Page 34 - Fundamentals of Probability and Statistics for Engineers

Basic Probability Concepts 17

probability of ‘heads’ in flipping a coin is 1/2. The relative frequency approach to
probability assignment is objective and consistent with the axioms stated in Section
2.2.1 and is one commonly adopted in science and engineering.
Another common but more subjective approach to probability assignment is
that of relative likelihood. When it is not feasible or is impossible to perform an
experiment a large number of times, the probability of an event may be assigned
as a result of subjective judgement. The statement ‘there is a 40% probability of
rain tomorrow’ is an example in this interpretation, where the number 0.4 is
assigned on the basis of available information and professional judgement.
In most problems considered in this book, probabilities of some simple but
basic events are generally assigned by using either of the two approaches. Other
probabilities of interest are then derived through the theory of probability.
Example 2.5 gives a simple illustration of this procedure where the probabilities
[
of interest, P(A B) and P(A C), are derived upon assigning probabilities to
[
simple events A, B, and C.

2.3 STATISTICAL INDEPENDENCE

Let us pose the following question: given individual probabilities P(A) and P(B)
of two events A and B, what is P(AB), the probability that both A and B will
occur? Upon little reflection, it is not difficult to see that the knowledge of P(A)
and P(B) is not sufficient to determine P(AB) in general. This is so because
P(AB) deals with joint behavior of the two events whereas P(A) and P(B) are
probabilities associated with individual events and do not yield information on
their joint behavior. Let us then consider a special case in which the occurrence
or nonoccurrence of one does not affect the occurrence or nonoccurrence of the
other. In this situation events A and B are called statistically independent or

simply independent and it is formalized by Definition 2.1.

Definition 2.1. Two events A and B are said to be independent if and only if

P AB† ˆ P A†P B†: 2:16†
To show that this definition is consistent with our intuitive notion of inde-
pendence, consider the following example.

Example 2.6. In a large number of trials of a random experiment, let n A and

n B be, respectively, the numbers of occurrences of two outcomes A and B, and
let n AB be the number of times both A and B occur. Using the relative frequency
interpretation, the ratios n A /n and n B /n tend to P(A) and P(B), respectively, as n
becomes large. Similarly, n AB /n tends to P(AB). Let us now confine our atten-
tion to only those outcomes in which A is realized. If A and B are independent,

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