Perception of Randomness: Subjective Probability of Alternation

Perception of Randomness: Subjective Probability of Alternation Yanlong Sun (Yanlong.Sun@uth.tmc.edu) Hongbin Wang (Hongbin.Wang@uth.tmc.edu) School of Biomedical Informatics, University of Texas Health Science Center Houston, TX 77030 USA Abstract We present a statistical account for the subjective probability of alternation in people’s perception of randomness. By ex- amining 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. Keywords: subjective randomness; probability of alternation; waiting time; perception of time and space. Introduction Much is known that subjective randomness—people’s intu- itive judgment on how an event or a series of events appears random—systematically deviates from the stochastic random- ness described by normative probability theories. Among many statistics describing such discrepancy, the probability of alternation (p A ) has been the most extensively studied in psychology literature (e.g., Budescu, 1987; Falk & Konold, 1997; Kahneman & Tversky, 1972; Kareev, 1992; Lopes & Oden, 1987; Nickerson, 2002; Sanderson, 2011). In a binary sequence generated by independent and stationary Bernoulli trials (for example, repeatedly tossing a fair coin), p A can be defined as the probability that the outcome of any single event is different from the preceding one. If the process is truly ran- dom (e.g., the same fair coin is being tossed independently), the probability of alternation has the expected value p A = .5. However, reviewed by Falk and Konold (1997, Table 1, p.304), in almost all of the studies with the tasks of recognizing or generating randomness, the modal subjective probability of alternation was approximately .60. That is, people tend to perceive sequences with p A ≈ .60 as the most random and generate random sequences with p A ≈ .60. One particular reason that the probability of alternation p A receives a great deal of attention in the studies on subjective randomness is that it is highly correlated with many other sequential statistics, such as the runs test and serial correlation. Together, these statistics cover a variety of empirical phe- nomena, for example, the perception of streaks in basketball shooting (Burns, 2004; Gilovich, Vallone, & Tversky, 1985; Sun & Wang, 2010b), the recency effect (Ayton & Fischer, 2004), the working memory capacity and detection of covari- ances in short sequences (Kareev, 1992), and the encoding of subjective complexity (Falk & Konold, 1997; Falk, Falk, & Ayton, 2009). (For a recent review, see Oskarsson, Van Boven, McClelland, & Hastie, 2009). To explain the biased probability of alternation in subjec- tive randomness, Falk and Konold (1997) developed a “diffi- culty predictor” (DP) as a measure of “subjective complexity”. Based on the concept that random sequences are irreducibly complex (i.e., algorithmic complexity, Kolmogorov, 1965), Falk and Konold propose that people’s sense of randomness is not based on the deviations from the equiprobability of pat- terns of the same length (i.e., “n-grams”), rather, it may be based on the difficulty level when people attempt to memo- rize or copy a sequence by its minimal description. Given any binary sequence, the difficulty predictor is defined by adding twice the number of alternating runs to the number of pure runs. For example, the following sequence is partitioned into 5 segments, where pure runs (streaks) are underlined and alternating runs are double underlined: H H T H T H T H T H T T T H H T H T H T H Thus, the DP score for this particular sequence is 1 + 2 + 1 + 1 + 2 = 7. By this measure, a perfect streak would be per- ceived as the most nonrandom because of its lowest DP score (i.e., the easiest to remember). In contrast, sequences with more alternating runs—hence greater p A —are more difficult to encode therefore tend to be perceived as more random. Overall, it has been demonstrated that DP correlates remark- ably well with participants’ ratings of randomness, memo- rization time, assessed difficulty of memorization, and copy- ing difficulty. And, the mean ratings of randomness show the classic preference for over-alternating sequences (Falk et al., 2009; Falk & Konold, 1997). However, Griffiths and Tenenbaum (2003) point out that DP remains a subjective mea- sure since its objective counterpart, algorithmic complexity, is not computable. Instead, Griffiths and colleagues propose to use Bayesian inferences to account for subjective random- ness (Griffiths & Tenenbaum, 2001, 2003; Hsu, Griffiths, & Schreiber, 2010). By this account, the subjective randomness of a particular sequence X is defined as random(X) = log p(X|random) p(X|regular) Then, the problem of judging randomness can be reduced to comparing two probabilities—whether the sequence is pro- duced by a random process (e.g., independent and stationary Bernoulli trials), or, by a process with some regularities. To specify p(X|regular), Griffiths and Tenenbaum (2003) de- velop a hidden Markov model that makes transitions between hidden states depending on whether a motif is to be repeated or altered, where a motif is a short pattern such as H, T, HT, or, TH. By maximizing p(X|regular), they obtain a set of param- eters that provide a better fit to the mean randomness ratings reported by Falk and Konold (1997). The difficulty predictor and the Bayesian account have the