Recent research has identified three invariants or identities that appear to hold in people’s probabilistic decision making: the addition law identity, the Bayes rule identity, and the QQ identity (Costello and Watts, 2014, Fisher and Wolfe, 2014, Costello and Watts, 2016, Wang and Busemeyer, 2013, Wang et al., 2014). Each of these identities represent specific agreement with the requirements of normative probability theory; strikingly, these identities seem to hold in people’s probability judgments despite the presence of strong and systematic biases against the requirements of normative probability theory in those very same judgments. We assess the degree to which two formal models of probabilistic reasoning (the ‘probability theory plus noise’ model and the ‘quantum probability’ model) can explain these identities and biases in probabilistic reasoning.