In probability theory, the statement "event S occurs almost surely" means by definition that probability of S occuring is 1. This is indeed confusing, but it's

In probability theory, the statement "event S occurs almost surely" means by definition that probability of S occuring is 1. This is indeed confusing, but it's completely justified, given constraints. When non-mathematicians talk about probability, the situation they usually have in mind is that they have n possible events that can occur, each is equally likely, for instance the fair dice throw. This leads to assigning each one of the events the same probability, in the case of dice it's 1/6. We also want to be able to assign the probability to the situation of some subset of events occurring, for instance the probability of the situation that number on a dice is even corresponds to a subset {2 was thrown, 4 was thrown, 6 was thrown}. By common sense, this should be 1/2, and generally the probability should be (number of events in the subset / number of all events). This approach is how probability is usually taught in high school. The problem is that this approach, while useful, is totally inadequate approach for lots and lots of things that people care about. For instance, say we want to investigate the probability of earthquake occurring. We want to be able to answer the question "what's the probability that we will not have an earthquake in the next t seconds?". Assuming that earthquake are totally unpredictable, we should expect that if we wait s seconds, the answer to this question is still the same, so in probability terms, P(next earthquake will not occur in next t+s seconds provided it will not occur in the next s seconds) = P(next earthquake will not occur in next t). It's really not obvious how to apply the finite set of equally likely events method, if it's at all possible. That's why most of the mathematicians use similar, but much more general approach. Just like before, we consider set of possible elementary events that can occur, but this time we don't assume it's finite, and we don't assign each one equal probability. Instead, we assign probabilities to subsets of events, so that it's subject to few simple rules, but otherwise in a completely arbitrary way. The rules are: probability of any of the events occurring is 1, and A_1, A_2, ... are pairwise disjoint sets of events, we assign to their union the sum of probabilities. For instance, we want to model the archer shooting the circular target, so we let events be the points on the target (there are infinity of them), and we let the probability of subset of events be its area (assuming that whole target has area one). Now it makes sense to ask, say, what's the probability of archer hitting arrow within half of a radius from the center of the target? (it's 1/4). But, since every point has area 0, the probability of an archer hitting any particular point is 0, thus probability of archer _not_ hitting that point is 1, meaning that archer will _almost surely_ not hit that point. Almost, because while for every point the probability of archer hitting it is 0, archer will hit _some_ point. That's why mathematicians say almost surely, and not surely, when talking about evens with probability 1.