It seems to me that there are at least two senses of "probable" which almost certainly need to be sharply distinguished, though there is a long philosophical history of confusing the two (one notable place; Hume clearly muddies the distinction in the section "Of Miracles" in the Enquiry Concerning Human Understanding, though the arguments remain plausible once the distinction is straightened out).
First, there is a mathematical sense of probability. On the basis of enough statistical data, one can make claims of the form "it's 90% likely that X", that it'll rain tomorrow, that a patient with a particular disease will die, that a kicker will make a field goal at a certain distance, and so forth. This kind of probability is likely unproblematic.
However, there seems to be another sense in which we say things are probable. When the evidence in favor of a particular claim is univocal, but not based on very many cases, then it is impossible to assign a statistical probability to the next case (unless it's of a category of claims to which we can give a statistical analysis, but this will surely not always be an option). And, of course, there's always the question of whether the data from which we construct our statistical analysis is any good; if we've put effort into getting good data, we're inclined to say it's probably good, but this is another case where we can't be making the statistical claim.
It seems to me that the non-statistical sense of probability is both indispensible and deeply problematic. We cannot help but rely on many claims for which we surely don't have certainty, but where are doubts are quite impossible to quantify statistically. Most of us think we probably aren't victims of Cartesian demon deception, for example; where are the studies on the relative frequencies of demon deception to back that one up? And for that matter, as already indicated, in order to make use of the statistical analysis, we need to count on our data being good, which is at best probable in the non-statistical sense.
But, indispensible though it is, this second sense of probability is deeply murky. How do we know that Cartesian demons are unlikely? That our perceptions are probably accurate in most cases? I fear that there is no easy answer. Certainly, there is no hope to solve the problem by appealing to the statistical notion of probability, though some have tried that; the non-statistical probability is, unfortunately, the more fundamental of the two, and not to be analyzed in terms of the less fundamental.