# What Are Complementary Events in Probability?

A compound event $A$ contains one or more of the outcomes in the sample space. The outcomes that are left are read as “not $A$”, and can be written as $\overline{A}$. $A$ and $A$ with a dash on top are then complementary events. Then you know that:

Rule

### ComplementaryEvents

 $P\phantom{\rule{-0.17em}{0ex}}\left(A\right)+P\phantom{\rule{-0.17em}{0ex}}\left(\overline{A}\right)=1$

Note! An important application is that

Example 1

You roll a die once. What’s the probability of getting two dots or more on the die?

Here you’re looking to find the probability of getting 2, 3, 4, 5 or 6 dots on the die. The easiest way to calculate this is to use the application above:

 $P\phantom{\rule{-0.17em}{0ex}}\left(\ge 2\right)=1-P\phantom{\rule{-0.17em}{0ex}}\left(<2\right)$

The event fewer than two dots is the same as getting one dot on the die, and the number of possibilities is then 1. That means you can write

 $P\phantom{\rule{-0.17em}{0ex}}\left(\ge 2\right)=1-P\phantom{\rule{-0.17em}{0ex}}\left(1\right)=1-\frac{1}{6}=\frac{5}{6}.$

The probability of getting at least two dots when you roll a die is then $\frac{5}{6}=0.833$.