Sample space

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In probability theory, the sample space of the experiment or random experiment is the set of all possible outcomes or outcomes of the experiment. The sample space is usually denoted using the specified notation, and the possible ordered results are listed as elements in the set. It is common to refer to the sample space by the label S ,?, Or U (for "universal set").

For example, if an experiment throws a coin, the sample space is usually set to {head, tail}. To throw two coins, the corresponding sample space is {(head, head), (head, tail), (tail, head), (tail, tail)}, usually written {HH, HT, TH, TT}. If the sample space is not consecutive, it becomes {{head, head}, {head, tail}, {tail, tail}}.

To cast a six-sided dice, the typical sample chamber is {1, 2, 3, 4, 5, 6} (where the flower is the number of pips facing up).

A well-defined sample space is one of three basic elements in the probabilistic model (probability space); the other two are a series of apparent events (sigma-algebra) and the probabilities assigned to each event (the probability measurement function).

Video Sample space

Multiple sample spaces

For many experiments, there may be more than one reasonable sample space, depending on what results are interesting for the experiment. For example, when drawing cards from a standard deck of fifty-two playing cards, one of the possibilities for sample space can be various ranks (Ace via King), while others can be clothing (clubs, diamonds, hearts, or spades). However, a more complete description of results can define denominations and suits, and sample spaces depicting each card can be constructed as Cartesian products from the two sample rooms mentioned above (this space will contain fifty-two possible outcomes). There is still another possible sample space, such as {right side up, top side down} if some cards have been reversed when shuffled.

Maps Sample space

Likely result same

Some probability treatments assume that the results of an experiment are always determined so that they have the same possibilities. However, there are experiments that are not easily explained by the sample space with the same possible outcomes - for example, if one has to throw the thumb over and over and observe whether it lands with an upward or downward point, there is no symmetry to indicate that two results must have possibly the same.

Although most random phenomena do not have the same possible outcomes, it can be helpful to define sample space in such a way that the results are at least as likely, since these conditions significantly simplify the probability calculations for events in the sample space. If each individual result occurs with the same probability, then the probability of each event becomes simple:

${\ displaystyle P (event) = {\ frac {\ text {number of results in event}} {\ text {number of results in sample space}}}}$

For example, if two dice are cast to produce two uniformly distributed integers, D ₁ and D ₂ , respectively in the range [1... 6], 36 the ordered pair (D ₁ , D ₂ ) is the sample space of a possible same event. In this case, the above formula holds, so the probability of a certain amount, say D ₁ D ₂ = 5 is easily shown to 4/36, since 4 of 36 results yield 5 as amount. On the other hand, the sample space of 11 possible numbers, {2,..., 12} does not have the same possible result, so the formula will give wrong result (1/11).

Simple random sample

In statistics, conclusions are made about the characteristics of the population by studying the sample of the individual population. To arrive at a sample that presents an unbiased estimate of the actual characteristics of the population, statisticians often try to study a simple random sample - that is, a sample in which each individual in the population has the same probability to enter. The result of this is that every possible individual combination that can be selected for the sample is also likely the same (ie, a simple random sample space of a given size of a given population consists of the same probability of a result).

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Larger sample space

In the basic approach to probability, each part of the sample space is usually called an event. However, this poses a problem when the sample space is continuous, so a more precise definition of an event is required. Under this definition only a measurable subset of the sample space, which is "-algebra over the sample space itself, is considered an event.

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See also

• Parameter space
• Possible spaces
• Space (math)
• Set (math)
• Events (probability theory)
• ? - algebra

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Note

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References

Source of the article : Wikipedia