PROBLEM SOLVING ON VENN DIAGRAMS
Question 1
A survey is taken of 120 people and it is found that 83 people are informed about daily news by reading
the newspaper and 69 by watching the TV. 45 are informed by both watching the TV and reading the
newspaper.
a. Create a Venn Diagram to represent this situation.
3 marks
b. Find the probability that a randomly selected person is informed of the news by:
i. watching TV but not by reading the newspaper
ii. either reading the newspaper or watching the TV
iii. reading the newspaper but not watching the TV
iv . neither reading the newspaper nor watching the TV.
4 marks
Question 2
In a group of 65 students, 46 listen to triple J and 40 listen to Fox FM. 14 listen to Fox FM but not Triple
J. Let J be the set of students who listen to triple J and let F be the set of students who listen to Fox FM.
a. Create a Venn diagram to reflect this information.
3 marks
b. i. Describe the students who belong to the set J  F
ii. If a student is selected at random, find the probability that this student belongs to the set
J  F
c. i. Find the probability that a randomly selected student listens to neither triple J nor Fox FM.
ii. How would we write using mathematical symbols the set of studens who “listen neither
to triple J nor Fox FM”
d. i. Find the probability that a randomly selected student listens to either triple J or FM.
ii. How would we write using mathematical symbols the set of students who “listen either to
triple J or to Fox FM” ?
6 marks
Question 3
In a group of 30 Year 10 students, 22 play chess and 18 play soccer. There are 3 who play neither of
these two games. Let C represent all students who play chess and S represent all those who play soccer.
a. Draw a Venn Diagram to reflect this information.
4 marks
A student from among the thirty is selected at random. Find:
i.
The probability that he plays both chess and soccer.
ii.
The probability that he plays chess but not soccer.
iii.
Find: Pr ( C  S )
iv.
Find: Pr( C   S )
4 marks
Question 4
A survey of 100 people at an icecream parlour found that:
49 liked chocolate
40 liked vanilla
20 liked vanilla and chocolate
11 liked all three flavours
14 liked vanilla and strawberry
17 liked exactly two flavours
Create a Venn Diagram to reflect this information, and use it to find:
4 marks
i.
The probability that a randomly selected customer likes only chocolate
ii.
The probability that a randomly selected customer likes chocolate and strawberry but not
vanilla
iii.
The probability that a randomly selected customer likes only one flavor of icecream.
3 marks
Question 5
In the kingdom of Mathmania there are 2000 mathematicans. 920 of these list
Algebra as their favourite topic, while 1446 list Trigonometry as their favourite
topic. There are 220 who do not list either Algebra or Trigonometry as their
favourite topic.
a. Create a Venn Diagram to illustrate this information. Let
A = {mathematicians who list Algebra as a favourite subject} and let
T = {mathematicians who list Trigonometry as a favourite subject}
4 marks
b. If a mathematician is selected at random, find the probability that he or she
has listed both Algebra and Trigonometry as his/her favourite subject.
1 mark
c. i. Find the probability that a randomly selected mathematician has listed
Algebra or Trigonometry as his/her favourite subject.
ii. Write in mathematical symbols the set :”mathematicians who listed
Trigonometry or Algebra as their favourtite subject”.
2 marks
d. i. Find Pr ( A T’)
ii. Describe the mathematicians who would be included in this set.
2 marks
e. i. Find the value of 𝑛(𝐴′ ∩ 𝑇)
ii. What does this number represent ?
2 marks
f. i. Find Pr⁡(𝐴′ ∩ 𝑇 ′ )
ii. How would you describe the mathematicians who are in the set 𝐴′ ∩ 𝑇 ′
2 marks
Question 6
At the Melbourne Cup it was observed that of the 22 horses racing, all the horses running had
one or more of the colours white, black and brown in their coat. It was observed that 2 had all
three colours in their coat, while 6 had brown and white. Three had brown and black colouring,
and three had black and white colouring only. In total, there were 10 horses with white
colouring, 14 horses with black colouring and 10 horses with brown colouring.
a. Create a Venn Diagram to illustrate this information.
5 marks
b. If a horse is selected randomly from the 22, what is the probability that it will be entirely
white?
c. Pr (Black ∪ Brown⁡ ∩ White′)
2 marks
One of the horses running is named Toofarinfront. Tom Waterhouse is offering odds of
11-2 for Toofarinfront to win.
d.
i.
According to Tom Waterhouse, what is the probability that Toofarinfront will win
the Melbourne Cup?
ii.
Jeremy is confident that Toofarinfront will win, and places a bet of $150 on
Toofarinfront with Tom Waterhouse. If Jeremy is right, and Toofarinfront wins,
what will Jeremy’s total payout be?
3 marks