Elementary event explained

In probability theory, an elementary event, also called an atomic event or sample point, is an event which contains only a single outcome in the sample space.[1] Using set theory terminology, an elementary event is a singleton. Elementary events and their corresponding outcomes are often written interchangeably for simplicity, as such an event corresponding to precisely one outcome.

The following are examples of elementary events:

\{k\},

where

k\in\N

if objects are being counted and the sample space is

S=\{1,2,3,\ldots\}

(the natural numbers).

\{HH\},\{HT\},\{TH\},and\{TT\}

if a coin is tossed twice.

S=\{HH,HT,TH,TT\}

where

H

stands for heads and

T

for tails.

\{x\},

where

x

is a real number. Here

X

is a random variable with a normal distribution and

S=(-infty,+infty).

This example shows that, because the probability of each elementary event is zero, the probabilities assigned to elementary events do not determine a continuous probability distribution.

Probability of an elementary event

Elementary events may occur with probabilities that are between zero and one (inclusively). In a discrete probability distribution whose sample space is finite, each elementary event is assigned a particular probability. In contrast, in a continuous distribution, individual elementary events must all have a probability of zero.

Some "mixed" distributions contain both stretches of continuous elementary events and some discrete elementary events; the discrete elementary events in such distributions can be called atoms or atomic events and can have non-zero probabilities.[2]

Under the measure-theoretic definition of a probability space, the probability of an elementary event need not even be defined. In particular, the set of events on which probability is defined may be some σ-algebra on

S

and not necessarily the full power set.

Further reading

Notes and References

  1. Book: Wackerly, Denniss. William Mendenhall. Richard Scheaffer. Mathematical Statistics with Applications. 2002 . Duxbury. 0-534-37741-6.
  2. Book: Kallenberg, Olav. Foundations of Modern Probability. 2nd. 2002. 9. Springer. New York. 0-387-94957-7.