Introduction
Often experiments are performed in order to produce
observations or measurements that assist us in arriving at conclusions. These
recorded informations in it’s original collected form are referred as “raw
data”. Mathematicians define experiment as any process or operation that
generates raw data. If a chemist runs an analysis several times under the same
experimental conditions, he will not get concurrent result, which indicates an
element of chance in the experimental procedure. It is these chance outcomes
that occur around us with which this chapter is basically concerned.
Random Experiment:
An experiment, whose all possible outcomes are known in
advance but the outcome of any specific performance cannot predicted before the
completion of the experiment, is known as random experiment.
An example of random experiment might be tossing of a
coin. This experiment consists of only two outcomes head or tail. Another
example might be launching of a missile and observing the velocity at specified
times. The opinions of voters concerning a new sales tax can also be considered
as outcomes of random experiment.
Sample-space and sample
point:
A set whose elements represent all possible outcomes of
a random experiment is called the sample space and is usually represented by
‘S’.
An element of a sample space is called a sample point.
Consider the experiment of tossing a die. If we are
interested in the number that shows on the top face, then sample space would be
S1 = {1, 2, 3, 4, 5, 6}.
If we are interested only in whether the number is even
or odd, then sample space is simply S2 = {even, odd}
Clearly more than one sample space can be used to
describe the outcomes of an experiment. In this case ‘S1’ provides
more information than ‘S2’. If we know which element in S1
occurs, we can tell which outcome in S2 occurs; however, a knowledge
of what happens in S2 in no way helps us to know which element in S1
occurs.
In general it is desirable to use a sample space that
gives the maximum information concerning the outcomes of the experiment.
Suppose three items are selected at random from a
manufacturing process. Each item is inspected and classified as defective or
non-defective. The sample providing the maximum information would be S1
= {NNN, NDN, DNN, NND, DDN, DND, NDD, DDD}.
A second sample space, although it provides, less
information, might be S2 = {0, 1, 2, 3}
Where the elements
represent no defectives, one defective, two defectives, or three defectives in
our random selection of three items.
Event
An event is a subset of sample – space.
In any sample space we may be interested in the
occurrence of certain events rather than in the occurrence of a specific
element in the sample space. For instance, we might be interested in the event
‘A’ that the outcome when die is tossed is divisible by 3. This will occur if
the outcome is an element of the subset A = {3, 6}. Clearly, to each event we
can assign a collection of sample point(s), which consistute a subset of the
sample space. This subset represents all the elements for which the event is
true.
For instance, given the subset A = {t | t < 5} of
sample space S = {t | t ≥ 0}, where ‘t’ is
the life in years of a certain electronic component, ‘A’ would represent the
event that the component fails before the end of fifth year.
Simple Event and Compound
Event
If an event is a set containing only one element of the
sample-space, then it is called a simple event.
A compound event is one that can be represented as a
union of sample points.
For instance, the event of drawing a heart from a deck
of cards is the subset A = {heart} of the sample space S = {heart, spade, club,
diamond}. Therefore A is a simple event. None the event B of drawing a red card
is a compound event since
B = {heart U diamond} = {heart, diamond}.
It must be noted that the union of simple events
produces a compound event that is still a subset of sample space. It should
also be noted that if 52 cards of the deck were the elements of sample space
rather than four suits, then event ‘A’ would also be compound event.