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.
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 , where the event occurs if a head is flipped, and if a tail is flipped. For this particular coin, it is assumed that the probability of flipping a head is , from which it follows that the complement event, that of flipping a tail, has probability .
Now, suppose an experiment were conducted where the coin is tossed repeatedly, with outcomes 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, where . i.e. each records the outcome of the th 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 flips is simply . Letting yields 0, since by assumption. The result is the same no matter how much we bias the coin towards heads, so long as we constrain to be strictly between 0 and 1. In fact, the same result even holds in non-standard analysis—where infinitesimal probabilities are allowed.[5]
Moreover, the event "the sequence of tosses contains at least one " 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, , would no longer be 0, while the probability of getting at least one tails, , would no longer be 1 (i.e., the event is no longer almost sure).