Event (probability theory)

In probability theory, an event is a set of outcomes of an experiment (a subset of the sample space) to which a probability is assigned.[1] A single outcome may be an element of many different events,[2] and different events in an experiment are usually not equally likely, since they may include very different groups of outcomes.[3] An event consisting of only a single outcome is called an elementary event or an atomic event; that is, it is a singleton set. An event that has more than one possible outcomes is called compound event. An event is said to occur if contains the outcome of the experiment (or trial) (that is, if ).[4] The probability (with respect to some probability measure) that an event occurs is the probability that contains the outcome of an experiment (that is, it is the probability that ). An event defines a complementary event, namely the complementary set (the event not occurring), and together these define a Bernoulli trial: did the event occur or not?

Typically, when the sample space is finite, any subset of the sample space is an event (that is, all elements of the power set of the sample space are defined as events).[5] However, this approach does not work well in cases where the sample space is uncountably infinite. So, when defining a probability space it is possible, and often necessary, to exclude certain subsets of the sample space from being events (see Events in probability spaces, below).

  1. ^ Leon-Garcia, Alberto (2008). Probability, statistics and random processes for electrical engineering. Upper Saddle River, NJ: Pearson. ISBN 9780131471221.
  2. ^ Pfeiffer, Paul E. (1978). Concepts of probability theory. Dover Publications. p. 18. ISBN 978-0-486-63677-1.
  3. ^ Foerster, Paul A. (2006). Algebra and trigonometry: Functions and Applications, Teacher's edition (Classics ed.). Upper Saddle River, NJ: Prentice Hall. p. 634. ISBN 0-13-165711-9.
  4. ^ Dekking, Frederik Michel; Kraaikamp, Cornelis; Lopuhaä, Hendrik Paul; Ludolf Erwin, Meester (2005). Dekking, Michel (ed.). A modern introduction to probability and statistics: understandig why and how. Springer texts in statistics. London [Heidelberg]: Springer. p. 14. ISBN 978-1-85233-896-1.
  5. ^ Širjaev, Alʹbert N. (2016). Probability-1. Graduate texts in mathematics. Translated by Boas, Ralph Philip; Chibisov, Dmitry (3rd ed.). New York Heidelberg Dordrecht London: Springer. ISBN 978-0-387-72205-4.