In thinking about what probability means, it’s important to sort out what is contingent and what relationship contingent events have to causality. Contingency simply means that what is could have been something else. That what will happen is not logical necessary. We can imagine the contingent. As with many things, Chesterton put it best:
You cannot IMAGINE two and one not making three. But you can easily imagine trees not growing fruit; you can imagine them growing golden candlesticks or tigers hanging on by the tail.
A necessity chain of events is one where “we cannot conceive [one thing] occurring without the other.” Statements of logic and mathematics are examples where we start by accepting certain facts and rules which then inevitably lead to other facts, where those facts are necessarily so. Thus empirical events are contingent.
Causality is the belief that (most? all?) events have a cause or causes which precedes them. It is a belief. There is no proof of it arising from simpler axioms. In particular, we have not and could not have deduced causality for contingent events from contingent events. Hume argued that we inferred the principle of causality from inspection of correlations; from noticing this event (almost) always followed this other happening. And since all events are contingent—they could have been otherwise—then our inference of causality is not a deduction, and thus doubt remains whether causality is real, especially for all events (each and every one of them).
There is also doubt that causality operates always and in all places and in the same way. Stating this is not questioning the belief, but it does give emphasis, and properly so, that it is a belief. You might assume however, given its overwhelming support, that philosophers are solidly behind causality. You would be wrong. The Stanford Encyclopedia quotes Bertrand Russell
The law of causality, I believe, like much that passes muster among philosophers, is a relic of a bygone age, surviving, like the monarchy, only because it is erroneously supposed to do no harm.
Because causality is not (or might not be) the same as determinism, many philosophers separate the ideas. It’s also important not to confuse predictability with determinism: all deterministic sequences are perfectly predictable given perfect knowledge of their starting points and the unbreakable and inflexible laws that govern their behavior. But we usually do not often mean necessarily flawless predictions when speaking of predictability.
If everything at some future time is a necessary consequence of the way things are now, and the way things are now are a necessary consequence of the way it was when time began (whenever that might have been, even never), then determinism rules.
Neurologists seeking publicity are often staunch determinists. We can’t hold the forlorn rule breaker accountable for his actions, says David Eagleman, for he has no free will, because each state of the criminal’s brain proceeds deterministically from each prior state. Presumably Eagleman believes that his brain has somehow escaped from determinism to bring us this news. He also implies that we his audience can also break free of determinism and modify our behavior (in a direction which acknowledges the supremacy of determinism).
Whether or not you think this is nonsense or a keen insight into the workings on the brain, it remains true that determinism has not been deduced and it too is a belief. That is, it is not certainly true given our observations of the contingent world. It is also not so that we can prove determinism, if operative, is always operative (no exceptions), or always has been, or that it applies to every imaginable kind of contingent event.
If anything observational evidence gives evidence against strict causality and determinism. We often witness events which have no apparent cause. We only believe a cause exists because we assume causality, which insists all events have a cause. Further, we often cannot say what were the relevant and exact initial conditions and unbreakable laws that led to an event—that led certainly to the event, to be clear. There is no probability in determinism. If an event is merely probable, given some evidence off initial conditions and operative laws, then the event has not been determined (not with respect to our knowledge).
So why do so many believe, and believe with all their hearts, that causality and determinism are necessary features of our universe? To ask this is not a proof of the opposite. But if you do believe, why?