Almost surely

Almost certainly is a concept of probability theory and the special case of the concept of almost everywhere from the measure theory. A random event that occurs with probability one, is called almost certain. According to an event called almost impossible if the probability of its occurrence is zero. These terms play for example in the almost sure convergence of random variables play an important role, as it occurs in the situation of the law of large numbers.

Definition

In a probability space is called an event almost surely if

Applies. It is almost impossible if

Applies.

An almost certain event occurs that is not a necessary, but on a set of measure one. In particular, the certain event is almost certain.

An almost impossible event can occur then, but only on a set of measure zero. The impossible event is also almost impossible.

Examples

For a uniform distribution on the interval applies:

• The probability of hitting exactly a certain number of random, is 0 even though this event is not impossible. Accordingly, the probability of hitting any number from the interval except a certain number, equal to 1, but this event will not occur with certainty.
• Also the chance to meet any rational number is 0, since there are only countably infinitely many rational numbers in that range, the amount that is just the Lebesgue measure 0. In contrast, the probability of hitting any irrational number equal to 1, although this event does not have to occur.