Venn Diagrams and Contingency Tables
In a group of 120 people, there are 50 women and 24 engineers. Of course some of the engineers are
women: 12 of them, in fact. All of the people who are not engineers are doctors. We can represent this
group in the following way, using a Venn diagram.
120
The rectangle indicates the entire group, and the numbers in the
various parts show the number of people who belong to that subsection of the whole group: when a number is inside the W oval
and also inside the E oval, then that number tells us how many
people are in W and also in E: in other words, how many people
are women and at the same time engineers.
W
38
12
12
E
Note that the total number of engineers is 24 (i.e. 12 + 12), and the total number of women is 50 (i.e. 12 +
38).
(a)
(b)
(c)
(d)
(e)
How many people in the group are neither engineers nor women?
How many people in the group are male doctors?
If a person from the group is selected at random, what is the probability that that person is
an engineer?
a male engineer?
a male doctor?
If one of the women is chosen at random what is the probability that she is an engineer?
If one of the engineers is chosen at random, what is the probability that it is a woman?
(Note the important difference between this and the previous question)
Exactly the same situation can be represented differently, in what is called a Contingency Table.
In a similar way to what we did above, we can work out what goes into the blank cells, realising that the
total in the E column must be 24, including those that are W and also those that are not.
E
W
Not E
12
50
24
120
Not W
When we have completed the table it looks like this:
E
Not E
W
12
38
50
Not W
12
58
70
24
96
120
Now we can easily see that the probability that a randomly chosen member of the group is a male doctor
is
58
No. of non  women non  engineers
 0,483
=
No. of all people in group
120
For the probability that an engineer chosen randomly will be a woman, we look at
12
No. of women engineers
=
 0,5
No. of engineers
24
Now that we have established exactly how the group of people is made up, let us consider two scenarios:
I: go the group and pick a person at random
II: go to the group and pick an engineer at random.
We can ask in which case you are more likely to get a woman. It is easy to see that the probabilities for
50
12
the two cases are
and
, i.e. 0,417 and 0,5. The fact that these probabilities are different show us
120
24
that if we want to estimate the likelihood that the next person to come out of the room is a woman, we
would change our mind if we knew the profession of the person. In other words, Gender and Profession
are not entirely separate issues. Our estimate of the likely gender would depend on our knowledge of the
profession.
We say that two aspects are independent if knowledge about one does NOT cause us to change our
estimates about the other – this is an informal definition but it works.
In a separate seminar, the numbers of women/men and
engineers/doctors were as shown in the table alongside.
E
W
Complete the table and then, showing reasoning, decide whether
gender and profession are dependent or independent for this second
group.
Not E
12
40
36
120
Not W
Your conclusion should have been that the two are independent. The proportion of women who are
engineers is the same as the proportion of all people who are engineers. So just as 0,3 of the women are
engineers, so are 0,3 of all the people engineers. Another way of understanding this is: 13 of all the people
are women, and 0,3 of all the people are engineers. So 0,3 of 13 of the people are woman who are also
engineers: that is 0,1 of them. (0,3 of 13 = 0,1)
The technical definition of independence is:
A and B are independent if P(A and B) = P(A) × P(B).
Contingency tables do not always use actual numbers; they can use percentages or, more often,
proportions, i.e. probabilities. Then the number at bottom right is 1.
The example above about the second seminar would then appear as
W
Not W
E
Not E
1
10
1
5
3
10
7
30
7
15
7
10
1
3
2
3
1
1
3
1
1
3
, P(E) =
and P(W and E) =
, which is ×
, i.e. the product of P(W) and
3
10
10
3 10
P(E). This indicates the independence of W and E.
Note that P(W) =
Draw up a contingency table for this seminar but using percentages, so that the number at bottom right
will be 100.
EXERCISES
(Thanks to Mr Bizony of Bishops for this work.)
Copy and complete the following contingency tables.
A
1.
B
Not A
5
P
2.
Q
12
11
X
42
Not X
B
Not A
21
NotB
0,16
0,6
100
A
4.
0,3
Not W
48
20
W
5.
30
Not Q
Not B
3.
Not P
22
30
1
Copy the Venn diagram alongside, and work
out the value of x, showing your reasoning.
60
F
x
11
8
G
6.
Find the value of x in the Venn diagram
alongside for each of the following cases:
(a)
(b)
(c)
(d)
7.
(a)
What is the probability that randomly chosen
member of A is also a member of B?
(b) What is the value of P(A and B)?
(c) What are the values of P(A) and P(B)?
(d) Decide, giving reasons, whether A and B are
independent
34
60
A
P(B) = 1/3
P(notA and notB) = 1/3
P(B and notA) = P(notA and notB)
A and B are independent
Use the information given in the Venn diagram
to draw up and complete a contingency table
for the situation.
60
32
x
8
B
80
A
20
15
B
30
8.
Copy and complete the contingency table of probabilities
alongside, and determine from it
whether A and B are independent.
B
Not B
A
Not A
0,7
0,12
0,4
1
9.
Among 100 pupils, 30 are red-headed and 50 are in Yellow House. Draw a Venn diagram for this
situation, on the assumption that half of the red-heads are in Yellow House. Then draw up a
contingency table for the same situation, and then finally draw up a contingency table of
probabilities.
10.
When the cricketers were asked about their winter sport, half of them had chosen rugby and
another 24 had chosen hockey. If 6 of them played neither hockey nor rugby, and nobody played
both, how many cricketers were there? Use a Venn diagram
11.
When the 80 rugby players were asked about their summer sport, 20 played cricket and tennis, 12
played neither. If the total number playing cricket is the same as the total number playing tennis,
how many played only cricket? Use a contingency table.
12.
I have 80 counters; 12 of them are red on one side and blue on the other side, 6 of them are white
on both sides, and half of them are red on both sides. How many are blue on both sides? What is
the probability that if a counter lying on the table shows one blue side, then it is blue on the other
side?
13.
Families were surveyed, all of whom had cars, and it was found that 28 % of the families had
white cars, and 4,2 % of the families had both white cars and TV. 85 % of the families had no TV.
Determine whether having TVs and having white cars are independent. Use a contingency table of
percentages.
14.
23 % of the population of Ayetown has flu, and 70% of the population had received an anti-flu
injection earlier in the year. 23 % of the people did not get the injection and have not developed
flu. Decide, giving reasons, whether you think the flu injection is effective protection.
15.
Bag A contains 2 White balls and 4 Red balls; Bag B contains 3 White balls and 3 Red ones.
Alfred draws a card at random from a normal pack: if it is a Spade then he draws one ball from
Bag A, but otherwise he draws one ball from Bag B.
(a)
Using a tree diagram, or otherwise, find the probability that he draws a Spade and gets a
Red ball; find also the probability that he draws a Spade and gets a White ball.
(b)
Draw up a contingency table of probabilities and complete it. Write down the probability
that Alfred ends up with a Red ball.
(c)
Now find the probability that if he ended up with a Red ball, he drew a Spade before that.
16.
In rain the Jaguars have a probability of 2/3 of winning their rugby match, but otherwise the
probability is 3/10. Given that the probability of rain on any match
Win
day is 1/6,
Rain
(a)
copy and complete the tree diagram
alongside
Lose
Win
(b)
make a contingency table of probabilities
for the Jaguars
Norain
(c)
calculate what percentage of all their
matches you think the Jaguars will win
(d)
If you hear that the Jaguars won their last game, what is the probability that it was
played in the rain?
Lose
Answers
A
Not A
B
5
7
12
Not B
6
2
8
11
9
20
X
Not X
W
0,06
0,24
0,3
Not W
0,54
0,16
0,7
0,6
0,4
1
1.
3.
5.
P
Not P
Q
30
22
52
Not Q
12
36
48
42
58
100
A
Not A
B
21
8
29
Not B
9
22
31
30
30
60
2.
4.
Copy the Venn diagram alongside, and work
out the value of x, showing your reasoning.
60
F
11 + x + 8 + 34 = 6, so x = 7
11
x
8
G
6.
Find the value of x in the Venn diagram
alongside for each of the following cases:
(a)
(b)
(c)
(d)
7.
P(B) = 1/3
8 + x = 20 so x = 12
P(notA and notB) = 1/3
32 + 8 + x = 40 so x = 0
P(B and notA) = P(notA and notB)
x = 60 – (32 + 8 + x) so x = 10
A and B are independent
8/60 = (40/60).(8+x)/60 giving x = 4
34
60
A
32
Use the information given in the Venn diagram
to draw up and complete a contingency table
for the situation.
What is the probability that randomly chosen
member of A is also a member of B? 15/35
(a) What is the value of P(A and B)? 15/80
(b) What are the values of P(A) and P(B)? 35/80; 30/80
(c) Decide, giving reasons, whether A and B are
independent
P(A and B) = 15/80; P(A).P(B) = (30/80).(35/80), not the
same so not independent
x
8
B
80
A
20
15
B
(a)
30
A
Not
A
B
Not B
15
20
35
15
30
45
30
50
80
8.
9.
Copy and complete the contingency table of probabilities
alongside, and determine from it
whether A and B are independent.
P(A).P(B) = 0,7 × 0,6 = 0,42 = P(A and B) so independent
B
Not B
A
0,42
0,28
0,7
Not A
0,18
0,12
0,3
0,6
0,4
1
Among 100 pupils, 30 are red-headed and 50 are in Yellow House. Draw a Venn diagram for this
situation, on the assumption that half of the red-heads are in Yellow House. Then draw up a
contingency table for the same situation, and then finally draw up a contingency table of
probabilities.
100
Y
Not Y
R
15
15
30
Not R
35
35
70
50
50
100
R
15
15
Y
Not Y
R
0,15
0,15
0,3
Not R
0,35
0,35
0,7
0.5
0,5
1
35
G
35
10.
When the cricketers were asked about their winter sport,
half of them had chosen rugby and another 24 had
chosen hockey. If 6 of them played neither hockey nor
rugby, and nobody played both, how many cricketers
were there? Use a Venn diagram 60 cricketers
60
R
30
0
24
H
6
11.
12.
When the 80 rugby players were asked about
their summer sport, 20 played cricket and tennis,
12 played neither. If the total number playing
cricket is the same as the total number playing tennis,
how many played only cricket? Use a contingency table.
24 play only cricket
I have 80 counters; 12 of them are red on one side and blue on
the other side, 6 of them are white
on both sides, and half of them are red on both sides. How many
are blue on both sides? 22 of them
T
Not T
C
20
24
44
Not C
24
12
36
44
36
80
R
Not R
B
12
22
34
Not B
40
6
46
52
28
80
TV
No TV
W
4,2
23,8
28
Not W
10,8
61,2
72
15
85
100
What is the probability that if a counter lying on the table shows
one blue side, then it is blue on the other side? 22/34
13.
Families were surveyed, all of whom had cars, and
cars, and 4,2 % of the families had both white cars and TV. 85
% of the families had no TV. Determine whether having TVs
and having white cars are independent. Use a contingency table
of percentages.
P(W and TV) = 4,2 %. P(W) × P(TV) = 0,28 × 0,15 =
4,2%, so independent
14.
15.
23 % of the population of Ayetown has flu, and
70% of the population had received an anti-flu injection earlier
Flu
in the year. 23 % of the people did not get the injection and have
not developed flu. Decide, giving reasons, whether you think
No Flu
the flu injection is effective protection.
P(F and I) = 0,16
P(F) × P(I) = 0,23 × 0,7 = 0,161
Very nearly equal, so F, I independent, i.e. jab makes no difference
Inj
No Inj
16
7
23
54
23
77
70
30
100
Bag A contains 2 White balls and 4 Red balls; Bag B contains 3 White balls and 3 Red ones.
Alfred draws a card at random from a normal pack: if it is a Spade then he draws one ball from
Bag A, but otherwise he draws one ball from Bag B.
W
1
3
(a)
Using a tree diagram, or otherwise, find the
probability that he draws a Spade and gets a Red ball;
find also the probability that he draws a Spade and
gets a White ball.
P(Spade and Red) = (1/4).(2/3) = 1/6
P(Spade and White) = (1/4).(1/3) = 1/12
Etc.
Spade
1
4
2
3
R
W
1
2
3
4
No
Spade
1
2
(b)
Draw up a contingency table of probabilities and
complete it. Write down the probability that Alfred ends
up with a Red ball.
R
W
Sp
1
6
1
12
1
4
No Sp
3
8
3
8
3
4
13
24
11
24
1
P(Red) = 13/24
(c)
R
Now find the probability that if he ended up with a Red ball, he drew a Spade before that.
1/ 6
4
Prob is

13 / 24
13
16.
In rain the Jaguars have a probability of 2/3 of winning their rugby match, but otherwise the
probability is 3/10. Given that the probability of rain on any match
day is 1/6,
(a)
copy and complete the tree diagram
W
2
3
(b)
make a contingency table of
probabilities for the Jaguars
Rain
1
6
1
3
3
10
5
6
(c)
(d)
Lose
L
Rain
1
9
1
18
1
6
W
No R
1
4
7
12
5
6
13
36
23
36
1
No Rain
7
10
Win
L
calculate what percentage of all their matches you think the Jaguars will win
P(win) =
13
so
36
approx 36% of games
If you hear that the Jaguars won their last game, what is the probability that it was played in the
1/ 9
4
rain? Prob is
=
13 / 36 13