In Steven Pinker's book Better Angels of Our Nature, he notes that
Probability is a matter of perspective. Viewed at sufficiently close range, individual events have determinate causes. Even a coin flip can be predicted from the starting conditions and the laws of physics, and a skilled magician can exploit those laws to throw heads every time. Yet when we zoom out to take a wide-angle view of a large number of these events, we are seeing the sum of a vast number of causes that sometimes cancel each other out and sometimes align in the same direction. The physicist and philosopher Henri Poincare explained that we see the operation of chance in a deterministic world either when a large number of puny causes add up to a formidable effect, or when a small cause that escapes our notice determines a large effect that we cannot miss. In the case of organized violence, someone may want to start a war; he waits for the opportune moment, which may or may not come; his enemy decides to engage or retreat; bullets fly; bombs burst; people die. Every event may be determined by the laws of neuroscience and physics and physiology. But in the aggregate, the many causes that go into this matrix can sometimes be shuffled into extreme combinations. (p. 209)
I am particularly interested in the bolded sentence, but I give the rest for context. My question: are there statistical ways of describing the two processes that Poincare described? Here are my guesses:
1) "A large number of puny causes add up to a formidable effect." The "large number of causes" and "add up" sound to me like the central limit theorem. But in (the classical definition of) the CLT, the causes need to be random variables, not deterministic effects. Is the standard method here to approximate these deterministic effects as some sort of random variable?
2) "A small cause that escapes our notice determines a large effect that we cannot miss." It seems to me like you could think of this as some sort of hidden Markov model. But the (unobservable) state transition probabilities in an HMM are just that, probabilities, which is by definition once again not deterministic.