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.
