# What is a simple event in probability? What is a complementary event?

Jan 30, 2017

See explanation.

#### Explanation:

A elementary event is (exactly speaking) a primary oobject which is not defined (as for example a point or a number).

More generally it is a "basic result" of a probability experiment. For example if a die is thrown the possible results (i.e. elementary events) are $1 , 2 , 3 , 4 , 5$ and $6$

A set of all elementary events is called a sample space and denoted by $\Omega$

An event is any subset of $\Omega$

A complementary event of $A$ is a set:

$A ' = \left\{\omega \in \Omega : \omega \notin A\right\}$

so the complementary event consists of all elementary events which are not in the event $A$

Example

I wrote earlier about an experiment of a die throw. Its sample space is (as I wrote) a set of numbers $1$-$6$

$\Omega = \left\{1 , 2 , 3 , 4 , 5 , 6\right\}$

Let $A$ be an event a prime number is thrown.

Then $A = \left\{2 , 3 , 5\right\}$, because in the set $\Omega$ only $2 , 3$ and $5$ are prime numbers

A complementary event would be $A '$ - a number which is not prime is thrown, so

$A ' = \left\{1 , 4 , 6\right\}$