Definition. An event which occurs with probability 1 is said to occur almost always.

Whether or not you believe such an event might occur depends on how you relate probability to your world. In a finite probability space, an event which occurs almost always always occurs. However, for infinite probability spaces we may imagine events which are possible but still occur with probability . The canonical example is to get an infinite sequence of heads while tossing a fair coin. We can imagine this happening (although we'd find it hard to believe), yet it has probability 0.

The "almost" in "almost always" is there to get around linguistic sophism and allow us to get some real mathematics done.

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