Event A constitutes a complementary event to Event B if it includes all of the possible results not included in Event B.

For example, in the case of throwing one of the dice:

• Event A consists of obtaining the result 6.

• Event B consists of obtaining the results 1, 2, 3, 4, or 5.

These two events are complementary events because all of the possible results not appearing in Event A are the same possibilities appearing in Event B, and vice versa.

Therefore, when Event B is a complementary event to Event A, then it automatically follows that Event A is a complementary event to Event B. If we group the results that define Event A (6) and the results that define Event B (1, 2, 3, 4, and 5) into a single group, then we obtain the entire sample space (1, 2, 3, 4, 5, and 6).

The probability that one of the complementary events will occur is therefore always 1 (or, 100%).

Every result that we obtain must cause one of the complementary events to occur.

Additional Examples of Complementary Events

Example 1:

The trial: Throwing one of the dice.

• Event A: An even number is obtained.

• Event B: An odd number is obtained.

Are these two events complementary events?

The answer is yes.

The explanation:

The possible results in Event A are 2, 4, and 6. We now ask ourselves what are the possible results not included in Event A.The answer is 1, 3, and 5, but these results are exactly the results that Event B defines. The events are therefore complementary.

Example 2

The trial: Throwing one of the dice.

• Event A: An even number is obtained.

• Event B: A number less than 4 is obtained.

Are these two events complementary events? The answer is no.

The explanation: The possible results of Event A are 2, 4, or 6. The possible events in Event B are

1, 2, or 3. Event B does not include all of the possible results not included in Event A, and it even includes one result that is also included in Event A (the result 2). This is an important rule:

If two events are complementary, then it is impossible for results to appear in both of the events.

Calculating Probability Using Complementary Events

This section shows how complementary events are used in calculating the probability. 

If Event A and Event B are complementary events, then the probability of Event A plus the probability of Event B is exactly equal to 1 (or, 100%). The following case demonstrates how we can use this information.

We will use the trial of throwing two dice simultaneously. We have already seen the sample space of this trial, and we have also seen that the size of the sample space is 36. We will define the event:

“Two different numbers are obtained.” If we pursue the ordinary course that we have learned, we will visually present the event to identify its size.

Presenting the event visually will look like this (see the next slide for further explanation):

The empty cells include the results in which the numbers are equal, or, in other words with the results that are not included in the event.

The size of the event is 30 since there are 30 pairs of dice showing different numbers.

We can therefore calculate the probability of the event:

The size of the event (30) is divided by the size of the sample space (36).

The probability is therefore 30/36 = 5/6. This is a long method because writing the event takes a long time.

We will consider a different method using complementary events: First, we note that the event we are talking about (i.e., different numbers are obtained) has a complementary event:

“Equal numbers are obtained”. If we consider the previous chart relating to our event, we can see that it actually represents the entire sample space of throwing two dice, except for the diagonal (from left to right) that includes the cases in which the numbers are identical.

In the preceding example, we already calculated the probability of obtaining identical numbers: i.e., 1/6. Our event, in which different numbers are obtained, is a complementary event to the event in which identical numbers are obtained.

The probability of our event (i.e., different numbers are obtained) is therefore: 1 – 1/6 = 5/6.