Probability is teleological and can describe natural things to the extent they are also. Here’s the argument:
Surprising things are chance events in the sense opposed to teleology.
Probability calculations seek to drive out surprise.
So probability calculations seek to advance the teleological and drive out the chance that is opposed to this.
You flip the fair coin a hundred times. You expect about 50-50 heads-tails with a margin of error of 5. Failure to get this is surprising in the sense we want to focus on. This allows logically for the following responses:
1.) You suspect the coin isn’t fair.
2.) You wonder if your model of probability is adequate.
3.) You treat the event as an outlier that rarely happens and that would wash out with a larger sample size or running the trial a few more times.
Notice that as long as #1 and #2 are ruled out #3 is all that’s left, but if we fail to disprove #3 we have to either retool #2 or add some epicycles to it, since we can’t develop a theory that assumes we are systematically deceived about what we’re observing, as a systematic belief in #1 would require. In other words, since we have to drop #1 as a long-term theoretical solution, all probability is a standoff between #2 and #3, with the one driving out the other.
#3 is chance in the sense that is opposed to teleology. This is why Aristotle defined chance as essentially rare, since by definition if it happens more than rarely it is taken as a fact that refutes your theory. Your model is built to predict outcomes, and this allows for prediction failure only in the case of outlier phenomena. Notice that when we call the #3 events rare this does not mean the same thing as “improbable”. All sorts of improbable events (like getting four of a kind, say) can be predicted by a theory. #3 events are improbable in the sense of being entirely outside the model that established both what will count as probable and improbable. The chance event, strictly speaking, has no probability whatsoever. While it is improbable in some sense (equivocal) of the term, in the technical sense we want to target now it is a-probable. And so we hit on a harmony between Aristotle’s account of chance as rare and TOF’s and Brigg’s antiphon-axioms that there is no probability outside the model and randomness is never a cause. Probability theory can never absolutely rule out chance, but it is (a) entirely outside the model and (b) lacks the sort of necessary connection that is the sine qua non of cause-effect relations. The reason why there is no probability outside the model is that, without the model, we have no way to distinguish #3 events, which are essential to any account of probability to distinguish.
In other words, while there is certainly some (equivocal) sense of chance or randomness that probability theory deals with, it is intrinsic to the theory that it exclude and seek to drive out chance in sense #3. But this is the only sense of “chance” or “randomness” that is teleology excludes. It is only in this sense of chance that expectation of outcomes is frustrated, i.e. when the order between conditions and outcomes becomes necessarily frustrated and so intrinsically rules out teleology. Since it is just this sort of thing that probability seeks to minimize, marginalize and rule out, probability theory is intrinsically teleological. Q.E.D.