A standard view in cognitive psychology is that people esti-mate probabilities using heuristics that do not follow proba-bility theory. We describe a model of probability estimationwhere people do follow probability theory in estimation, butare subject to random error or noise. This model predicts thatpeople’s conditional probability estimates will agree closelywith probability theory for certain noise-cancelling expres-sions, but deviate from probability theory for other expres-sions. We describe an experiment which strongly confirmsthese predictions, suggesting that people estimate conditionalprobabilities in a way that follows standard probability theory,but is subject to the biasing effects of random noise.

Recent research has identified three invariants or identities thatappear to hold in people’s probabilistic decision making: theaddition law identity, the Bayes rule identity, and the QQidentity (Costello and Watts, 2014, Fisher and Wolfe, 2014,Costello and Watts, 2016b, Wang and Busemeyer, 2013, Wanget al., 2014). Each of these identities represent specific agree-ment with the requirements of normative probability theory;strikingly, these identities seem to hold in people’s probabilityjudgments despite the presence of strong and systematic bi-ases against the requirements of normative probability theoryin those very same judgments. We assess the degree to whichtwo formal models of probabilistic reasoning (the ‘probabilitytheory plus noise’ model and the ‘quantum probability’ model)can explain these identities and biases in probabilistic reason-ing.

We test contrasting predictions of two recent models of proba-bility judgment: the quantum probability model (Busemeyeret al., 2011) and the probability theory plus noise model(Costello and Watts, 2014). Both models assume that peo-ple estimate probability using formal processes that follow orsubsume standard probability theory. The quantum probabil-ity model predicts people’s estimates should agree with oneset of probability theory identities, while the probability the-ory plus noise model predicts a specific pattern of violation ofthose identities. Experimental results show just the form of vi-olation predicted by the probability theory plus noise model.These results suggest that people’s probability judgments donot follow quantum probability: instead, they follow the rulesof standard probability theory, with the systematic biases seenin those judgments due to the effects of random noise.

This paper investigates a fundamental conflict in the literatureon people’s probability estimation. Research on ‘perception’of probability shows that people are accurate in their estimatesof probability of various simple events from samples. Equally,however, a large body of research shows that people’s probabil-ity estimates are fundamentally biased, and subject to reliableand striking fallacies in reasoning. We investigate this con-flict in an experiment that examines the occurrence of the con-junction fallacy in a probability perception task where peopleare asked to estimate the probability of simple and conjunc-tive events in a presented set of items. We find that people’sprobability estimates are accurate, especially for simple events,just as seen in previous studies. People’s estimates also showhigh rates of occurrence of the conjunction fallacy. We showhow this apparently contradictory result is consistent with arecent model of probability estimation, the probability theoryplus noise’ model.