Extending Probability
Using Venn Diagrams to Find Probabilities

Venn diagrams can be used to help find probabilities when events are not mutually exclusive.

This Venn diagram illustrates the situation for a pack of cards where the set A is the aces and the set H is the hearts. Note that the total of the entries is 52, and the complete set is denoted by ε.

From the diagram

p(Ace of Hearts) =
p(Ace or Heart) =
=
=

Worked Examples

1

In a class of 30 children there are 20 who can swim and 25 who can ride a bike. There is no child who cannot swim nor ride a bike.

Draw a Venn diagram and find the probabilities that a child selected at random:

(a)

can swim and ride a bike,

(b)

cannot swim but can ride a bike,

(c)

can swim but cannot ride a bike.

Show me

Let R represent the set of children that can ride a bike and S represent the set of children that can swim.

As each child belongs to R or S, a 0 can be placed as shown.

As 20 + 25 = 45

and 45 – 30 = 15

there must be 15 children in the overlap, so the diagram can be completed as shown.

The probabilities can now be found:

p(can swim and ride a bike) =
=
p(cannot swim but can ride a bike) =
=
p(can swim but cannot ride a bike) =
= .
2

In a group of 40 students 6 are left-handed, 18 have size 7 feet and 2 are left-handed with size 7 feet. Draw a Venn diagram and find the probability that a student is :

(a)

left-handed or has size 7 feet,

(b)

not left-handed and does not have size 7 feet,

(c)

not left-handed and has size 7 feet.

Show me

The number 2 is placed in the overlap

In F we have 18 − 2 = 16,

In L we have 6 − 2 = 4.

In the other space we have 40 − (4 + 2 +16) = 18.

Now the probabilities can be calculated.

p(left-handed or size 7 feet) =
=
=
p(not left-handed and does not have size 7 feet) =
=
p(not left-handed and has size 7 feet) =
=

Exercises

The Venn diagrams represent the ways the events A and B can take place.

For each case find

(a)

p(A and B)

(i) (ii) (iii) (iv)
(b)

p(A but not B)

(i) (ii) (iii) (iv)
(c)

p(A or B)

(i) (ii) (iii) (iv)
(d)

p(B but not A)

(i) (ii) (iii) (iv)

When a card is selected at random from a full pack two possible events are:

R : A red card is selected.

Q : A queen is selected.

Draw a Venn diagram and use it to find the probabilities that a card is:

(a)
a red queen,
(b)
red or a queen,
(c)
not red or a queen.

In a survey 100 people were asked whether they watched snooker or cricket when it was on TV.

20 watched neither,   80 watched snooker,   40 watched cricket.

Draw a Venn diagram and find the probabilities that a person selected at random from the sample:

(a)
watched both cricket and snooker.
(b)
watched cricket but not snooker,
(c)
watched snooker but not cricket.

The following Venn diagrams show the ways in which the events A, B and C can take place.

For each diagram find:

(a)

p(A and B and C)

(i) (ii) (iii) (iv)
(b)

p(A or B or C)

(i) (ii) (iii) (iv)
(c)

p(A or B)

(i) (ii) (iii) (iv)
(d)

p(A and B)

(i) (ii) (iii) (iv)
(e)

p(A or C)

(i) (ii) (iii) (iv)
(f)

p(A)

(i) (ii) (iii) (iv)

When a person is selected at random, the probability that they are left handed is 0.08 and the probability that they are female is 0.52. If the probability that a female is left-handed is 0.042, find the probability that a person selected at random is female or left-handed.