# Complement of an Event

If event
$A$
is a
subset
of a
sample space
$S$
, the
**
complement
**
of
$A$
contains the elements of
$S$
that are
**
not
**
members of
$A$
. The symbol for the complement of an event
$A$
is
$\stackrel{\xaf}{A}$
.

**
Example 1:
**

Consider the experiment of rolling a single die.

$S=\left\{1,2,3,4,5,6\right\}$
. Let
$A$
be the event that the roll yields a number greater than
$4$
.
$A=\left\{5,6\right\}$
. Then
$\stackrel{\xaf}{A}=\left\{1,2,3,4\right\}$
.

Remember: $P\left(S\right)=1$ . $A\cup \stackrel{\xaf}{A}=S$ , so $P\left(A\cup \stackrel{\xaf}{A}\right)=1$ . Since $A$ and $\stackrel{\xaf}{A}$ are mutually exclusive , $P\left(A\cup \stackrel{\xaf}{A}\right)=P\left(A\right)+P\left(\stackrel{\xaf}{A}\right)=1$ . Therefore, $P\left(\stackrel{\xaf}{A}\right)=1-P\left(A\right)$

**
Example 2:
**

When it is not raining, the probability of the New Orleans Saints winning a football game is $\frac{7}{10}$ . What is the probability of the Saints winning if it is raining?

$P$ (Saints winning with rain)

= $1-$ $P$ (Saints winning without rain)

= $1-\frac{7}{10}=\frac{3}{10}$