Sample space , Sample point & Event

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 can not be 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 constitute 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 & 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. However 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.

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