A deterministic mathematical model is meant to yield a single solution describing the outcome of some "experiment" given appropriate inputs. A probabilistic model is, instead, meant to give a distribution of possible outcomes (i.e. it describes all outcomes and gives some measure of how likely each is to occur).It should be noted that a probabilistic model can be quite useful even for a person who believes the entire universe to be deterministic. This utility arises because even a deterministic process may have so many variables that any model that attempts to account for them all is too cumbersome to work with. For example, a coin toss might be deterministic if one could precisely measure everything about the flip, the coin, the floor, the air currents, the tides, the precise location on earth, etc. In practice, this level of deterministic modeling is impossible, so stochastic models are used instead. On the other hand, if one takes quantum mechanics seriously, everything has some level of non-deterministic behavior.

Similarly, deterministic models can be used to great effect even in real-world process that is clearly stochastic. For example, the heat equation works great in many situations despite the fact that it ignores the "random" motion of the atoms involved. Usually, in these scenarios, the distribution of possible final answers is so sharply peaked (i.e. has such a small variance) that there is no need to complicate the model by forcing it to calculate the distribution rather than just a single value.