Video Transcript
Out of a group of 100 people, 46
have dogs, 41 have cats, and 28 have rabbits. 12 of the people have both dogs and
cats, 10 have both cats and rabbits, and nine have both dogs and rabbits. Eight of the people have dogs,
cats, and rabbits. There are four parts to this
question. Find the probability of randomly
selecting a person who has dogs, cats, and rabbits. Find the probability of randomly
selecting a person who has only dogs and rabbits. Find the probability of selecting a
person who has pets. And find the probability of
selecting a person who does not have pets.
In all four parts, we’re asked to
give our answer as a fraction in its simplest form. The first two parts can be answered
directly from the question. We are told that the group contains
100 people, and eight of them have dogs, cats, and rabbits. This means that the probability of
randomly selecting a person who has dogs, cats, and rabbits is eight out of 100. Dividing both the numerator and
denominator by four, this simplifies to two out of 25 or two twenty-fifths. We are also told in the question
that nine of the people have both dogs and rabbits. This is a subset of dogs, cats, and
rabbits, those that have all three animals. And we know that eight people have
dogs, cats, and rabbits. We can therefore subtract eight
from nine to calculate the number of people that have just dogs and rabbits. This is equal to one. And we can therefore conclude that
the probability of randomly selecting a person who has only dogs and rabbits is one
out of 100.
An alternative way to visualize
this would be to use a Venn diagram. We will now clear some space to
draw this. Our Venn diagram has three
sections, one for dogs, one for cats, and one for rabbits. The parts of the Venn diagram where
these sections overlap represent the people with two or more pets. As eight people have all three
pets, we can place an eight in the intersection of the dogs, cats, and rabbits. As nine people have both dogs and
rabbits and eight of these have all three pets, only one person has only a dog and a
rabbit. In the same way, we are told 10
people have both cats and rabbits. This means that two people have
only cats and rabbits. Likewise, since a total of 12
people have both dogs and cats, four of these will have only dogs and cats.
Next, we see that 28 people have
rabbits. This means that the sum of the
numbers in the rabbit section must equal 28, and there are therefore 17 people who
have only rabbits. Repeating this for cats and dogs,
we see that 27 people have only cats and 33 people have only dogs. In order to calculate the number of
people who have pets, we can find the sum of the seven numbers in the Venn diagram
currently. This is equal to 92, and this means
that the probability of randomly selecting a person who has pets is 92 out of
100. As the numerator and denominator
are both divisible by four, this simplifies to 23 out of 25. The probability of selecting a
person who has pets is twenty-three twenty-fifths.
As the sum of all of our numbers in
the Venn diagram must equal 100, we can calculate the number that lies outside the
three sections by subtracting 92 from 100. There are eight people in the group
who do not have pets. Therefore, the probability of
randomly selecting a person who does not have pets is eight out of 100. Once again, this fraction can be
simplified by dividing the numerator and denominator by four. The probability of randomly
selecting a person who does not have pets is two twenty-fifths.
