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In probability theory, one says that an event happens **almost surely** (sometimes abbreviated as **a. s.**) if it happens with probability one. In other words, the set of possible exceptions may be non-empty, but it has probability zero.

In probability experiments on a finite sample space, there is often no difference between *almost surely* and *surely*. However, the distinction becomes important when the sample space is an infinite set, because an infinite set can have non-empty subsets of probability zero.

An example of the use of this concept is the strong version of the law of large numbers.

**Almost never** describes the opposite of *almost surely*: an event that happens with probability zero happens *almost never*.

**Definition**[]

In a probability space , an event happens *almost surely* if

- .

It happens *almost never* if

- .

**Examples**[]

**Interval**[]

For a uniform distribution on the interval , the probability of randomly picking exactly a certain number is 0, although this event is not impossible. Correspondingly, the probability of picking any number except is 1, but this event will not necessarily occur.

**Daily Kit Box**[]

Consider the case where the Tech card of a ** Daily Kit Box †** is revealed. As the card can only be Advanced or Mid-Tech, the corresponding sample space is , where the event occurs if the card is Advanced Tech, and if it is Mid-Tech. The probability of getting Advanced Tech is from which it follows that the complementary event, not getting Advanced Tech (getting Mid-Tech), has the probability .

In an experiment where an infinite amount of Tech cards from a Daily Kit Box is revealed repeatedly, any infinite sequence of Advanced and Mid-Tech is a possible outcome of the experiment. However, any *particular* infinite sequence of Advanced and Mid-Tech has probability zero of being the exact outcome of the (infinite) experiment.

To see why, note that the probability of getting only Advanced Techs with 2 repetitions is

- .

The probability of getting 3 Advanced Techs with 3 repetitions is

- ,

and that of getting Advanced Techs with repetitions is . As the probability gets smaller the higher the number of repetitions is, letting tend to yields zero:

Note that the result is the same no matter the value of , so long as we constrain to be greater than 0, and less than 1. In particular, the event "the sequence contains at least one " happens almost surely (i. e., with probability 1).

However, if instead of an infinite number of repetitions we stop revealing Tech cards after some finite time, say a million turns, then the all-Advanced Tech sequence has non-zero probability. The all-Advanced Tech sequence has probability , while the probability of getting at least one Mid-Tech is and the event is no longer *almost sure*.

**See also**[]

- Convergence of random variables, for "almost sure convergence"
- Law of large numbers
- Bernoulli process with a generalized example for the
**Daily Kit Box †**