Universe (probabilities)

In Theory of probability, a universe , often noted $\ Omega, U$ or $S$ , is the whole of all the possible results which can be obtained during a random Expérience.

Definitions

With each element $\ omega$ of the universe, i.e. with each possible result of the experiment considered, we can associate the subset $\ {\ Omega \}$ made up of this element, called elementary event. In a more general way, very part of the universe is simply called a event.

One also speaks about space of the elementary events or about space of observable the , or about space sample .

For example, if we launch a part, we have two possible results: pile or face . the random experiment considered is then: “ 1 to launch part ”. We can define the universe associated with this experiment, which gathers all the possible results: $\ Omega$ ≡ { pile , face }. For an experiment of launching die, we would choose the universe $\ Omega$ ≡ {1, 2,3,4,5,6}.

With any discrete universe (finished and/or countable), one can associate a probability, which is entirely determined by the values that it takes on the elementary events (and when the universe is not discrete, one calls very event part which one can define the probability).

Thus, with each event a probability of realization is associable (for example, for the throw of die, with each event of the universe {1, 2,3,4,5,6} is associated a probability equal to 1/6). Any definition of probability starts with the search for a universe of all the realizable events and with the precise definition of all the events useful to its resolution.

The research of the universe consists in representing in a single way the possible results of the experiment by mathematical objects (numbers, p-lists, p-lists of distinct elements, started from a unit, permutations, continuations,…) to form a unit.

Choice of the universe

For certain types of experiments, we can define several different universes. For example, when we draw a chart from a play of 52 charts, we can be interested in the row of the chart in the play and define the universe as the whole of the entireties from 1 to 52; in addition, we can be interested in the color of the chart obtained and define the universe as being the unit {spade, heart, square, clover}. To have a complete description of a result, we would be brought to specify the color and the row of the chart, and to define in this case the universe as the Cartesian Produit these two units: $\ Omega$ ≡ {spade, heart, square, clover} × {2, 3,4,5,6,7,8,9,10, servant, lady, king, ace}.

To choose the universe, we must also take account of the probabilities which enter the definition of the random experiment. For example, it is possible to consider a universe on which there is equiprobability , i.e. on which the probability is uniform (for example, for the throw of die, if the die is not faked, there is equiprobability for each event in {1, 2,3,4,5,6}.

See too

Probability
Probability (elementary mathematics)

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