1.6.1 Complement Rule

The complement rule states that for an event , the probability of its complement  is:

           

Complement Rule Proof Recall from the complement laws of sets (section 1.3.7.3), the union of a set  and its complement set , is equal to the universal set , represented as:                  (1.1) In the context of probability, set  is the event and the universal set  is the sample set, since it contains all the possible outcomes of the experiment, thus from equation 1.1, it follows:                 (1.2)  and  are disjoint events, so, by the axiom of additivity:         (1.3) Also, by the axiom of normalization:         (1.4) Substituting 1.4 into 1.3 yields:         Hence:

Example 1.27. Consider an airport where, historically, 6 out of 10 flights have departed on time. So, what is the probability of a randomly selected flight NOT departing on time (i.e., getting delayed)?"

Let’s start by defining the event  and its complement :

• : Probability of the flights departing on time
• : Probability of the flights NOT departing on time

The probability of  can be calculated as:

        

        

By complement rule:

        

        

We get the same result if we calculate the probability of directly:

        

        

        

 

Thus, the probability of a flight NOT departing on time is 40%. Hopefully, the airport has a plan in place to address this issue.