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Deterministic mathematical models are models in which the outcomes are uniquely determined by the input values and the relationships between them. There is no uncertainty or randomness involved in the predicted results. Deterministic models are commonly used in situations where the system being modeled has well-defined and predictable behavior.

Probabilistic mathematical models, on the other hand, incorporate randomness and uncertainty in their predictions. These models assign probabilities to the outcomes as a way of acknowledging that the actual outcome may not be exactly what the model predicts. Probabilistic models are often used in situations where the system being modeled is influenced by random factors or if there is uncertainty in the input data.

The main differences between deterministic and probabilistic models can be summarized as follows: 1. Deterministic models provide a single, unique outcome for a given set of input values, while probabilistic models assign probabilities to different possible outcomes. 2. Deterministic models assume that the system being modeled has well-defined and predictable behavior, whereas probabilistic models take into account the randomness and uncertainty present in the system. 3. Deterministic models are typically easier to analyze and understand compared to probabilistic models, due to their straightforward cause-and-effect relationships. 4. Probabilistic models are more flexible and adaptable in the face of uncertainty, making them a better choice for modeling complex systems with variable or changing conditions. In summary, deterministic mathematical models present an exact and predictable outcome whereas probabilistic mathematical models consider uncertainties and randomness in their predictions. The choice between using deterministic or probabilistic models depends on the nature of the system being modeled and the level of uncertainty involved.