Human estimates of the probabilities of combinations of events show well-established violations of probabilitytheory, most notably the conjunction and disjunction fallacies. These violations have led researchers to conclude that therules of probability are too complex for most people to use, and that cognitively-easier approximations such as averagingare used instead. Unlike previous work that has assumed that individuals use only a single combination rule, we collectedrepeated estimates of conjunctions and disjunctions and investigated whether individuals consistently used a single rule orused a repertoire of rules in a trial-by-trial Bayesian analysis. When not incentivized, most participants were best describedas randomly selecting a combination rule on each trial, and the correct rule was the most often used. Despite this, whenincentivized to use their single-best strategy participants were more likely to use the incorrect averaging rule. People do notseem to understand their own strategies well.

Do people use a single strategy or sample from multiple strategies when estimating the conjunction and disjunction of two independent events? Here we address this tension directly by comparing individual level Bayesian simulations of multi and single strategy models using data from a frequency estimation experiment. Participants were shown two statements describing attributes and asked to estimate how many people had either one attribute, conjunction, or disjunction of attributes. In our Bayesian simulations we compare models in which participants either adopt a single strategy or sample from a set of strategies when forming estimates of both conjunctions and disjunctions. We compared every permutation of models in which a participant is responding based on a single component, a weighted average of the two events, probability theory, or combination of strategies. Our findings show that people sample from multiple strategies and are sampling from the normative strategy some of the time.

When asked to combine two pieces of evidence, one diagnostic and one non-diagnostic, people show a dilution effect: the addition of non-diagnostic evidence dilutes the overall strength of the evidence. This non-normative effect has been found in a variety of tasks and has been taken as evidence that people inappropriately combine information. We investigated the dilution effect using simple perceptual stimuli but unlike in previous work we asked participants to judge likelihoods ratios, allowing us to assess not just ordinal relationship between judgments but also whether each individual judgment was accurate. We found the dilution effect, but surprisingly it was not due to inaccurate combination of diagnostic and non-diagnostic information. People were accurate at judging diagnostic evidence combined with non-diagnostic evidence, but overestimated the strength of diagnostic evidence alone. We explain this within-participants dilution effect as the result of inference about missing features rather than incorrect combination of information.