Probabilistic representation refers to the ability to perceive, judge, and infer about uncertainties. Human infants, children, and several nonhuman animals, such as great apes, monkeys, and birds, are sensitive to probabilistic information, as well as capable of making probabilistic judgments and inferences, even learning based on probabilities, suggesting inter-species consistency of probabilistic representation. Also, human and nonhuman animals are capable of integrating different-domain information into probabilistic representation, such as spatio-temporal information, physical constraints, and mental states, suggesting domain-generality of probabilistic representation. Domain-generality and inter-species consistency indicate that probabilistic representation is of great importance for organisms’ survival and reproduction. However, how human represent probabilities remains controversial for a long time. Understanding what cognitive mechanisms underlie probabilistic representation and how children acquire the ability to represent probabilities would be of great importance for understanding how human learn and reason about the world, as probabilities must be the core information of the world in which human and animals live.

Existing empirical research mostly used two paradigms to examine probabilistic representation, violation of expectation (VoE) and two-alternative forced-choice (2AFC) task. Violation of expectation paradigm was mostly applied to infant and nonhuman animal studies, as infants and nonhuman animals were not able to give meaningful verbal responses. Two-alternative forced-choice paradigm was mostly applied to child and nonhuman animal studies. Compared with two-alternative forced-choice paradigm, violation of expectation paradigm could only indicate infants’ sensitivity to probabilities, as it always showed all the information to capture infants’ visual attention patterns toward expected and unexpected consequences, instead of making infants choose uncertain choices based on their predictions.

Researchers have examined human and nonhuman animals’ probabilistic representation by the two paradigms mentioned above, and have discussed how human and nonhuman animals represent probabilities from different theoretical perspectives. Two theoretical explanations have been proposed to construct the cognitive developmental mechanisms of probabilistic representation, numerical processing, and logic inferences. On the one hand, theoretical perspective based on numerical processing, or intuitive statistics, suggests that we represent probabilities based on numerical information, such as the proportions. Some research indeed showed that children and nonhuman animals exhibited similar characteristics to numerical processing in probabilistic representations, and numerical representation acuity was positively correlated with performances of probabilistic representation. Children showed increasingly better abilities to represent probabilities based on numerical information with age. On the other hand, according to the theoretical perspective based on logic inferences, we represent probabilities by enumeration of all the possible and exclusive consequences, which was referred to as modal concept or modal logic. However, some researchers argued that we were not born with the ability to represent probabilities with modal logic. Instead, they proposed that infants and toddlers represented probabilities by simulating a random consequence from all the possible consequences till age three, whereas older children and adults represent probabilities by enumerating all the consequences, or modal logic.

Combined with the empirical evidence and existing theoretical explanations, we expound the theoretical explanations for the cognitive developmental mechanisms of probabilistic representation, including different perspectives from numerical processing and logical inferences. This article proposed an integrative hypothesis of probabilistic representation, and suggested the possible directions for future research. Additionally, the two aforementioned proposals may not be exclusive to each other. Each explanation focused on one aspect of probability representation. Therefore, we proposed the integrated theory to construct the cognitive developmental mechanism of probability representation. Logic inferences, or modal logic, allows us to understand that all the possible consequences might happen and they are exclusive. Meanwhile, numerical processing helps us estimate the numerical information of probabilities. In general, both modal logic and numerical processing underly human’s probabilistic representation. Future research could focus on explore the integrated cognitive developmental mechanism of probability representation and apply different approaches in this field.