Sensory-motor systems are able to extract statistical regularities in dynamic environments, allowing them to generate quicker responses and anticipatory behavior oriented towards expected events. Anticipatory smooth eye movements (aSEM) have been observed in primates when the temporal and kinematic properties of a forthcoming visual moving target are fully or partially predictable. However, the precise nature of the internal model of target kinematics which drives aSEM remains largely unknown, as well as its interaction with environmental predictability. In this study we investigated whether and how the probability of target speed or acceleration is taken into account for driving aSEM. We recorded eye movements in healthy human volunteers while they tracked a small visual target with either constant, accelerating or decelerating speed, keeping the direction fixed. Across experimental blocks, we manipulated the probability of the presented target motion properties, with either 100% probability of occurrence of one kinematic condition (fully-predictable sessions), or a mixture with different proportions of two conditions (mixture sessions). We show that aSEM are robustly modulated by the target kinematic properties. With constant-velocity targets, aSEM velocity scales linearly with target velocity across the blocked sessions, and it follows overall a probability-weighted average in the mixture sessions. Predictable target acceleration/deceleration does also have an influence on aSEM, but with more variability across participants. Finally, we show that the latency and eye acceleration at the initiation of visually-guided pursuit do also scale, overall, with the probability of target motion. This scaling is consistent with Bayesian integration of sensory and predictive information.