The complement rule works off of the idea that two parts make a whole. In probability, the "whole" refers to all possible outcomes. I find it's easiest to think of this as being 100%, which we know as a decimal value is simply 1. Thus, the sum of the probabilities of all possible outcomes must equal 1.
There are a few situations where this rule comes in handy. Let's take a look at those.
The most common application of this rule is when we see probabilities that use the phrasing of "at least 1". For example, let's say a group of 25 students had to indicate if they were eating lunch at school or not (yes/no). You want to find the probability of getting at least 1 person that will be eating at school.
Applying the thought process we've been using, there are 26 possible outcomes: 0-25. The desired outcomes include outcomes 1 through 25. That means we would need to compute the probability for each of those value individually, which is tedious!
Looking at this another way, we can see that while 1-25 are desired outcomes, the outcome of 0 is not desired. Using the logic of two parts make a whole, we can apply the following logic to approach this problem:
[probability of desired outcomes] + [probability of not desired outcomes] = 1
We can rearrange this to state that the [probability of desired outcomes] = 1 - [probability of not desired outcomes]. Therefore, if we find the probability of getting 0 people, we can subtract that from 1 to find the probability for this scenario. Let's say the probability that no one each lunch at school is .06. That means the probability of at least one person eating lunch at school is 1 - .06 = .94 or 94%.
This approach works with any situation where you can divide the outcomes into desired (successful) and not desired (failure) outcomes. We'll explore scenarios similar to this more on the binomial probability tab.
Another handy time to use this rule is to find a missing probability from a probability distribution table. Let's take a look at the following example:
In this table, the "X" column denotes the possible outcomes. The "Probability" column denotes the probability of achieving each outcome. Notice that the table does not provide the probability for "orange".
Applying an understanding of basic probability rules, we can compute the missing probability. We know that the probability column must sum to 1 in order to be a proper probability distribution, therefore, we can add up the probabilities given and subtract from 1 to find the missing probability value:
.21 + .17 + .36 + .14 = .88
1 - .88 = .12
Therefore, the probability of getting orange must be .12.