478 CHAPTER^8 Discrete Probability
we believe that the 36 outcomes are equally likely, we choose p(co) to be 1/36 for each
(0 E S22.
The important point here is that mathematics does not tell us exactly how to assign
probabilities, nor does it provide us with a precise definition of "experiment" or tell us
what sample space to choose. In fact, mathematics does not even offer a definition for
"probability of an outcome" in terms of our intuitive ideas about likelihood of occurrence
of uncertain phenomena. It is entirely up to us to supply the meanings and interpretations,
to define the sample spaces, and to make the probability assignments. This we do on the
basis of what seems to be reasonable to us. Once we have done this, mathematics can help
us to determine the logical consequences of our assumptions.8.1.2 Basic Definitions
Here we give the mathematical definitions of the terms we have been using. The definitions
make no reference to the notions of likelihood, randomness, and uncertainty that we all
have in mind when we talk about probability.
Definition 1. A discrete sample space is a nonempty set that has only a finite or count-
ably infinite number of elements.
The word discrete in the definition refers to the fact that the set has only a finite or
countably infinite number of elements. Although it is meaningful to consider processes
having more than countably many outcomes, all the sample spaces in this chapter will be
discrete. Hence, from now on, we will usually just say "sample space" instead of "discrete
sample space." Definition 1 does not say that a sample space must be the set of possible
outcomes of an experiment, but this is the interpretation we will have in mind when we
elect to call a set a sample space.
Definition 2. An outcome is an element of a sample space.Definition 3. An event is a subset of a sample space.Sample spaces will be denoted by Q's and outcomes by w's. Events will generally
be denoted by E's, although other capital letters will sometimes be used. According to
Definition 3, both the entire sample space Q and the empty set 0 are events. An event Ecan also consist of a single outcome a) E Q, in which case we write E = {o}.
Example 1. Suppose we take[Saturday, Sunday, Monday, Wednesday, Friday)as a sample space QŽ. Then, Wednesday is an outcome, but Thursday is not. Wednesday
can also be regarded as the event E = {Wednesday}. The weekend can also be regarded as
an event in Q2, since{Saturday, Sunday} C Q
(Watch out for the notation: {Wednesday) is a subset of Q2 and denotes an event, whereas
Wednesday is an element of Q2 and denotes an outcome.)^0
Suppose a probability experiment with sample space Q2 is executed. Then, an event
E C f2 is said to occur if the outcome wo of the experiment belongs to E.