Video Transcript
Out of a group of 55 people, 26
like bananas, 25 like apples, and 18 like oranges. Eight of the people like both
bananas and apples, nine like both apples and oranges, and seven like both bananas
and oranges. Five of the people like bananas,
apples, and oranges. Find the probability of randomly
selecting a person from the group that does not like any of these three fruits. Give your answer as a fraction in
its simplest form.
To answer this question, we will
sketch a Venn diagram of the group of people. Let’s call the group of 55 people
the sample space 𝑆, the group of people who like bananas 𝐵, the group of people
who like apples 𝐴, and the group of people who like oranges 𝑂.
We know that five people from the
group like bananas, apples, and oranges. So the intersection of these three
sets has five members. This means the overlapping part of
the three circles in the Venn diagram contains five outcomes. We also know that eight people like
both bananas and apples. Removing the five who like all
three fruits, we have three people that like bananas and apples but not oranges. We add this number to the Venn
diagram.
We also know that nine people like
both apples and oranges. Removing the five people who like
all three fruits, we have four people that like apples and oranges but not
bananas. We add this new information to the
Venn diagram. We have also been told that seven
people like both bananas and oranges. Removing the five people who like
all three fruits, we have two people that like bananas and oranges but not
apples. We use this to update the Venn
diagram once again.
Now we will follow the same process
to fill in the rest of the Venn diagram. We know that 26 people like
bananas. We need to remove the number of
people who like bananas we have already accounted for. This includes the five that like
all three fruits, the three that like only bananas and apples, and the two that like
only bananas and oranges. Thus, we subtract 10 from the total
of 26. This leaves 16 people who like only
bananas. So we write 16 in the empty part of
circle 𝐵.
We also know that 25 people like
apples. Now we need to remove the five that
like all three fruits, the three that like only bananas and apples, and the four
that like only apples and oranges. So we subtract 12 from the total of
25. This leaves 13 people who like only
apples. We add this new information to the
Venn diagram.
Finally, we know that 18 people
like oranges. We need to remove the five that
like all three fruits, the four that like only apples and oranges, and the two that
like only bananas and oranges. This means we subtract 11 from the
total of 18. We’re left with seven people who
like only oranges. This completes the values needed to
fill in the Venn diagram.
Now that the Venn diagram is
completely filled in, we can calculate the total number of people who like bananas,
apples, or oranges. The number of people accounted for
in the Venn diagram that like one or more of the three fruits is five plus three
plus four plus two plus 16 plus 13 plus seven, which equals 50.
We are asked to find the
probability of randomly selecting a person from the group of 55 that does not like
any of these three fruits. Since 50 of the 55 people like one
or more of these three fruits, that leaves five people that do not like any of
them. So we write the number five in the
sample space outside of the Venn diagram.
To find the desired probability, we
divide the number of people that do not like any of these three fruits by the total
number of people in the sample space, that is, five over 55, which simplifies to one
over 11. Therefore, the probability of
randomly selecting a person from the group that does not like any of these three
fruits is one eleventh.
