# 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 $\stackrel{¯}{A}$ .

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 $\stackrel{¯}{A}=\left\{1,2,3,4\right\}$ .

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

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)

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