If an event is sure, then it will always happen. No other event (even events with probability 0) can possibly occur. If an event is almost sure, then other events are theoretically possible in a given sample space, however as the cardinality of the sample space increases, the probability of any other event asymptotically converges toward zero. Thus, one can never definitively say for any sample space that other events will never occur, but can in the general case assume this to be true. In this respect, the concept is similar to that of a mathematical limit.
Next, consider the event that "the dart hits the diagonal of the unit square exactly". The probability that the dart lands on any subregion of the square is equal to the area of that subregion. But, since the area of the diagonal of the square is zero, the probability that the dart lands exactly on the diagonal is zero. So, the dart will almost surely not land on the diagonal. Nonetheless the set of points on the diagonal is not empty and a point on the diagonal is no less possible than any other point.
The same may be said of any point on the square. Any such point P will contain zero area and so will have zero probability of being hit by the dart. However, the dart clearly must hit the square somewhere. Therefore, in this case, it is not only possible or imaginable that an event with zero probability will occur; one must occur. Thus, we would not want to say we were certain that a given such event would not occur, but rather almost certain.
The infinite sequence of all heads (H-H-H-H-H-H-…), ad infinitum, is possible in some sense (it does not violate any physical or mathematical laws to suppose that tails never appears) but it is very, very improbable. In fact, the probability of tails never being flipped in an infinite series is zero. Thus, though we cannot definitely say tails will be flipped at least once, we can say there will almost surely be at least a single tails flip in an infinite sequence of flips.
However, if instead of an infinite number of flips we stop flipping after some finite time, say a million flips, then the all-heads sequence has non-zero probability. The all-heads sequence has probability 2−1,000,000, thus the probability of getting a tails is 1 − 2−1,000,000 < 1, and the event is no longer almost sure.
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