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Unit 1: Sample Space of A Random Experiment

measure the age of a person, in the idealised situation we can think of age being any number Notes
between 0 and . Of course, no one has met a person. with infinite age of for that matter who is
more than 150 years old. Nevertheless, most of the actuarial and demographic studies are
carried out assuming that there is no upper bound on age. Thus, we may say that the sample
space of the experiment of finding out the age of an arbitrarily selected person is the interval ]0,
[. Since the elements of the interval ]10, [ cannot be arranged in a sequence, such a sample
space is not a discrete sample space.
Some other examples where non-discrete sample spaces are appropriate are (i) the price of
wheat, (ii) the amount of ozone in a volume of space, (iii) the length of a telephone conversation,
(iv) the duration one spends in a queue, (v) the yield of rice in our country in one year.
In all these examples, it is necessary to deal with non-discrete sample spaces. However, we’ll
defer the study of probability theory for such experiments to the next block.

1.3 Events

We have described a number of random experiments till now. We have also identified the
sample spaces associated with them. In the study of random experiments, we are interested not
only in the individual outcomes but also in certain events. As you will see later, events are
subsets of the sample space. In this section we shall formalise the intuitive concept of an event
associated with a random experiment which has a discrete sample space. We shall also study
methods of generating new events from specified ones and study their inter-relationships.

Consider the experiment of inspecting three items (Experiment 5). The sample space has the
eight points,
GGG, GGB, GBG, BGG, BBG, BGB, GBB, BBB.

We label these points  ,  , ... ,  , respectively.
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Suppose we are interested in those outcomes which correspond to the event of obtaining exactly
one good item in the three inspected items. The corresponding’sample points are  = BBG,
5
 = BGB and  = GBB. Thus, the subset {  ,  ,  } of the sample space corresponds to the
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“event” A that only one of the inspected items is good.
On the other hand, consider the subset C = {  ,  ,  ,  } consisting of the points BBG, BGB, GBB,
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BBB. We can identify the subset C with the event “There are at least two bad items.”
This discussion suggests that we can associate a subset of the sample space with an event and an
event with a subset. This leads us to the following definition.
Definition 4 : When the sample space of an experiment is discrete, any subset of the sample space
is called an event.
Thus, we alsq consider the empty set as an event.
You will soon find that the two extreme events,  and , consisting, respectively, of no points
and all the points of R are most uninteresting. But we need them to complete our description of
the class of all events. In fact,  is called the impossible event and  is called the sure event, for
reasons which will be obvious in the next unit. Also, note that an individual outcome , when
identified with the singleton (), constitutes an event.
The following example will help you in understanding events.

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