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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 $\bar{A}$ .

Example 1:

Consider the experiment of rolling a single die.
$S = {1, 2, 3, 4, 5, 6}$ . Let $A$ be the event that the roll yields a number greater than $4$ . $A = {5, 6}$ . Then $\bar{A} = {1, 2, 3, 4}$ .

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

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)

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