Mathematical Methods for Physics and Engineering : A Comprehensive Guide

PROBABILITY

iiiiii

iv

A B

S

Figure 30.1 A Venn diagram.

be (we will return to the probability of an outcome in due course) but rather

will concentrate on the classification of possible outcomes. It is clear that some

sample spaces are finite (e.g. the outcomes of throwing a die) whilst others are

infinite (e.g. the outcomes of measuring people’s heights). Most often, one is not

interested in individual outcomes but in whether an outcome belongs to a given

subsetA(say) of the sample spaceS; these subsets are calledevents. For example,

we might be interested in whether a person is taller or shorter than 180 cm, in

which case we divide the sample space into just two events: namely, that the

outcome (height measured) is (i) greater than 180 cm or (ii) less than 180 cm.

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, together with one or more closed curves inside

it. The interior of each closed curve then represents an event. Figure 30.1 shows

a typical Venn diagram representing a sample spaceSand two eventsAand

B. Every possible outcome is assigned to an appropriate region; in this example

there are four regions to consider (marked i to iv in figure 30.1):

(i) outcomes that belong to eventAbut not to eventB;

(ii) outcomes that belong to eventBbut not to eventA;

(iii) outcomes that belong to both eventAand eventB;

(iv) outcomes that belong to neither eventAnor eventB.

A six-sided die is thrown. Let eventAbe ‘the number obtained is divisible by 2 ’ and event

Bbe ‘the number obtained is divisible by 3 ’. Draw a Venn diagram to represent these events.

It is clear that the outcomes 2, 4, 6 belong to eventAand that the outcomes 3, 6 belong
to eventB. Of these, 6 belongs to bothAandB. The remaining outcomes, 1, 5, belong to
neitherAnorB. The appropriate Venn diagram is shown in figure 30.2.

In the above example, one outcome, 6, is divisible by both 2 and 3 and so

belongs to bothAandB. This outcome is placed in region iii of figure 30.1, which

is called theintersectionofAandBand is denoted byA∩B(see figure 30.3(a)).

If no events lie in the region of intersection thenAandBare said to bemutually

exclusiveordisjoint. In this case, often the Venn diagram is drawn so that the

closed curves representing the eventsAandBdo not overlap, so as to make