns possess some statistical properties that make them uniquely distinguishable from other patterns. Because of the uncontaminated continuity, streak patterns have the largest amount of self-overlap, resulting in the longest waiting time and the largest variance of interarrival times. We then discuss the psychological implications of pattern time such as in memory encoding and perception of randomness.

Occurrence and nonoccurrence of random sequences: Comment on Hahn and Warren (2009)

Psychological Review, 2010

On the basis of the statistical concept of waiting time and on computer simulations of the "probabilities of nonoccurrence" (p. 457) for random sequences, Hahn and Warren (2009) proposed that given people's experience of a finite data stream from the environment, the gambler's fallacy is not as gross an error as it might seem. We deal with two critical issues in Hahn and Warren's argument, a possible ambiguity in distinguishing the events of occurrence and nonoccurrence, and an incomplete consideration of the context in which the statistics of waiting time are defined. Our analyses show that the statistics of waiting time and the probabilities of nonoccurrence, once correctly interpreted, do not vindicate the error in the gambler's fallacy.

Expectations of clumpy resources influence predictions of sequential events

Evolution and Human Behavior, 2011

When predicting the next outcome in a sequence of events, people often appear to expect streaky patterns, such as that sport players can develop a "hot hand", even if the sequence is actually random. This expectation, referred to as positive recency, can be adaptive in environments characterized by resources that are clustered across space or time (e.g., expecting to find multiple berries on separate bushes). But how strong is this disposition towards positive recency? If people perceive random sequences as streaky, will there be situations in which they forego a payoff because they prefer an unpredictable random environment over an exploitable but alternating pattern? To find out, 238 participants repeatedly chose to bet on the next outcome of one of two sequences of (binary) events, presented next to each other. One sequence displayed events at random while the other sequence was either more streaky (positively autocorrelated) or more alternating (negatively autocorrelated) than chance. The degree of autocorrelation varied in a between-subject design. Most people preferred to predict purely random sequences over those with moderate negative autocorrelation and thus missed the opportunity for above-chance payoff.

Assessing the “bias” in human randomness perception

Human randomness perception is commonly described as biased. This is because when generating random sequences humans tend to systematically under-and over-represent certain sub-sequences relative to the number expected from an unbiased random process. In a purely theoretical analysis we have previously suggested that common misperceptions of randomness may actually reflect genuine aspects of the statistical environment, once cognitive constraints are taken into account which impact on how that environment is actually experienced. In the present study we provide a preliminary test of this account, comparing human-generated against unbiased process-generated binary sequences. Crucially we apply metrics to both sets of sequences that reflect constraints on human experience. In addition, sequences are compared using statistics that are shown to be more appropriate than a standard expected value analysis. We find preliminary evidence in support of our theoretical account and challenge the notion of bias in human randomness perception.

Randomization in individual choice behavior

Psychological Review, 1997

There is ample evidence that people cannot generate random series when instructed to do so. Rather, they produce sequences with too few symmetries and long runs and too many alternations among events. The authors propose a psychological theory to account for these findings, which assumes that subjects generate nonrandom sequences that locally represent theoretical random series subject to a constraint on their short-term memory. Closed-form expressions are then derived for the major statistics that have been used to test for deviations from randomness. Results from 3 experiments with 1 and 3 equiprobable alternatives support the model on both the individual and group levels. Our ability to discriminate between random and nonrandom events is fundamental to the process of induction, which, in turn, is essential for survival (Lopes, 1982). Without this ability we cannot detect successfully nonrandomness (signals) against a background of randomness (noise), discern patterns in human decision behavior (e.g., the controversy regarding the "hot hand" effect in basketball discussed by Gilovitch, Vallone, & Tversky, 1985), and recognize departures from chance probability to perceive causal relationships between events (Ayton & Wright, 1987). Numerous studies of subjective randomization, which are mostly concerned with our ability to discriminate between random and nonrandom events or produce random sequences when specifically instructed to do so, have concluded that this cognitive ability is seriously limited. To illustrate the findings that these studies typically reveal, consider an experiment reported by Green (1982), who used a recognition task (Bar-Hillel & Wagenaar, 1991) in which subjects are presented individually with predetermined sequences of binary events and asked to choose the ''truly random sequences" or rate all the series according to their "degree of randomness." Green presented a very simple coin-tossing problem to 2,930 English school pupils aged 11 to 16 years. The problem stated that a teacher asks each of 2 female students, named Clare and Susan, to toss a coin many times and record every time whether the coin landed heads (1) or tails (0). Two sequences of 150 Os and Is were then presented, one produced by Clare and the other by Susan. The students were informed that ' 'one girl performed the task honestly, by actually tossing the coin 150 times," whereas the other girl "cheated and just made it up.'' They were then asked to inspect the two sequences, decide which student cheated, and specify the reasons for their decision. Green stated five hypotheses regarding different properties

Disentangling the effects of alternation rate and maximum run length on judgments of randomness

2011

Binary sequences are characterized by various features. Two of these characteristics---alternation rate and run length---have repeatedly been shown to influence judgments of randomness. The two characteristics, however, have usually been investigated separately, without controlling for the other feature. Because the two features are correlated but not identical, it seems critical to analyze their unique impact, as well as their interaction,

Perception of Randomness: Subjective Probability of Alternation

We present a statistical account for the subjective probability of alternation in people's perception of randomness. By examining the spatio-temporal distances between pattern events, specifically, the frequency and delay of binary patterns in a Markov chain, we obtain some normative measures to calibrate people's expectation of randomness. We suggest that it can be fruitful to study subjective randomness in the context of human object representation and perception of time and space.

Fair coins tend to land on the same side they started: Evidence from 350,757 flips

arXiv (Cornell University), 2023

Many people have flipped coins but few have stopped to ponder the statistical and physical intricacies of the process. In a preregistered study we collected 350,757 coin flips to test the counterintuitive prediction from a physics model of human coin tossing developed by Diaconis, Holmes, and Montgomery (DHM;. The model asserts that when people flip an ordinary coin, it tends to land on the same side it started-DHM estimated the probability of a same-side outcome to be about 51%. Our data lend strong support to this precise prediction: the coins landed on the same side more often than not, Pr(same side) = 0.508, 95% credible interval (CI) [0.506, 0.509], BF same-side bias = 2359. Furthermore, the data revealed considerable between-people variation in the degree of this same-side bias. Our data also confirmed the generic prediction that when people flip an ordinary coin-with the initial side-up randomly determined-it is equally likely to land heads or tails: Pr(heads) = 0.500, 95% CI [0.498, 0.502], BF heads-tails bias = 0.182. Furthermore, this lack of heads-tails bias does not appear to vary across coins. Additional exploratory analyses revealed that the within-people same-side bias decreased as more coins were flipped, an effect that is consistent with the possibility that practice makes people flip coins in a less wobbly fashion. Our data therefore provide strong evidence that when some (but not all) people flip a fair coin, it tends to land on the same side it started. Our data provide compelling statistical support for the DHM physics model of coin tossing. 1 Some even assert that a biased coin is a statistical unicorn-everyone talks about it but no one has actually encountered one . Physics models support this assertion as long as the coin is not bent or allowed to spin on the ground .