Survey Problems: An Application of Venn Diagrams
Sometimes we may be interested in how many elements are in the union or intersection of sets, but not know the actual elements of each set. This is common in Surveying. We can often use a Venn diagram in which the number of elements in each region of the diagram are indicated (rather than a list of the elements themselves) to help in solving such problems. To simplify our notation, we will denote the number of people in a set, say A, as n(A).
Example
A survey asks 200 people “What beverage do you drink in the morning”, and offers choices:
- Tea only
- Coffee only
- Both coffee and tea
Suppose 20 report tea only, 80 report coffee only, and 40 report both.
- How many people drink coffee in the morning?
- How many people drink neither tea or coffee?
Show Solution
This question can most easily be answered by creating a Venn diagram in which the number of elements in each region (rather than the elements themselves) are identified.
- We see that we can find the people who drink coffee by adding those who drink only coffee to those who drink both: 80+40 = 120 people.
- We can also see that those who drink neither are those not contained in the any of the three other groupings. We know that the cardinality of the universal set is 200 since that is the number of people who were surveyed. Thus we need to subtract the total number who drank coffee, tea, or both (80+40+20=140) from the cardinality of the universal set, 200.
200 – 140 = 60 people who drink neither.
Example
A survey asks: Which online services have you used in the last month:
- Twitter
- Facebook
- Have used both
The results show 40 of those surveyed have used Twitter, 70 have used Facebook, and 20 have used both. If 100 people were surveyed, how many have used neither Twitter or Facebook?
Show Solution
Let T be the set of all people who have used Twitter, and F be the set of all people who have used Facebook. Notice that while the number of people in set F is 70 and the number of people in set T is 40, the number of people in set F ⋃ T is not simply 70 + 40, since that would count those who use both services twice.
To find the number of people in set F ⋃ T, denoted n(F ⋃ T), we can add n(F) and n(T), then subtract those in intersection that we’ve counted twice.
That is,
n(F ⋃ T) = n(F) + n(T) – n(F ⋂ T)
n(F ⋃ T) = 70 + 40 – 20 = 90.
Now, to find how many people have not used either service, we’re looking for the number of people in (F ⋃ T)c. Since the universal set contains 100 people and n(F ⋃ T) = 90 people, then the number of people in (F ⋃ T)c must be the other 10 people.
important properties
The previous example illustrated the following two important properties:
-
- n(A ⋃ B) = n(A) + n(B) – n(A ⋂ B)
- n(Ac) = n(U) – n(A)
Notice that the first property can also be written in the following equivalent form:
n(A ⋂ B) = n(A) + n(B) – n(A ⋃ B)
Example
Fifty students were surveyed, and asked if they were taking a social science (SS), humanities (HM) or a natural science (NS) course the next quarter.
| 21 were taking a SS course |
26 were taking a HM course |
| 19 were taking a NS course |
10 were taking HM and NS |
| 7 were taking SS and NS |
9 were taking SS and HM |
| 3 were taking all three |
7 were taking none |
How many students are only taking a SS course?
Show Solution
It might help to look at a Venn diagram. You should draw out this diagram and label the numbers we find as we go.
From the given data, we know:
- There are 3 students in region e.
- There are 7 students in region h.
- Since 7 students are taking a SS and NS course, we know that n(d) + n(e) = 7; since there are 3 students in region e, there must be 7 – 3 = 4 students in region d.
- Similarly, since there are 10 students taking HM and NS, which includes regions e and f, there must be 10-3 = 7 students in region f.
- Since 9 students are taking SS and HM, there must be 9 – 3 = 6 students in region b.
Now, we know that 21 students are taking a SS course. This includes students from regions a, b, d, and e. Since we know the number of students in all but region a, we can determine that 21 – 6 – 4 – 3 = 8 students are in region a.
We conclude that 8 students are taking only a SS course.
Example
One hundred fifty people were surveyed and asked if they believe in UFOs, ghosts, and/or Bigfoot.
| 43 believe in UFOs |
8 believe in ghosts and Bigfoot |
| 25 believe in Bigfoot |
5 believe in UFOs and Bigfoot |
| 44 believe in ghosts |
2 believe in all three |
| 10 believe in UFOs and ghosts |
|
How many people surveyed believe in none of these things?
Show Solution
Again, we begin with a Venn diagram as shown.
Starting with the intersection of all three circles, we work our way out.
- Since 10 people believe in UFOs and Ghosts, and 2 believe in all three, that leaves 8 that believe in only UFOs and Ghosts.
- Similarly, 3 believe in only UFOs and Bigfoot while 6 believe in only Ghosts and Bigfoot.
- Using the same process, we work our way out, filling in all the regions. Once we have, we can add up all those regions, getting 91 people in the union of all three sets.
- This leaves 150 – 91 = 59 who believe in none of these things.
