Introduction To Probability

Introduction

Probability theory is the most important branch of statistics that is concerned with random phenomena. It is the fact that the Statistics begins with the innovation of the theory of probability.According to Prof. Ya-Lin-Chou, "Statistics is the science of decision making with calculated risks in the face of uncertainty."Probability theory is so indispensable in modern Statistics.This is why ,the study of the methods of collection, analysis of statistical data and presentation is not enough to cover the entire operational techniques of Statistics.In order to cover the entire operational techniques and to make certain decisions under uncertainty, we need to study the theory of probability.

The probability theory is an essential mathematical tool developed mainly to describe and measure such uncertainties of outcomes of any performance of a random experiment.The uncertainty of occurrence of such an event is expressed in terms of a number in probability theory,Generally,Probability means the chance that some things will happen.But probability is expressed in a number between 0 and 1 whereas chance is the probability multiplied by 100 and expressed in percentage.

In the today's modern age, the use of probability theory has been expanded to the physical,natural,medical,biological,and social sciences, in engineering and in the business world.Also, it is extensively applied in the quantitative analysis of various problems arising in the economics,industry,commerce etc. It plays a vital role in insurance and statistical quality control in industry. Thus, the probability theory is the basis of decision making and Statistical Inference which is the very important branch of statistics.

Some Important Terms of Probability

Random experiment

An experiment is any process or mechanism which generates a set of data,If an experiment is performed repeatedly under essentially identical conditions and it does not give unique results(outcomes) but may give any one of the various possible outcomes,then such an experiment is called a random experiment.In short ,an experiment is said to be a random experiment is its outcome cannot be predicted with certainty.

For example,Rolling a die is a random experiment as the face value is unknown until anyone faces(1,2,3,4,5,6) is turn up.

Trial

The task of performing a random experiment is called a trial.

For example,tossing a coin is a trial.

Event

An outcome (a result ) or a set of outcomes of a random experiment is called an event.An event is denoted by capital letters A,B,C etc. or E₁,E₂,E₃ etc.

For example, in tossing a coin,getting a head or a tail is known as an event.Similarly, in throwing a die,getting 1,2,3.4.5 or 6 is an event.

Simple or Elementary Event

A simple or an elementary event is nothing but an outcome of a random experiment.In other words,an event which can not be decomposed in more than one event is known as the simple or elementary event.

For example,when a coin is tossed,the event of getting a head is an elementary event.

Composite or Compound Event

An event is said to be a composite or compound or decomposable event if it can be decomposed in more than one simple events.In other words,the combination of two or more elementary events is called a composite or compound event.

For example ,when two dice are thrown simultaneously,the event of getting a sum of 7 points,viz.A=(3,4),is a composite event as it can be decomposed into simple events one A₁=(3),the event of showing 3 by the first die and the other,A₂=(4),the event of showing 4 by the second die.

Null or Impossible Event

An event is said to be a null or impossible event if it can never happen in a random experiment.

For example,in throwing a die,the event of getting 8 points or a face numbered 8 is an impossible event.Similarly,if two cards are drawn at random from a full pack of cards,then the event of drawing two aces of a spade is an impossible event.And if a fruit is picked up, the event of getting an apple from an orange tree is another example of the impossible event.

Sure or Certain Event

A sure or certain event is one which is sure to happen in a random experiment.

For example,If a ball is drawn from a bag containing white balls only,then the event that 'it is a white ball'' is a sure event.

Equally Likely Events

Events are said to be equally likely if the chance (or probability) of happening of every event is equal (same) in a random experiment, i.e. if none of the events is expected to occur more in preference to the other.

For example, In tossing of an unbiased coin,the event of turning up of 'head' and 'tail' are equally likely.

Mutually Exclusive Events

Events are said to be mutually exclusive or disjoint if they cannot happen simultaneously in the same experiment in others words if the happening of any one event excludes the happening of other (all other) events in the same trial ,these events are called mutually exclusive events.

For example ,when we draw a card from a full pack of cards ,one and only one of the 52 cards can appear at a time and therefore the 52 different outcomes are mutually exclusive.

Favourable Events

Favourable events or case to an event are defined as the outcomes in an experiment which are favourable to that event.In short,Favourable events to an event A are those events which correspond to the occurrence of the event A.

For example, in the tossing of two coins,the number of case favourable to event ''happening at least one head " are 3, viz . HH,HT,TH.

Exhaustive Events

The events associated with a random experiment are said to be exhaustive events if they include all possible outcomes of the experiment.

For example,If we draw a ball from a box containing 20 balls,the exhaustive events are 20. If we draw 2 balls at random from the box having 20 balls, the exhaustive events are ²⁰C₂=190,where C stands for combinations.

References

Sukubhattu,Narendra Prasad. Probability and Inference-I. Asmita Books Publishers & Distributors (P)Ltd. 2013