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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 $ (\Omega, \mathcal{F}, P) $, an event $ E \in \mathcal{F} $ happens *almost surely* if

- $ P(E) \; = \; 1 $.

It happens *almost never* if

- $ P(E) \; = \; 0 $.

## **Examples**

### **Interval**

For a uniform distribution on the interval $ [0,1] \subset \mathbb R $, the probability of randomly picking exactly a certain number $ x \in [0,1] $ is 0, although this event is not impossible. Correspondingly, the probability of picking any number except $ x $ 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 $ \{A, M\} $, where the event $ \{A\} $ occurs if the card is Advanced Tech, and $ \{M\} $ if it is Mid-Tech. The probability of getting Advanced Tech is $ p = 0.3 $ from which it follows that the complementary event, not getting Advanced Tech (getting Mid-Tech), has the probability $ 1 - p = 0.7 $.

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

- $ p \cdot p = p^2 = 0.3^2 = 0.09 $.

The probability of getting 3 Advanced Techs with 3 repetitions is

- $ p \cdot p \cdot p = p^3 = 0.027 $,

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

- $ \lim_{n\to\infty}0.3^n = 0 $

Note that the result is the same no matter the value of $ p $, so long as we constrain $ p $ to be greater than 0, and less than 1. In particular, the event "the sequence contains at least one $ M $" 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 $ p^{1,000,000}\neq 0 $, while the probability of getting at least one Mid-Tech is $ 1 - p^{1,000,000} $ 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 †**