Lesson Video: Complementary Events Mathematics

Video Transcript

In this video, we will learn how to find the probability of complementary events. We will begin by recalling some probability rules that we should already know and then define what we mean by a complementary event.

We know that for any event 𝐴, the probability of event 𝐴 occurring, written 𝑃 of 𝐴, must be greater than or equal to zero and less than or equal to one. If the probability is equal to zero, it is impossible, and if the probability equals one, it is certain to happen. A probability can be written as a fraction, a decimal, or a percentage. If it is written as a percentage, then the probability of 𝐴 is greater than or equal to zero percent and less than or equal to 100 percent. We also know that the sum of the probabilities of all possible outcomes of an event is equal to one.

For example, let’s assume we have four red balls and three blue balls in a bag. The probability of selecting a red ball is four out of seven or four-sevenths. The numerator is the number of successful outcomes, in this case, the four red balls. The denominator is the number of possible outcomes, in this case, seven as there are seven balls in total in the bag. The probability of selecting a blue ball is three-sevenths as there are three blue balls in the bag. As there are only two colors, the probability of selecting a red ball plus the probability of selecting a blue ball must equal one. Four-sevenths plus three-sevenths is equal to seven-sevenths, which equals one.

The complement of an event is all the possible outcomes that are not our event. This leads us to a third rule. The probability of the complement of event 𝐴 — in other words, not 𝐴, which is written 𝐴 bar — is given by the probability of not 𝐴 is equal to one minus the probability of 𝐴. The complement is also sometimes written as 𝐴 prime. Using the example above, we can now calculate the probability of not selecting a red ball, the complement of red. This is equal to one minus four-sevenths as four-sevenths was the probability of selecting a red ball. One minus four-sevenths is equal to three-sevenths. The probability of not selecting a red ball is three-sevenths.

We notice that this is the same as the probability of selecting a blue ball. This is because there were only two possible options when selecting a ball from the bag, either red or blue. The probability of not selecting a red ball will be equal to the probability of selecting a blue ball and vice versa. We will now look at some questions where we need to calculate the probability of a complementary event.

If the probability of an event happening is 11 out of 30 or eleven thirtieths, what is the probability of the event not happening?

We know that the probability of an event not happening is known as the complementary event. This is denoted by 𝐴 bar or 𝐴 prime, where the probability of 𝐴 bar is equal to one minus the probability of 𝐴. In this question, we are told the probability of the event happening is 11 out of 30. If we let this be event 𝐴, then the probability of 𝐴 is equal to eleven thirtieths or 11 out of 30. We can then calculate the probability of the event not happening by subtracting this from one. One whole one is the same as 30 out of 30 or thirty thirtieths. As the denominators are the same, we simply subtract the numerators, giving us 19 out of 30 or nineteen thirtieths. The probability of the event not happening is therefore equal to 19 out of 30.

Our next question involves calculating probability using decimals and percentages.

If the probability that a student passes an exam is 39 percent, what is the probability that the student fails?

As there are only two possibilities in this scenario, the student could either pass or fail, they are the complements of each other. We know that the probability of any complementary event, denoted 𝐴 bar, is equal to one minus the probability of 𝐴. When dealing with percentages, the one is equal to 100 percent. As the probability of the student passing is 39 percent, the probability of the student failing will be 100 percent minus 39 percent. This is the same as saying that the student does not pass the exam and is equal to 61 percent.

We could also write this answer as a fraction or a decimal. As percentages are out of 100, this can be written as a fraction as 61 out of 100 or sixty-one one hundredths. As a decimal, this is equal to 0.61 as the line in a fraction means divide and 61 divided by 100 is 0.61. The probability that the student fails the exam is 61 percent, 61 out of 100, or 0.61.

An alternative method here would be to convert 39 percent into the fraction thirty-nine one hundredths or the decimal 0.39 first. We could then subtract either of these from one to calculate the complement, which is equal to sixty-one one hundredths or 0.61.

Our next two questions are more complicated problems in context.

A box contains 56 balls. The probability of selecting at random a red ball is five-sevenths. How many balls in the box are not red?

The event of selecting a red ball and the event of selecting a ball that is not red are complementary events. The probability of the complement of 𝐴, denoted 𝐴 bar, is equal to one minus the probability of 𝐴. In this question, we are told that the probability of selecting a red ball, 𝑃 of 𝑅, is equal to five-sevenths. This means that the probability of selecting a ball that is not red is one minus five-sevenths. This is equal to two-sevenths. The probability of an event occurring and its complement must always sum to one.

We are also told in this question that there are 56 balls in the box. Two-sevenths of these 56 balls are not red, so we need to calculate two-sevenths of 56. As the word “of” in mathematics means multiply, we need to multiply two-sevenths by 56. 56 is the same as 56 over one. We can then cross cancel or cross simplify by dividing 56 and seven by seven. This gives us two over one multiplied by eight over one, which is equal to 16 over one. We multiply the numerators and denominators separately. As this is equal to 16, we can conclude that 16 of the 56 balls in the bag are not red.

An alternative method here would be to calculate the number of red balls first. We can do this by working out five-sevenths of 56. This is equal to 40, so we have 40 red balls in the box. This means that the rest of the balls must not be red. 56 minus 40 is equal to 16. This once again proves that there are 16 balls in the box that are not red.

A class has 45 students. The probability of choosing at random a student whose age is 10 or less is two-thirds. How many students in the class are 11 or older?

There are two possibilities when selecting a student in this question. They could be aged 10 or less, or 11 or older. These are known as complementary events. We know that the probability of any complementary event, 𝐴 bar, occurring is equal to one minus the probability of 𝐴, the event itself occurring. We are told in the question that the probability of choosing a student whose age is 10 or less is equal to two-thirds. This means that the probability of selecting a student who is not 10 or less is equal to one-third as one minus two-thirds is one-third. This is the same as saying the probability of selecting a student who is 11 or older is one-third. One-third of the 45 students are aged 11 or older.

We can calculate this by multiplying one-third by 45. Multiplying any number by one-third is the same as dividing the number by three. We know that four divided by three is equal to one remainder one. 15 divided by three is equal to five. As 45 divided by three is equal to 15, one-third multiplied by 45 is also 15. There are 15 students in the class who are aged 11 or older. An alternative method here would be to calculate two-thirds of 45 first. This is the number of students who are aged 10 or less. As one-third multiplied by 45 is 15, two-thirds multiplied by 45 is 30. There are 30 students in the class whose age is 10 or less.

As 30 students are aged 10 or less, we can subtract this from 45 to calculate the number of students that are 11 or older. Once again, this gives us an answer of 15 students.

Our final question in this video involves using a two-way frequency table.

The table represents the data collected from 200 conference attendees of different nationalities. Find the probability that a randomly selected participant does not speak English.

The rows in our table tell us whether the participant was male or female. The columns tell us which language they speak, whether they speak Arabic, English, or French. We are told in the question that there are a total of 200 attendees. If we let 𝐸 be the event that the conference attendee speaks English, we can calculate the probability of event 𝐸. This will be the number of attendees that speak English out of the total number of attendees.

There are 35 men who speak English and 30 women, giving us a total of 65 people. The probability that a randomly selected participant speaks English is 65 out of 200 or sixty-five two hundredths. We are interested in the probability that the participant does not speak English. This is known as the complement. We know that the probability of any complementary event, 𝐴 bar, occurring is equal to one minus the probability of 𝐴. In this question, the probability of 𝐸 bar, the participant not speaking English, is equal to one minus 65 out of 200. This is equal to 135 out of 200.

We can simplify this fraction by dividing the numerator and denominator by five. 135 divided by five is 27 and 200 divided by five is equal to 40. The probability that a randomly selected participant does not speak English is 27 out of 40 or twenty-seven fortieths. We could also write this answer as a decimal by firstly considering the fraction 135 out of 200. Dividing the denominator by two gives us 100. If we divide the numerator by two, we get 67.5 as a half of 100 is 50 and a half of 35 is 17.5. Dividing 67.5 by 100 gives us 0.675. The probability that the randomly selected participant does not speak English, written as a decimal, is 0.675. We could also write this as a percentage by multiplying 100, giving us 67.5 percent.

We will now summarize the key points from this video. For any event 𝐴, if 𝑃 of 𝐴 is the probability of event 𝐴 occurring, the following rules exist. When the probability is written as a fraction or a decimal, 𝑃 of 𝐴 must be greater than or equal to zero and less than or equal to one. As a percentage, it must lie between zero and 100 percent. The sum of the probabilities of all outcomes must equal one. We also found that the probability of the complement of 𝐴, denoted 𝐴 bar, is equal to one minus the probability of 𝐴. The complement means the probability of the event not occurring. It is also important to note that the complement of an event is also sometimes written as 𝐴 prime.