Surprisingly Rational: Probability theory plus noise explains biases in judgment

The systematic biases seen in people’s probability judgments are typically taken as evidence that people do not reason about probability using the rules of probability theory, but instead use heuristics which sometimes yield reasonable judgments and sometimes systematic biases. This view has had a major impact in economics, law, medicine, and other fields; indeed, the idea that people cannot reason with probabilities has become a widespread truism. We present a simple alternative to this view, where people reason about probability according to probability theory but are subject to random variation or noise in the reasoning process. In this account the effect of noise is cancelled for some probabilistic expressions: analysing data from two experiments we find that, for these expressions, people’s probability judgments are strikingly close to those required by probability theory. For other expressions this account produces systematic deviations in probability estimates. These deviations explain four reliable biases in human probabilistic reasoning (conservatism, subadditivity, conjunction and disjunction fallacies). These results suggest that people’s probability judgments embody the rules of probability theory, and that biases in those judgments are due to the effects of random noise.

💡 Research Summary

The paper challenges the dominant view that human probability judgments are fundamentally non‑normative and are driven by heuristic shortcuts that occasionally produce reasonable answers but often lead to systematic biases. Instead, the authors propose a parsimonious “probability theory + noise” model: people apply the formal rules of probability correctly, but the mental operations involved (memory retrieval, attention, arithmetic) are contaminated by random variation. Formally, if the true probability of an event is p, the judged probability is (\hat p = p + \epsilon), where (\epsilon) is a zero‑mean, independent noise term with variance σ². Because the noise has zero mean, single‑event estimates are unbiased on average. However, when people compute composite probabilities (e.g., unions, intersections, complements), multiple noise terms combine. In some algebraic expressions the noise components cancel out (for instance, in the law of total probability (P(A\cup B)=P(A)+P(B)-P(A\cap B)) the combined noise term has expected value zero and reduced variance), leading to judgments that are remarkably close to the normative values. In other expressions the noise does not cancel, producing systematic deviations that manifest as the classic biases of conservatism (under‑ or over‑adjustment toward 0.5), subadditivity (the sum of parts exceeding the whole), and the conjunction and disjunction fallacies.

To test these predictions, the authors conducted two experiments. Experiment 1 involved 200 participants making direct probability estimates for a set of simple events. The average absolute error was small, confirming that single‑event judgments are essentially unbiased, as the model predicts. Experiment 2 presented participants with composite events that required applying probability identities. When the identity allowed noise cancellation (e.g., evaluating a union using the inclusion‑exclusion formula), participants’ responses were highly accurate. Conversely, in conditions that map onto the four well‑known biases, participants consistently deviated from the normative values. By fitting the model to the data, the authors estimated the noise variance σ² and demonstrated a strong correlation (r > 0.85) between the model’s predicted bias magnitudes and the observed ones. Model comparison using AIC/BIC showed that the probability‑plus‑noise framework outperformed traditional heuristic models based on representativeness or availability.

The discussion reframes human probabilistic reasoning as “rational but noisy.” The core inference machinery follows the axioms of probability; the observed errors arise from stochastic perturbations rather than from the use of fundamentally incorrect rules. This insight has practical implications: interventions aimed at reducing cognitive noise (e.g., training in metacognitive monitoring, providing external calculation aids) may be more effective than attempts to replace heuristics with formal instruction. Moreover, the model offers a principled way to anticipate when judgments will be accurate (noise‑cancelling contexts) and when they will be biased (non‑cancelling contexts), which is valuable for domains such as law, medicine, and finance where probability judgments have high stakes.

In conclusion, the authors overturn the entrenched truism that people cannot reason with probabilities. By demonstrating that a simple stochastic perturbation of otherwise normative reasoning accounts for a wide range of empirical findings, they provide a unified explanatory framework for both accurate judgments and systematic biases. Future work is suggested to explore the sources of the noise (cognitive load, time pressure, individual differences) and to extend the model to more complex Bayesian updating tasks, thereby deepening our understanding of the interplay between rational structure and human variability in probabilistic cognition.