Basic terms in the probability theory

Basic terms in probability theory are outcomes, sample space, events, and probability measure.

Outcomes and sample space . When you conduct a random experiment, there is a set of possible results of this experiment. This results are called outcomes. All possible outcomes together form a set, which is called sample space.

It is customary to denote outcomes with \(\omega\) and sample space with \(\Omega\): \[ \Omega = \{ \omega \text{ such that } \omega \text{ is an outcome of random experiment} \}. \]

Events and events space. Subsets of the sample space \(\Omega\) are called events. We collect events which are interesting for us in another set \(\mathcal F\), that is, \(\mathcal F\) is a set of subsets of \(\Omega\). Note that the set \(\mathcal F\) must contain not only events we are interested in but also unions and intersections of all of them. Besides, if an event is in \(\mathcal F\), the complement of this event must be also in \(\mathcal F\). Sets such as \(\mathcal F\) are called $σ$-algebras.

Important. Note that \(\omega\) is an outcome, while \(\{\omega\}\) is a possible event.

Probability measure. Finally, each event can be measured with a number from zero to unity. Function that takes as input an event and returns number from zero to unity, is called probability measure, or just simply probability. In general, measure in mathematics is a function that takes a set and returns its size as a nonnegative number.

Probability space. Three quantities—sample space, $σ$-algebra of events, probablity measure— form a triplet called probability space: \[ (\Omega, \mathcal F, \P). \]

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