definition of sample space

Within probability statistics, the sample space is defined as the set of all possible outcomes that are obtained when performing a random experiment (the one whose outcome cannot be predicted).

The most common denotation of the sample space is by the Greek letter omega: Ω. Among the most common examples of sample spaces we can find the results of tossing a coin (heads and tails) or rolling a dice (1, 2, 3, 4, 5 and 6).

Multiple sample spaces

In many experiments it may be the case that several possible sample spaces coexist, leaving the person conducting the experiment to choose the one that best suits them according to their interests.

An example of this would be the experiment of drawing a card from a standard 52-card poker deck. Thus, one of the sample spaces that could be defined would be that of the different suits that make up the deck (spades, clubs, diamonds and hearts), while other options could be a range of cards (between two and six, for example ) or the figures in the deck (jack, queen and king).

One could even work with a more precise description of the possible results of the experiment by combining several of these multiple sample spaces (drawing a figure from the suit of hearts). In this case, a single sample space would be generated, which would be a Cartesian product of the two previous spaces.

Sample space and probability distribution

Some approaches to probability statistics assume that the different results that can be obtained from an experiment are always defined so that they all have the same probability of happening.

However, there are experiments in which this is really complicated, being very complex to construct a sample space where all the results have the same probability.

A paradigmatic example would be to throw a thumbtack in the air and observe how many times it falls with its tip down or up. The results will show a clear skewness, so it would be impossible to suggest that both results have the same probability of happening.

Probability symmetry is the most common when analyzing random phenomena, but that does not mean that it is very helpful to be able to construct a sample space in which the results are at least approximately similar, since this condition is basic in order to simplify the calculation of probabilities. And it is that, if all the possible results of the experiment have the same probability of happening, then the study of probability is greatly simplified.

Photos: iStock - Moncherie

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