Explainer: Complementary Events

In this explainer, we will learn how to find the probability of complementary events.

In the context of probability, an event is a possible outcome or set of outcomes. If we think about the days of the week, for example, every day in the calendar is 1 of 7 days of the week. In many countries, the days Saturday and Sunday are classified as the weekend and an event might be “born on the weekend.” There are two possible outcomes in this set: {Saturday, Sunday}.

If we assume that births are equally likely to occur on any given day of the week, the probability that a person chosen at random was born on the weekend is 27, since there are 2 weekend days and 7 days in total every week.

The complement of an event is all the possible outcomes that are not our event. So, in our days of the week example, the complement to weekend days is the set {Monday, Tuesday, Wednesday, Thursday, Friday}, that is, all the days that are not weekend days. The complement of an event and the event itself cannot happen at the same time. In our example, we know that it cannot be both a weekday and a weekend day at the same time. We can say that the events “weekday” and “weekend day” are mutually exclusive: there is no overlap between them.

Let us translate what we know so far into some probability rules.

Some Probability Rules

For any event 𝐴, if 𝑃(𝐴) is the probability of event 𝐴 occurring, the following rules exist:

  • Rule 1: 0𝑃(𝐴)1.
  • Rule 2: The sum of the probabilities of all possible outcomes is equal to 1.
  • Rule 3: The probability of the complement, 𝐴, of event 𝐴, or “Not 𝐴”, is given by 𝑃𝐴=1𝑃(𝐴).

Note

The complement of event 𝐴 is sometimes also written as 𝐴.

Let us look at an example of working out the probability of a complementary event.

Example 1: Finding Probabilities for Complementary Events

The probability that an event occurs is 16. What is the probability that the event does not occur?

Answer

If we know the probability of an event occurring, we can work out the probability that it does not occur, that is, the probability of its complement, by subtracting its probability from 1: 𝑃𝐴=1𝑃(𝐴)=116=56.

Thus, the probability that the event does not occur is 56.

In our next example, we will work out the probability of a complementary event where the probabilities are decimals.

Example 2: The Probability of a Complementary Event

The probability that a student passes their mathematics exam is 0.73. What is the probability that they fail?

Answer

If the probability that a student passes their exam is 0.73, then the probability that they fail is 𝑃()=1𝑃()=10.73=0.27.FailPass

Hence, the probability that the student fails their mathematics exam is 0.27. We can convert this into a percentage by multiplying our answer by 100: 100×0.27=27%, and we interpret this as follows: the student has a 27% chance of failing their mathematics exam.

Now let us try an example of working out the probability of a complementary event on a fair spinner.

Example 3: Complementary Probabilities for Events on a Spinner

What is the probability of the pointer landing on a section which is not green on the pictured fair spinner?

Answer

To find the probability that the spinner lands on a section which is not green, we need to know how many sections there are in total and how many of those are not green.

There are 8 sections in total on the spinner and 5 of those are not green. If the probability that the spinner lands on a green section is 𝑃(𝐺), then 𝑃()=𝑃𝐺==58.notgreennumberofnongreensectionstotalnumberofsections

Note that we could also have calculated this by counting the number of green sections and using the rule 𝑃𝐺=1𝑃(𝐺). That is, 𝑃𝐺=138=58.

The probability that the spinner lands on a section that is not green is therefore 58.

Example 4: Working Out Complementary Probability Using the Total Probability = 1

A survey of 48 students reveals that 28 of the students prefer playing football and 11 prefer basketball. The rest of the students prefer tennis. If a student is chosen at random, what is the probability that they prefer tennis?

Answer

Our aim is to find the probability that a student chosen at random prefers tennis. We know that there are 48 students in total and that 28 prefer football and that 11 prefer basketball.

We can calculate the probability that a student prefers tennis in two different ways.

  1. The first method is to work out the probability that a student chosen at random does not prefer tennis and subtract this from 1. The number of students who do not prefer tennis is 28()+11()=39footballbasketball. Therefore, the probability that a student chosen at random does not prefer tennis is 𝑃()==3948.nottennisnumberofnon-tennisplayerstotalnumberofstudents Now we can use the fact that 𝑃()=1𝑃()tennisnottennis to find the probability we seek: 𝑃()=1𝑃()=13948=948.tennisnottennis Hence, the probability that a student chosen at random prefers tennis is 948. Since both the numerator and the denominator are multiples of 3, we can write this in its simplest form as 316.
  2. Our second method of finding the probability that a student chosen at random prefers tennis is simply to work out the number of students who prefer tennis.
    Then we divide this number by the total number of students. The probability that a student chosen at random prefers tennis is therefore 948, which simplifies to 316, as before.

In our next example, we will calculate the probability of a complementary event using information from a table.

Example 5: Complementary Probabilities from a Table

The table represents data on the first languages of 200 attendees of a computer game convention of different nationalities.

ArabicEnglishFrenchTotal
Man453545125
Woman4030575
Total856550200

Find the probability that a randomly selected participant does not have English as their first language.

Answer

There are two ways to find the probability that a participant chosen at random does not have English as their first language.

  1. The first method is to work out the number of participants who do not have English as their first language. Then, dividing by the total number of participants gives us the required probability.
    The number of participants who do not have English as their first language is 85()+50()=135ArabicFrench. The total number of participants is 200. Out of the total 200, 135 do not have English as their first language, so, the probability that a participant chosen at random does not have English as their first language is 135200.
  2. The second method for finding the probability that a participant does not have English as their first language is to find the probability that a participant does have English as their first language and subtract this from 1.
    Hence, the probability that a participant chosen at random does not have English as their first language is 135200=0.675.

Note that we can work out the percentage of participants who do not have English as their first language by multiplying our result by 100: 135200×100=67.5, and we can interpret this as follows: 67.5% of participants do not have English as their first language.

Our final example is a little more abstract.

Example 6: Complementary Probabilities in a Venn Diagram

Suppose 𝐴 and 𝐵 are two events. Given that the probability of 𝐴 or 𝐵 occurring is 0.67, find the probability that neither 𝐴 nor 𝐵 occurs.

Answer

To find the probability that neither 𝐴 nor 𝐵 occurs, let us put the information we have into Venn diagrams. There are two possible scenarios: either 𝐴 and 𝐵 are mutually exclusive (they cannot occur at the same time and are disjoint events (left-hand picture below)), or they are not mutually exclusive and they can occur at the same time (the right-hand picture below), and they have some overlap.

Recall that in probability “𝐴 or 𝐵” means either 𝐴 or 𝐵 or both and is written as 𝐴𝐵. This is called the union of 𝐴 and 𝐵. In our Venn diagrams, all of the green colored areas belong to 𝐴𝐵. Everything in green is either 𝐴 or 𝐵 or both.

Everything else (i.e., everything in the blue areas in the diagrams) is neither 𝐴 nor 𝐵 nor both. We know that the sum of all possible outcomes is equal to 1 and we know that the probability of 𝐴𝐵=𝑃(𝐴𝐵)=0.67or (green). Hence, the probability of neither 𝐴 nor 𝐵 is given by 𝑃𝐴𝐵=1𝑃(𝐴𝐵)=10.67=0.33.

Let us remind ourselves of the main probability rules associated with complementary events.

Key Points

For any event 𝐴, if 𝑃(𝐴) is the probability of event 𝐴 occurring, the following rules exist:

  • Rule 1: 0𝑃(𝐴)1.
  • Rule 2: The sum of the probabilities of all possible outcomes is equal to 1.
  • Rule 3: The probability of the complement, 𝐴, of event 𝐴, or “Not 𝐴”, is given by 𝑃𝐴=1𝑃(𝐴).

Note

The complement of event 𝐴 is sometimes also written as 𝐴.

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