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Combinatorial Analysis

A common graphical representation of the outcomes of an experiment is the Venn diagram.
A Venn diagram usually consists of a rectangle, the interior of which represents the sample space,
togetherwithoneormoreclosedcurvesinsideit. Theinteriorofeachclosedcurvethenrepresents
an event. Fig. 1.1 shows a typical Venn diagram representing a sample space S and two events A
and B. Every possible outcome is assigned to an appropriate region; in this example there are four
regions to consider (marked i to iv in Fig. 1.1):
(i) outcomes that belong to event A but not to event B;
(ii) outcomes that belong to event B but not to event A;
(iii) outcomes that belong to both event A and event B;
(iv) outcomes that belong to neither event A nor event B.

A B

i iii ii

iv
S

Figure 1.1 – A Venn diagram

Example 1.11. A six-sided die is thrown. Let event A be ’the number obtained is
divisible by 2’ and event B be ’the number obtained is divisible by 3’. Draw a
Venn diagram to represent these events. ,

Solution. It is clear that the outcomes 2, 4, 6 belong to event A and that the outcomes 3, 6
belong to event B. Of these, 6 belongs to both A and B. The remaining outcomes, 1, 5, belong
to neither A nor B. The appropriate Venn diagram is shown in Fig. 1.2.

A 4 B

2 6 3

1 5

S
Figure 1.2 – The Venn diagram for the outcomes of the die-throwing trials described in the worked
example

In the above example, one outcome, 6, is divisible by both 2 and 3 and so belongs to both A and B.
This outcome is placed in region iii of Fig. 1.1, which is called the intersection of A and B and is
denoted by A ∩ B (see Fig. 1.3). If no events lie in the region of intersection then A and B are said
to be mutually exclusive or disjoint. In this case, often the Venn diagram is drawn so that the

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