Page 33 - Fundamentals of Probability and Statistics for Engineers

16 Fundamentals of Probability and Statistics for Engineers

and, in the case of n events,

!
n n n n n n n
[ X XX X X X
P A j ˆ P A j † P A i A j †‡ P A i A j A k †
jˆ1 jˆ1 iˆ1 jˆ2 iˆ1 jˆ2 kˆ3 2:15†
i<j i<j<k
‡ 1† n 1 P A 1 A 2 ...A n †;

where A j , j ˆ 1, 2, .. . , n, are arbitrary events.

Example 2.5. Let us go back to Example 2.4 and assume that probabilities
[
[
P(A), P(B), and P(C) are known. We wish to compute P(A B) and P(A C).
[
Probability P(A C), the probability of having either 50 or fewer cars turn-
ing left or between 80 to 100 cars turning left, is simply P(A) ‡ P(C). This
follows from Axiom 3, since A and C are mutually exclusive. However,
P(A B), the probability of having 60 or fewer cars turning left, is found from
[
P A [ B†ˆ P A†‡ P B† P AB†

The information given above is thus not sufficient to determ ine this probability
and we need the additional information, P(AB), which is the probability of
having between 40 and 50 cars turning left.
With the statement of three axioms of probability, we have completed the
mathematical description of a random experiment. It consists of three funda-
mental constituents: a sample space S, a collection of events A, B, ... , and the
probability function P. These three quantities constitute a probability space
associated with a random experiment.

2.2.2 ASSIGNMENT OF PROBABILITY

The axioms of probability define the properties of a probability measure, which are
consistent with our intuitive notions. However, they do not guide us in assigning
probabilities to various events. For problems in applied sciences, a natural way to
assign the probability of an event is through the observation of relative frequency.
Assuming that a random experiment is performed a large number of times, say n,
then for any event A let n A be the number of occurrences of A in the n trials and
define the ratio n A /n as the relative frequency of A. Under stable or statistical
regularity conditions, it is expected that this ratio will tend to a unique limit as n
becomes large. This limiting value of the relative frequency clearly possesses the
properties required of the probability measure and is a natural candidate for
the probability of A. This interpretation is used, for example, in saying that the

TLFeBOOK