Almost surely
In probability theory, an event is said to happen almost surely (sometimes abbreviated as a.s.) if it happens with probability 1 (or Lebesgue measure 1). In other words, the set of possible exceptions may be non-empty, but it has probability 0. The concept is analogous to the concept of "almost everywhere" in measure theory. In probability experiments on a finite sample space, there is often no difference between "almost surely" and "surely" (since having a probability of 1 often entails including all the sample points). However, this distinction becomes important when the sample space is an infinite  set, because an infinite set can have non-empty subsets of probability 0. Some examples of the use of this concept include the strong and uniform versions of the law of large numbers, and the continuity of the paths of Brownian motion. The terms almost certainly (a.c.) and almost always (a.a.) are also used. Almost never describes the opposite of "almost surely": an event that happens with probability zero happens "almost never". Formal definition. Let formula_1 be a probability space. An event formula_2 happens "almost surely" if formula_3. Equivalently, formula_4 happens almost surely if the probability of formula_4 not occurring  is zero: formula_6. More generally, any event formula_7 (not necessarily in formula_8) happens almost surely if formula_9 is contained in a null set: a subset formula_10 in formula_11 such that The notion of almost sureness depends on the probability measure formula_12. If it is necessary to emphasize this dependence, it is customary to say that the event formula_4 occurs "P"-almost surely, or almost surely "formula_14". Illustrative examples. In general, an event can happen "almost surely", even if the probability space in question includes outcomes which do not belong to the event—as the following examples illustrate. Arriving on time. As  Douglas Adams wrote: the only time you cannot arrive, is on time. Thus, even though you in theory could arrive EXACTLY when you intended to, you would "almost surely" arrive at a different time, for example, a quarter of a nanosecond too early. Throwing a dart. Imagine throwing a dart at a unit square (a square with an area of 1) so that the dart always hits an exact point in the square, in such a way that each point in the square is equally likely to be hit. Since the square has area 1, the probability that the  dart will hit any particular subregion of the square is equal to the area of that subregion. For example, the probability that the dart will hit the right half of the square is 0.5, since the right half has area 0.5. Next, consider the event that the dart hits exactly a point in the diagonals of the unit square. Since the area of the diagonals of the square is 0, the probability that the dart will land exactly on a diagonal is 0. That is, the dart will "almost never" land on a diagonal (equivalently, it will "almost surely" not  land on a diagonal), even though the set of points on the diagonals is not empty, and a point on a diagonal is no less possible than any other point. Tossing a coin repeatedly. Consider the case where a (possibly biased) coin is tossed, corresponding to the probability space formula_15, where the event formula_16 occurs if a head is flipped, and formula_17 if a tail is flipped. For this particular coin, it is assumed that the probability of flipping a head is formula_18, from which it follows that the complement event, that of flipping a tail, has probability formula_19. Now , suppose an experiment were conducted where the coin is tossed repeatedly, with outcomes formula_20 and the assumption that each flip's outcome is independent of all the others (i.e., they are independent and identically distributed;"i.i.d"). Define the sequence of random variables on the coin toss space, formula_21 where formula_22. "i.e." each formula_23 records the outcome of the formula_24th flip. In this case, any infinite sequence of heads and tails is a possible outcome of the experiment. However, any particular infinite sequence of heads and tails has probability 0 of being the exact outcome of the (infinite) experiment. This is because  the "i.i.d." assumption implies that the probability of flipping all heads over formula_25 flips is simply formula_26. Letting formula_27 yields 0, since formula_28 by assumption. The result is the same no matter how much we bias the coin towards heads, so long as we constrain formula_29 to be strictly between 0 and 1. In fact, the same result even holds in non-standard analysis—where infinitesimal probabilities are not allowed. Moreover, the event "the sequence of tosses contains at least one formula_30" will also happen almost surely (i.e., with probability 1). But if instead of an infinite number of flips , flipping stops after some finite time, say 1,000,000 flips, then the probability of getting an all-heads sequence, formula_31, would no longer be 0, while the probability of getting at least one tails, formula_32, would no longer be 1 (i.e., the event is no longer almost sure). Asymptotically almost surely. In asymptotic analysis, a property is said to hold "asymptotically almost surely" (a.a.s.) if over a sequence of sets, the probability converges to 1. For instance, in number theory, a large number is asymptotically almost surely composite, by the prime number theorem; and in random graph theory, the statement "formula_33  is connected" (where formula_34 denotes the graphs on formula_25 vertices with edge probability formula_29) is true a.a.s. when, for some formula_37 In number theory, this is referred to as "almost all", as in "almost all numbers are composite". Similarly, in graph theory, this is sometimes referred to as "almost surely".