Complementary Events

What are Complementary Events?

In probability theory, the complement of any event $A$ is the event $[\text{not}\ A]$, i.e. the event in which $A$ does not occur. The event $A$ and its complement $[\text{not}\ A]$ are mutually exclusive and exhaustive, meaning that if one occurs, the other does not, and that both groups cover all possibilities. Generally, there is only one event $B$ such that $A$ and $B$ are both mutually exclusive and exhaustive; that event is the complement of $A$ . The complement of an event $A$ is usually denoted as $A'$, $A^c$ or $\bar{A}$.

Simple Examples

A common example used to demonstrate complementary events is the flip of a coin. Let's say a coin is flipped and one assumes it cannot land on its edge. It can either land on heads or on tails. There are no other possibilities (exhaustive), and both events cannot occur at the same time (mutually exclusive). Because these two events are complementary, we know that $P(\text{heads}) + P(\text{tails}) = 1$.

Coin Flip

Often in sports games, such as tennis, a coin flip is used to determine who will serve first because heads and tails are complementary events.

Another simple example of complementary events is picking a ball out of a bag. Let's say there are three plastic balls in a bag. One is blue and two are red. Assuming that each ball has an equal chance of being pulled out of the bag, we know that $P(\text{blue}) = \frac{1}{3}$ and $P(\text{red}) = \frac{2}{3}$. Since we can only either chose blue or red (exhaustive) and we cannot choose both at the same time (mutually exclusive), choosing blue and choosing red are complementary events, and $P(\text{blue}) + P(\text{red}) = 1$.

Finally, let's examine a non-example of complementary events. If you were asked to choose any number, you might think that that number could either be prime or composite. Clearly, a number cannot be both prime and composite, so that takes care of the mutually exclusive property. However, being prime or being composite are not exhaustive because the number 1 in mathematics is designated as "unique. "