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In probability theory, an event happens almost surely (a.s.) if it happens with probability one. The concept is analogous to the concept of "almost everywhere" in measure theory. It is often encountered in questions that involve infinite time, infinite-dimensional spaces such as function spaces, or infinitesimals. Probability theory is the branch of mathematics concerned with analysis of random phenomena. ...
In probability theory, an event is a set of outcomes (a subset of the sample space) to which a probability is assigned. ...
In measure theory (a branch of mathematical analysis), one says that a property holds almost everywhere if the set of elements for which the property does not hold is a null set, i. ...
In mathematics, a measure is a function that assigns a number, e. ...
Infinity is a word carrying a number of different meanings in mathematics, philosophy, theology and everyday life. ...
2-dimensional renderings (ie. ...
In mathematics, a function space is a set of functions of a given kind from a set X to a set Y. It is called a space because in many applications, it is a topological space or a vector space or both. ...
In mathematics, an infinitesimal, or infinitely small number, is a number that is smaller in absolute value than any positive real number. ...
Formal definition
Let (Ω, F, P) be a probability space. An event E in F happens almost surely if P(E) = 1. Alternatively, an event E happens almost surely if the probability of E not occurring is zero. In mathematics, the definition of the probability space is the foundation of probability theory. ...
In probability theory, an event is a set of outcomes (a subset of the sample space) to which a probability is assigned. ...
An alternate definition from a measure theoretic-perspective is that (since P is a measure over Ω) E happens almost surely if E = Ω almost everywhere. In mathematics, a measure is a function that assigns a number, e. ...
In measure theory (a branch of mathematical analysis), one says that a property holds almost everywhere if the set of elements for which the property does not hold is a null set, i. ...
"Almost sure" versus "sure" The difference between an event being almost sure and sure is the same as the subtle difference between something happening with probability 1 and happening always.
If an event is sure, then it will always happen. No other event (even events with probability 0) can possibly occur. If an event is almost sure, then there are other events that could happen, but they happen almost never, that is with probability 0.
Example: Throwing a dart For example, imagine throwing a dart at a unit square, and imagine that this square is the only thing in the universe. There is physically nowhere else for the dart to land. Then, the event that "the dart hits the square" is a sure event. No other alternative is imaginable.
Next, consider the event that "the dart hits the diagonal of the unit square exactly". The probability that the dart lands on any subregion of the square is equal to the area of that subregion. But, since the area of the diagonal of the square is zero, the probability that the dart lands exactly on the diagonal is zero. So, the dart will almost surely not land on the diagonal, or indeed any other given line or point. Notice that even though there is zero probability that it will happen, it is still possible.
Example: Tossing a coin Suppose that a coin is flipped again and again. The infinite sequence of all heads (H-H-H-H-H-H-…), ad infinitum, is possible in some sense—it does not violate any physical or mathematical laws to suppose that tails never appears—but it is very, very improbable. In fact, such a sequence has probability zero. Thus, there will almost surely be at least a single tails flip in an infinite sequence of flips.
However, if instead of an infinite number of flips we stop flipping after some finite time, say a million flips, then the all-heads sequence has non-zero probability. If heads and tails are equiprobable, then the all-heads sequence has probability 2-1,000,000, thus the probability of getting a tails is 1-2-1000000≠1, and the event is no longer almost sure. In the theory of probability and statistics, a Bernoulli trial is an experiment whose outcome is random and can be either of two possible outcomes, called success and failure. ...
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