PROBABILITY

Introduction

Probability is the branch of mathemtics that deals with the measurement of chance. At busy roads we oftnely see sign boards showing "Accident pron Area".

Do you know how traffic controllers identified the road pron to accidents?

They decide it on the basis of some data of traffic movment on the road from earlier experiences.

We do this type of predictions at many places. Like it is more likely to rain today,

team India may win this T-20 cricket match,chances of the train reaching at the destinatin on time are quite high,the topper of the last unit test will

most likely top in annual examination too etc. In the last two examples we find that the chance of the train reaching on time and the chance of the the student

topping again are rated as very high.But which one of these has more chance to occur in comparison to other? To find an answer to this question we must give a numerical

measurment to chance. The measurment of chance is called probability.

Random Experiments

The probability is calculated for the events or happenings which have factors of certaininty and uncertinity. Such happenings are called random expereiments. For example tossing of a coin is a random experiment as here any one of two known events i.e. occurence of head or tail can happen.

      

For example the random withdrawl of a ball from an urn containing all red color balls of same size is not a random experiment as there is no abmiguity in telling in advance the outcome i.e. The ball drawn will be red.

Events

An outcome or a collection of outcomes of a random experiment is called an event . For example,

In throwing a die the possible outcomes are { 1, 2, 3, 4, 5, 6}

For the event “odd number appears on the upper face of the die” the outcomes 1, 3, 5 are in favour of the event. Similarily, for the event “a multiple of 3 occurs” the outcomes 3 and 6 are in favour of the event.

Note that each individual outcome is called an elementary event.

Probabilityof an Event:

Probability of an event is obtained by following three methods

• By using Statistical data

• By using all the possible outcomes (Set Theoretical Probability)

• By using axioms laid by A.N.Kolmogrov

Statistical Method

Probability of an event is calculated on the basis of the experience of repeating the experiment. The probality is measured as the ratio of the number of times that event occurred to the total number of times the experiment is repeated

Example

Let us find the probability of getting 6 in tossing a die

Toss a die 50 times and note down the number of sixes occur. Let the number of sixes be 17.

We will say that in this experiment the probability of getting a six is 17/50

Then the number of trials of the experiment are increased to a large number and if the ratio of number of times the event occurred to the total number of trials tend to a number then that number is called the probability of the event.

Lim as n tends to infinity= probability of the event E

Limitations of Statistical method

For different number of trials of an experiment the sequence of ratios may not approach to a number. In that case we will not be able to find the probability.Also finding the limit when the number of trials tend to infinity depends upon the convergence of the sequence of the ratio of number of times the event occurs to the total number of trials.

Thus another method of evaluating probability was evolved.

Set theoretic probability

In this method we find all the possible outcomes of the random experiment. We also find the number of outcomes favouring the event whose probability we want to find. We also assume that all the possible outcomes are equally likely ( i.e. all have equal chance to occur). Then the probability of an event E is denoted by P(E) and is evaluated as P(E) = Number of oput comes in favour of the event / Total number of outcomes For example

What is the probability of an ace in drawing a card from a pack of playing cards?

Event: Ace card on drawing a card

Total number of outcomes: 52 ( any of 52 cards can come)

Number of favourable outcomes: 4

Therefore, required probability = =

Limitations of Set theoretic probability

To find the probability we assumed that all the outcomes of the experiment are equally likely to happen. This means that all outcomes has equal probability. Thus, to define probability we used probability. Which logically is not a valid definition

A.N.Kolmogrov

To overcome this logical incompetence of the definition of probability Russian mathematician gave another probability theory, in the year 1933, named as Axiomatic theory.. In this theory A.N.Kolmogov assigned probabilities to variopus events of a random experiment on the basis of some axioms. You will learn about this in next lesson.