Quite often the word experiment describes an experimental setup, while the word trial applies to actually executing the experiment and obtaining an outcome.

A formal theory of probability has been developed in the 1930s by the Russian mathematician A. N. Kolmogorov.

Thus we require that the function be non-negative and that its values never exceed 1. The subsets of Ω for which P is defined are called events. 1-element events are said to be elementary. The function is required to be defined on the empty subset Φ and the whole set Ω:

This says in particular that both Φ and Ω are events. The event Φ that never happens is impossible and has probability 0. The event Ω has probability 1 and is certain or necessary.

If Ω is a finite set then usually the notions of an impossible event and an event with probability 0 coincide, although it may not be so. If Ω is infinite then the two notions practically never coincide. A similar dichotomy exists for the notions of a certain event and that with probability 1. Examples will be given shortly.

The probability function (or, as it most commonly called, measure) is required to satisfy additional conditions. It must be additive: for two disjoint events A and B, i.e. whenever A ∩ B = Φ,

which is a consequence of a seemingly more general rule: for any two events A and B, their union A∪B and intersection A∩B are events and

Note, however, that (2') can be derived from (2). Indeed, assuming that all the sets involved are events, events A - B and A ∩ B are disjoint as are B - A and A ∩ B. In fact, all three events A - B, B - A, and A ∩ B are disjoint and the union of the three is exactly A∪B. We have,

For an infinite space Ω, we require σ-additivity: for mutually disjoint sets A_i, i = 1, 2, ...,, their union is an event and

In general, the collection of events is assumed to be a σ-algebra, which means that the complements of events are events and so are the countable unions and intersections.

Probability is a monotone function - the fact that jibes with our intuition that a larger event, i.e. an event with a greater number of favorable outcomes, is more likely to occur than a smaller event.

Disjoint events do not share favorable outcomes and, for this reason, are often called incompatible or, even more assertively, mutually exclusive.

In addition to a sample space, an experiment may be profitably associated with what is known as a random variable. X is a random variable if its values are not determined with certainty but come from a sample space defined by a random experiment. If x is a possible outcome of an experiment, we often write P(x) for the probability P({x}) of the elementary event {x}. In terms of random variable X, the same quantity is described as P(X = x), the probability that the random variable X takes the value of x.

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