Page 16 - 4660

Probability

1.2. Probability

In the previous section we discussed Venn diagrams, which are graphical representations of
the possible outcomes of experiments. We did not, however, give any indication of how likely each
outcome or event might be when any particular experiment is performed. Most experiments show
some regularity. By this we mean that the relative frequency of an event is approximately the same
on each occasion that a set of trials is performed. For example, if we throw a die N times then
we expect that a six will occur approximately N/6 times (assuming, of course, that the die is not
biased). The regularity of outcomes allows us to define the probability, P(A), as the expected
relative frequency of event A in a large number of trials. More quantitatively, if an experiment has
a total of n S outcomes in the sample space S, and n A of these outcomes correspond to the event
A, then the probability that event A will occur is

n A
P(A) = . (2.1)
n S

Events. Classification of events

Definition 2.1. Realization of some definite complex of conditions is called a
trial (or test). A possible result of trial is called an event.

Definition 2.2. An event that can occur or can not occur in a given trial is
termed a random event.

Definition 2.3. An event which can not come in a given trial is called an
impossible event.

Definition 2.4. An event which necessarily comes in a given trial is termed a
certain event.

Definition 2.5. Events A 1 , A 2 , . . . , A n that can not occur at the same time are
mutually exclusive or incompatible. Otherwise events are named joint.

Definition 2.6. Events A 1 , A 2 , . . . , A n are said to be uniquely possible if one of
them necessarily occurs in a given trial. ✓

Definition 2.7. Events A 1 , A 2 , . . . , A n form a total group if they are unique and
incompatible. ✓

Definition 2.8. Two events A and A which form a total group are called
mutually opposite.

Definition 2.9. We will say that an event B implies an event A if in the given
trial the occurrence of the event B implies that of the event A. ✓

16