Engineering Optimization: Theory and Practice, Fourth Edition

11.2 Basic Concepts of Probability Theory 633

phenomena are chance dependent and one has to resort to probability theory to describe

the characteristics of such phenomena.

Before introducing the concept of probability, it is necessary to define certain terms

such as experiment and event. Anexperimentdenotes the act of performing something

the outcome of which is subject to uncertainty and is not known exactly. For example,

tossing a coin, rolling a die, and measuring the yield strength of steel can be called

experiments. The number of possible outcomes in an experiment may be finite or

infinite, depending on the nature of the experiment. The outcome is a head or a tail

in the case of tossing a coin, and any one of the numbers 1, 2, 3, 4, 5, and 6 in the

case of rolling a die. On the other hand, the outcome may be any positive real number

in the case of measuring the yield strength of steel. Aneventrepresents the outcome

of a single experiment. For example, realizing a head on tossing a coin, getting the

number 3 or 5 on rolling a die, and observing the yield strength of steel to be greater

than 20,000 psi in measurement can be called events.

Theprobabilityis defined in terms of the likelihood of a specific event. IfE

denotes an event, the probability of occurrence of the eventEis usually denoted by

P (E). The probability of occurrence depends on the number of observations or trials.

It is given by

P (E)= lim

n →∞

m

n

(11.1)

wheremis the number of successful occurrences of the eventEandnis the total

number of trials. From Eq. (11.1) we can see that probability is a nonnegative number

and

0 ≤P (E)≤ 1. 0 (11.2)

whereP (E)=0 denotes that the event is impossible to realize whileP (E)= 1. 0

signifies that it is certain to realize that event. For example, the probability associated

with the event of realizing both the head and the tail on tossing a coin is zero (impossible

event), while the probability of the event that a rolled die will show up any number

between 1 and 6 is 1 (certain event).

Independent Events. If the occurrence of an eventE 1 in no way affects the probabil-

ity of occurrence of another eventE 2 , the eventsE 1 andE 2 are said to bestatistically

independent. In this case the probability of simultaneous occurrence of both the events

is given by

P (E 1 E 2 ) =P(E 1 )P (E 2 ) (11.3)

For example, ifP (E 1 ) =P(raining at a particular location) = 0.4 andP (E 2 )=

P(realizingthe head on tossing a coin)= 0 .7, obviouslyE 1 andE 2 are statistically

independent and

P (E 1 E 2 ) =P(E 1 )P (E 2 ) = 0. 28

11.2.2 Random Variables and Probability Density Functions

An event has been defined as a possible outcome of an experiment. Let us assume that

a random event is the measurement of a quantityX, which takes on various values in