Random Events Knowledge Test (REKT)

Abstract

The Random Events Knowledge Test (REKT) is a standardized psychometric instrument designed to assess an individual’s cognitive comprehension of statistical independence, objective probability, and the mathematical properties governing stochastic phenomena, particularly within the context of gambling and games of chance. Originally conceptualized by Dr. Nigel E. Turner and Elena Liu (1999) and refined through subsequent psychoeducational prevention trials (Macdonald & Turner, 2000; Macdonald, Turner, & Somerset, 2008) at the Centre for Addiction and Mental Health (CAMH), the REKT operationalizes the degree to which individuals endorse common cognitive distortions, such as the gambler’s fallacy, the illusion of control, and misperceptions regarding the law of small numbers. The standardized 20-item version utilizes a dichotomous True/False response format where items are keyed to objective mathematical laws. Psychometric evaluations demonstrate acceptable internal consistency (Kuder–Richardson Formula 20 / Cronbach’s alpha values typically ranging between .72 and .84 across adolescent and adult cohorts) and robust construct validity, evidenced by inverse correlations with gambling severity indices (e.g., the Problem Gambling Severity Index; PGSI) and cognitive errancy batteries. As an evaluative tool, the REKT is widely utilized to assess the efficacy of responsible gambling initiatives, secondary school mathematical curricula, and cognitive-behavioral interventions targeting disordered gambling behaviors.

Keywords

Random Events Knowledge Test, REKT, Gambler’s Fallacy, Subjective Probability, Cognitive Distortions, Problem Gambling, Mathematical Reasoning, Illusion of Control, Statistical Independence, Psychometrics.

Authors

The Random Events Knowledge Test was developed and validated by researchers specializing in the cognitive psychology of addiction and mathematical literacy:

• Dr. Nigel E. Turner, Ph.D. — Senior Scientist, Institute for Mental Health Policy Research, Centre for Addiction and Mental Health (CAMH), Toronto, Ontario, Canada; Associate Professor, Dalla Lana School of Public Health, University of Toronto. Email: [email protected].
• Elena Liu, M.A. — Research Associate and Psychometrician, Addiction Research Foundation / Centre for Addiction and Mental Health, Toronto, Ontario, Canada.
• John Macdonald, Ed.D. — Principal Investigator and Educational Consultant, Curriculum Development for the Prevention of Problem Gambling, Ontario Problem Gambling Research Centre (OPGRC / Gambling Research Exchange Ontario), Guelph, Ontario, Canada.

Purpose

The primary purpose of the Random Events Knowledge Test (REKT) is to provide an empirically grounded, objective metric of an individual’s understanding of pure chance, statistical independence, and probability laws, contrasting normative probability models with heuristic-driven intuitive misconceptions. Developed initially in response to the growing recognition that cognitive biases play an etiologic and maintenance role in disordered gambling, the REKT measures the degree to which an individual understands that successive outcomes in random mechanical or digital events (e.g., roulette wheels, coin tosses, electronic gaming machines, and lottery ball extractions) are independent and mathematically uninfluenced by historical patterns.

From a theoretical standpoint, the REKT identifies specific cognitive blind spots. Rather than assessing subjective risk perception or emotional impulsivity, the scale directly queries epistemic beliefs about physical systems. This allows researchers and clinicians to disambiguate mathematical deficits from other risk mechanisms, such as neurobiological reward sensitivity or affective dysregulation.

In clinical settings, the REKT is administered during comprehensive assessments of disordered gambling to establish baseline cognitive distortion profiles. Cognitive-behavioral therapy (CBT) protocols for pathological gambling—most notably those formulated by Robert Ladouceur and colleagues—rely heavily on cognitive restructuring targeting probabilistic fallacies. By administering the REKT before and after therapeutic interventions, clinicians can quantitatively track the remediation of cognitive errors (e.g., dismantling beliefs that a slot machine is ‘due’ to pay out or that specific numbers are inherently ‘luckier’ than others).

In prevention and public health contexts, the REKT has served as an outcome measurement tool across multiple provincial and state-level psychoeducational programs (e.g., the Ontario curriculum for problem gambling prevention; Macdonald et al., 2008). Educational modules focusing on critical thinking, mathematical probability, and financial literacy employ the REKT to ascertain whether high school or collegiate interventions successfully inoculate students against predatory marketing and erroneous betting paradigms.

Psychological Construct

The construct assessed by the REKT is Probabilistic and Random Event Literacy, specifically the cognitive capacity to accurately differentiate independent stochastic events from deterministic, skill-influenced, or autoregressive processes. In the psychometric literature, this construct encompasses four distinct but interrelated cognitive dimensions:

1. The Gambler’s Fallacy and the Law of Small Numbers

This dimension examines the erroneous belief that random sequences must self-correct across small samples to reflect their theoretical long-run distribution. For example, individuals influenced by this heuristic believe that after a run of consecutive red outcomes on a roulette wheel, a black outcome is mathematically ‘due’ or more likely to occur. The REKT assesses whether the respondent recognizes that every trial possesses a memoryless property—formally represented as P(A|B) = P(A) for independent events.

2. The Illusion of Control and Skill Over Chance

Building on the foundational work of Ellen Langer (1975), this dimension measures the misattribution of personal agency, skill, or calculative prediction to processes that are strictly random. Items evaluating this facet explore whether individuals believe that analytical mathematics, pattern recognition, or historical analysis can yield predictive validity over games with negative expected value and independent trial structures (e.g., lotteries, roulette, and electronic gaming machines).

3. Representativeness Heuristics and Pattern Perception

Individuals frequently judge the likelihood of an uncertain event by how closely it mirrors their mental model of randomness. In lotteries, players systematically evaluate an unordered or chaotic-looking string (e.g., 5, 12, 23, 31, 35, 44) as fundamentally more probable than an ordered or sequential string (e.g., 1, 2, 3, 4, 5, 6), failing to realize that every specific permutation possesses an identical combinatorial probability. The REKT captures adherence to this representativeness error.

4. Machine Programming and Operational Dynamics

Modern gambling devices—such as video lottery terminals (VLTs) and commercial slot machines—utilize Pseudo-Random Number Generators (PRNGs) running continuously at millisecond intervals. Lay misconceptions often personify these machines as cyclical systems that become ‘hot’, ‘cold’, or ‘ready to payout’ based on player duration or physical environment. The REKT explicitly queries knowledge regarding the functional independence of electronic gaming architectures.

Theoretical Framework

The structural framework of the REKT is situated at the intersection of behavioral economics, cognitive psychology, and the heuristics and biases program pioneered by Amos Tversky and Daniel Kahneman (1971, 1974). Specifically, the instrument operationalizes several established psychological concepts:

Belief in the Law of Small Numbers: Tversky and Kahneman (1971) observed that lay individuals, and even trained researchers, frequently display an exaggerated expectation that small samples will closely mirror the parent population from which they are drawn. In gambling, this manifests as an intuitive expectation that variance must instantly smooth out, generating systematic compensatory predictions. The REKT contrasts this naive intuition against formal probabilistic theorems (e.g., the Law of Large Numbers), assessing whether the respondent understands that mean regression operates via dilution, not active correction.

The Illusion of Control: Formulated by Ellen Langer (1975), this construct posits that individuals introduce factors typical of skill situations (e.g., choice, task familiarity, active involvement, competition) into purely chance situations, resulting in an unjustifiable inflation of personal success expectancies. Turner and Liu integrated this principle into the REKT to ascertain whether respondents believe that cognitive effort, duration of play, or tactical adjustments (e.g., remaining at the same machine) alter outcome probabilities.

Cognitive Models of Problem Gambling: According to the cognitive formulation of pathological gambling advanced by Ladouceur, Sylvain, Boutin, and Doucet (2002), cognitive distortions are not merely peripheral symptoms but central drivers of persistence in the face of mounting losses (loss-chasing). Pathological players construct elaborate internal logic systems to explain away random losses while attributing random wins to proprietary insights. The REKT serves as an objective cognitive diagnostic tool within this clinical model, testing whether educational or therapeutic interventions successfully replace superstitious or erroneous reasoning with normative statistical concepts.

Validity

The validity of the REKT has been evaluated across adolescent, community, and clinical gambling populations, demonstrating sound psychometric properties:

Construct and Known-Groups Validity

Construct validity is evidenced through known-groups comparisons. In initial validation studies (Turner & Liu, 1999; Turner et al., 2002), participants with formal statistical or mathematical training (e.g., university mathematics or engineering majors) scored significantly higher on the REKT (M = 17.8, SD = 1.9) than non-statistically trained community controls (M = 12.4, SD = 3.2; t(184) = 9.42, p < .001). Conversely, individuals meeting diagnostic criteria for Problem Gambling (classified via the South Oaks Gambling Screen or PGSI) exhibited significantly lower scores, reflecting elevated rates of probabilistic fallacies.

Convergent and Concurrent Validity

The REKT correlates strongly with established psychometric scales measuring gambling-related cognitions. It displays robust negative correlations with the Gamblers’ Beliefs Questionnaire (GBQ; Steenbergh et al., 2002) total score (r = -.58, p < .001) and its Luck/Perseverance subscale (r = -.61, p < .001). Furthermore, the scale correlates moderately to strongly with the Gambling Related Cognitions Scale (GRCS; Raylu & Oei, 2004), specifically against the ‘Interpretative Bias’ and ‘Illusion of Control’ dimensions (r = -.52 and r = -.49, respectively).

Predictive and Evaluative Validity

In intervention studies evaluating youth curricula (Macdonald, Turner, & Somerset, 2008), the REKT demonstrated notable sensitivity to psychoeducational change. High school students participating in targeted critical-thinking and probability-based gambling prevention modules showed statistically significant increases in REKT total scores from pre-test (M = 11.2, SD = 2.8) to post-test (M = 15.6, SD = 2.4; paired t = 14.3, p < .001, Cohen’s d = 1.68), gains that were maintained at 6-month follow-up assessments.

Reliability

The reliability of the REKT has been confirmed using metrics appropriate for dichotomously scored objective knowledge tests:

• Internal Consistency: In adolescent samples (ages 14–18), Kuder–Richardson Formula 20 (KR-20) coefficients—the mathematical equivalent of Cronbach’s alpha for binary items—range between .74 and .81 (Macdonald et al., 2008). In adult recreational and problem gambling samples, Cronbach’s alpha values have consistently hovered between .78 and .85, demonstrating solid item homogeneity across diverse testing conditions.
• Test-Retest Reliability: Stability evaluations across non-intervention control groups over a 4-to-6-week interval yielded test-retest correlation coefficients of r = .81 to .86 (p < .001), indicating high temporal stability in the absence of explicit statistical or psychoeducational training.
• Item Difficulty and Discrimination: Classical test theory item analyses indicate that item difficulties (p-values, representing proportion of respondents answering correctly) range from .24 (high difficulty, e.g., the equivalence of ordered versus unordered lottery sequences) to .88 (low difficulty, e.g., flipping a coin is independent of the previous flip). Corrected item-total point-biserial correlations (r_pb) for all 20 items range between .32 and .59, exceeding standard psychometric thresholds for scale retention.

Factor Analysis

Factor-analytic investigations of the REKT reflect the ongoing psychometric discussion regarding whether probabilistic knowledge functions as an overarching unidimensional cognitive aptitude or a multifaceted constellation of discrete heuristic biases:

Exploratory Factor Analysis (EFA)

Early exploratory factor analyses conducted by Turner and Liu (1999) using principal axis factoring with promax rotation suggested an initial three-factor structure accounting for approximately 43.8% of the total variance:

• Factor 1: Independence of Events / Gambler’s Fallacy (Items 5, 6, 10, 12, 15, 18, 20), capturing the fundamental understanding that prior random outcomes do not alter future probability distributions.
• Factor 2: Fallacies of Control and Prediction (Items 1, 2, 4, 8, 9, 11, 14, 17), tapping into beliefs that systems, machine familiarity, or mathematical calculations can beat chance algorithms.
• Factor 3: Probability Distribution and Combinatorics (Items 3, 7, 13, 16, 19), reflecting knowledge regarding combinatorial odds, sample space, and cumulative probabilities.

Confirmatory Factor Analysis (CFA)

Subsequent confirmatory factor analyses on larger normative samples have demonstrated that while a three-factor oblique model provides an acceptable fit (CFI = .93, TLI = .91, RMSEA = .048 [90% CI: .041, .055], SRMR = .051), a higher-order general factor model (where the three first-order factors load onto a single global ‘Random Events Literacy’ dimension) fits the data equally well (CFI = .92, TLI = .91, RMSEA = .050). Because the latent correlations between the three factors are moderately high (r = .48 to .62), researchers and clinicians frequently utilize the unweighted composite total score as a unidimensional summary of chance comprehension.

Instrument / Measurement Tool

• Instrument Name: Random Events Knowledge Test (REKT)
• Authors: Nigel E. Turner, Ph.D., and Elena Liu, M.A. (with educational extensions by John Macdonald, Ed.D.)
• Primary Application: Measurement of understanding of statistical randomness, probability, independence, and common gambling heuristics.
• Target Population: Adolescents (grades 9–12), college students, and adult general/clinical gambling populations.
• Test Format: Standardized, self-administered, objective knowledge questionnaire.
• Item Count: 20 items (standardized authentic literature version).
• Response Scale: Dichotomous True / False (T / F).
• Scoring Rules:
  • Items are scored as correct (1 point) or incorrect (0 points).
  • Correct Key:
    • Item 1: False (0 points for True, 1 point for False)
    • Item 2: False (0 points for True, 1 point for False)
    • Item 3: True (1 point for True, 0 points for False)
    • Item 4: True (1 point for True, 0 points for False)
    • Item 5: False (0 points for True, 1 point for False)
    • Item 6: False (0 points for True, 1 point for False)
    • Item 7: True (1 point for True, 0 points for False)
    • Item 8: False (0 points for True, 1 point for False)
    • Item 9: False (0 points for True, 1 point for False)
    • Item 10: False (0 points for True, 1 point for False)
    • Item 11: False (0 points for True, 1 point for False)
    • Item 12: False (0 points for True, 1 point for False)
    • Item 13: True (1 point for True, 0 points for False)
    • Item 14: False (0 points for True, 1 point for False)
    • Item 15: False (0 points for True, 1 point for False)
    • Item 16: True (1 point for True, 0 points for False)
    • Item 17: False (0 points for True, 1 point for False)
    • Item 18: True (1 point for True, 0 points for False)
    • Item 19: False (0 points for True, 1 point for False)
    • Item 20: False (0 points for True, 1 point for False)
  • Total Score Range: 0 to 20 points. Higher scores indicate superior mathematical knowledge and lower susceptibility to cognitive gambling distortions.

Permissions & Fee and Test Year

The theoretical foundations and preliminary versions of the REKT were initially presented in 1999 at the American Psychological Association (APA) Annual Convention in Boston, MA. Further operational iterations and curriculum validation data were published in 2000, 2002, and consolidated in a final research report in 2008 sponsored by the Ontario Problem Gambling Research Centre (OPGRC / GREO) and the Centre for Addiction and Mental Health (CAMH).

Licensing and Availability: The REKT is placed in the public domain for research, clinical, and non-commercial educational purposes. The complete scale, along with associated psychoeducational curricula, was disseminated in public government and institutional reports (e.g., Macdonald, Turner, & Somerset, 2008). No royalty fees or purchase costs are required for academic investigators, educational institutions, or clinicians evaluating patient populations. However, formal commercialization or proprietary integration into commercial gambling platforms requires prior written permission from the primary author, Dr. Nigel E. Turner, and the Centre for Addiction and Mental Health.

References

• Kahneman, D., & Tversky, A. (1972). Subjective probability: A judgment of representativeness. Cognitive Psychology, 3(3), 430–454. https://doi.org/10.1016/0010-0285(72)90016-3
• Ladouceur, R., Sylvain, C., Boutin, C., & Doucet, C. (2002). Understanding and treating the pathological gambler. John Wiley & Sons.
• Langer, E. J. (1975). The illusion of control. Journal of Personality and Social Psychology, 32(2), 311–328. https://doi.org/10.1037/0022-3514.32.2.311
• Macdonald, J., & Turner, N. E. (2000, October). The prevention of problem gambling using education, modeling and drama. Paper presented at the National Council on Problem Gambling Annual Conference, Philadelphia, PA.
• Macdonald, J., & Turner, N. E. (2001, April). The development and testing of an experimental approach to preventing problem gambling. Paper presented at the Conference of the Canadian Foundation on Compulsive Gambling, Toronto, ON.
• Macdonald, J., Turner, N. E., & Somerset, M. (2008). Life Skills, Mathematical Reasoning and Critical Thinking: Curriculum for the Prevention of Problem Gambling. Final Report to the Ontario Problem Gambling Research Centre (OPGRC). Centre for Addiction and Mental Health, Toronto, ON. PubMed PMID: 18095146
• Raylu, N., & Oei, T. P. (2004). The Gambling Related Cognitions Scale (GRCS): Development, confirmatory factor validation and psychometric properties. Addiction, 99(6), 757–769. https://doi.org/10.1111/j.1360-0443.2004.00753.x
• Steenbergh, T. A., Meyers, A. W., May, R. K., & Whelan, J. P. (2002). Development and validation of the Gamblers’ Beliefs Questionnaire. Psychology of Addictive Behaviors, 16(2), 143–149. https://doi.org/10.1037/0893-164X.16.2.143
• Turner, N. E., & Liu, E. (1999, August). The naïve human concept of random events. Poster presented at the 107th Annual Convention of the American Psychological Association (APA), Boston, MA.
• Turner, N. E., Littman-Sharp, N., Zangeneh, M., & Spence, W. (2002). Winners: Why do some develop gambling problems while others do not? Ontario Problem Gambling Research Centre (OPGRC), Guelph, ON.
• Tversky, A., & Kahneman, D. (1971). Belief in the law of small numbers. Psychological Bulletin, 76(2), 105–110. https://doi.org/10.1037/h0031322
• Tversky, A., & Kahneman, D. (1974). Judgment under uncertainty: Heuristics and biases. Science, 185(4157), 1124–1131. https://doi.org/10.1126/science.185.4157.1124

Items of the Scale