Theory of probability is a study of random or unpredictable nature of experiments. It is helpful in investigating the important features of these experiments. A preliminary knowledge of set theory and permutations and combinations is a pre-requisite for the study of this topic. We shall first introduce some key concepts and terminology that is necessary to develop the theory.

Definition

An experiment that can be repeated any number of times under identical conditions is which:

1. All possible outcomes of the experiment are known in advance.
2. The actual outcome in a particular case is not known in advance, is called a random experiment. We shall now give some examples of random experiments.

Examples

1. In an experiment of tossing an unbiased coin, we have only two possible outcomes Head (H) and Tail (T). In a particular trial, one does not know in advance, the outcome. This experiment can be performed any number of times under identical conditions. Therefore it is a random experiment .

2. Rolling a fair die is also a random experiment. If we denote the six faces of the cubic die with the numbers 1, 2, 3, 4, 5 and 6, then the possible outcome of the experiment is one of the numbers 1, 2, 3, 4, 5 or 6 appearing on the uppermost face of the die. The six faces of a die may also contain dots in numbers 1, 2, 3, 4, 5, 6. In any case, we shall identify the faces of a die, hereafter with the numbers 1, 2, 3, 4, 5, 6.

3. Tossing a fair coin till a tail appears is also a random experiment. The experiments such as:

   1. Measuring the acceleration due to gravity using a compound pendulum.
      
   2. Measuring the volume of a gas by increasing the pressure, keeping the temperature fixed, are not random experiments.
      

Basic Terms and their definitions

1. Any possible outcome of a random experiment 'E' is called an elementary or simple event .
2. The set of all elementary events (possible outcomes) of a random experiment 'E' is called the sample space S, associated with E. That is, S is the sample space of a random experiment E if (a) every element of S is an outcome of E and (b) every performance of E results in an outcome that corresponds to exactly one element of S.
3. An elementary event is a point of the sample space.
4. A subset 'D' of S, is called an event. That is, a set of elementary events is called an event.
5. An event 'D' is said to happen (or occur) if an outcome of the experiment belongs to D. Otherwise, we say that D has not happened (or not occurred).
6. The complement of an event D, denoted by D', is the event given by D'=S-D, which is called the complementary event of D.
7. The empty set “Φ” and the set S, being trivial subsets of S, are events called impossible event and certain (definite) event respectively.

Mutually Exclusive Events

Two or more events are said to be mutually exclusive if the occurrence of one of the events prevents the occurrence of any of the remaining events. Thus events E1,E2... Ek are said to be mutually exclusive if Ei ∩ Ej=Φ for i != j, 1<=i, j<=k.

Equally Likely Events

Two or more events are said to be equally likely (or equiprobable) if there is no reason to expect any one of them to happen in preference to the others.

Exhaustive Events

Two or more events are said to be exhaustive if the performance of the experiment always results in the occurrence of at least one of them. Thus events E1,E2,...Ek are said to be exhaustive if E1UE2UE3....UEk=S The following examples illustrate these concepts:
Example
1 In the experiment of throwing a die, the event
E: Occurrence of an even number (on the face of the die) and
E2: Occurrence of an odd number are mutually exclusive events. They are also exhaustive.

Classical (or Mathematical) definition of probability

If a random experiment results in an exhaustive, mutually exclusive and equally likely elementary events and m of them are favourable to the happening of an event E, then the probability of occurrence of E (or simply probability of E) denoted by P(E) is defined by P(E)=m/n
From this definition it is clear that, for any event E, 0≤P(E) ≤1.
Since the number of elementary events or outcomes not favourable to this event is (n-m), the probability of non-occurrence of the event E, denoted by P(E'), is given by

P(E')= (n-m)/n =1-(m/n)=1-P(E)
P(E)+P(E')=1

Limitations of the classical definition of probability

The classical definition of probability has the following limitations:

1. If the outcomes of experiment are not equally likely, then the probability of an event in such an experiment is not defined.
   For instance, the probability of a student passing an examination is not 1/2, as the outcomes of pass and failure in an examination are not equally likely.
   

1. If the random experiment contains infinitely many outcomes, then this definition can not be applied to find the probability of an event in such an experiment. For example, either in the random experiment of tossing a coin until tail appears or choosing a natural number, there are infinite number of outcomes.
   In order to overcome these deficiencies, we now consider the relative frequency approach to the definition of probability.
   

Relative frequency (or statistical or emperical) definition of probability

Suppose a random experiment is repeated n times, out of which an event E occurs m(n) times. Then the ratio rn=m(n)/n mix called ** the n-th relative frequency of the event E.

Now consider the sequence r1,r2,.....rn. If rn tends to a definite limit, say I as n tends to infinity.
i.e, lim(n→∞) rn=l ⁡, then 'I' is defined to be the probability of the event E and we write P(E)= lim(n→∞) rn=l

Deficiencies of the relative frequency definition of probability

From the above definition we observe the following deficiencies:

1. Repeating a random experiment infinitely many times is practically impossible.
2. The sequence of relative frequencies is assumed to tend to a definite limit, which may not exist.
   
3. The values r1,r2,.....rn are not real variables. It is therefore not possible to prove the existence and the uniqueness of the limit of rn as n→∞, by applying methods used in calculus.

Example Problem

Q.number x is drawn arbitrarily from the set {1,2,3....... 100}.Find the probability that
(x + 100/x) > 29 ?
Solution: The total points of the sample space are 100. Let A be the event that an x selected (drawn) at random from the set S = (1,2,3,,100) has the property (x+100/x)>29.
Now x+ 100/x>29
x²-29x+100 > 0
(x-4) (x-25) > 0
x>25 or x Since x ∈ S, it follows that A = {1, 2, 3, 26, 27, ..., 100)
Thus the number of cases favourable to A is 78 .
The required probability: P(A) = 78/100= 0.78.