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0 pointsvog14y ago0 comments

Probability 1 is as guaranteed as anything probabilistic can reasonably hope to be, isn’t it?

Unfortunately, it isn't. This is the reason why in mathematics, we say that "possibility 1" means that an event is "almost sure", which is different from a "sure" event.

For instance, you can play a game where you roll a dice over an over again. You win if you get 100 times a 6 in sequence. If you play this game without any time limit, your possibility to win is exactly 1, which means that you can be "almost sure" you'll win in the end. However, you can't be sure, because there is still the possibility that you don't get 100 times a 6 in an eternity.

Note that this only happens in infinite probability spaces. In finite spaces, "almost sure" and "sure" are equivalent.

Also note that the same holds for "probability 0" which means "almost impossible", not to be confused with "impossible".

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nhaehnle14y ago

Your example is misleading, because the same can be said about the game where you roll a dice over and over again, and you win if you get a 6.

The probability that the game ends is 1, but it is not guaranteed to end. Yet no reasonable person will worry about this in practice.

Your example with 100 times 6 in a row is conceptually exactly the same, just the expected time until the game terminates is much, much larger.

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