Does the baby already have a sense of probability that enables it to visualize probability distributions, update them by applying Bayes' rule, and use them to generate predictions that it compares with data received from the outside world?
For Jean Piaget and Bärbel Inhelder(La genèse de l'idée de hasard chez l'enfant, 1951), the answer can only be negative. The child cannot understand the irreversibility of chance before the stage of "reversible operations". According to Piaget, the idea of chance does not appear until the age of 7-11, "but elementary operations of order (ordering a sequence ABC...) are far from sufficient to constitute a complete scheme of permutations properly so called". This mastery would appear around age 12.
Piaget and Inhelder rely on a series of experiments in which 4-5 year-olds respond effectively without taking probabilities into account. But their mistake was to rely exclusively on children's verbal introspection (or their drawings), rather than on the implicit knowledge that can be detected in their gaze, their surprise, or their spontaneous choice. Using these techniques, contemporary cognitive science demonstrates that babies are competent at probabilistic calculation. A twelve-month-old baby, seeing for the first time an urn containing three blue objects and one yellow one, expresses surprise when a random draw sees the yellow object come out rather than one of the blue ones. His surprise is directly proportional to the improbability of the observed outcome (Teglas et al., 2011). His ability to anticipate is probabilistic, but not frequential. According to Luca Bonatti, he doesn't need to observe many draws to calculate their probabilities: his probabilistic deduction is based exclusively on an internal model of the logic of the situation. This internal model combines spatial, temporal and probabilistic sources of information. The baby is probably performing a probabilistic mental simulation, relying on the principles of solidity and spatio-temporal continuity of physical objects. The hypothesis of an internal simulation accounts for a whole series of experiments on the infant's sense of probability, concept of object, occlusion and sense of number (Teglas et al., 2011).
