Definition

Probability means the chance or the likelihood of
occurrence of an event. It is a numerical value that is lying between 0 and 1.  When the event is impossible probability of occurring that event  is 0, when the event is sure, probability of occurring that event is 1 and all other events will have probability between 0 and 1.

There are mainly 3 definitions of probability

  1. Theoretical or classical definition of probability.
  2.  Empirical or Statistical definition of probability
  3. Axiomatic definition of probability.

Theoretical or classical definition of probability

Let a random experiment produce only a finite number of  mutually exclusive and equally likely outcomes. Then the probability of an event A is defined as
                P(A) = Number of favorable outcomes to A/ Total number of outcomes
Theoretical probability is also known as Classical or A Priori probability. Classical probability is also known as statistical probability.
Example: -
Consider the experiment of tossing a coin.What is the probability of getting a tail?
Solution: -
We know that the probability of an event A is given by
P(A) = Number of favorable outcomes to A/ Total number of outcomes
The Sample space  of the experiment of tossing a coin is given by,
S = {Head, Tail}
Number of favorable outcomes to tail = 1
Total number of outcomes = 2
So P(getting a tail) = 1/2

Empirical  definition of probability

Let A be an event of a random experiment. 
Let the experiment be repeated n number of times out of which A occurs f
times.  Then f/n is called frequency ratio.  The limiting value of the
frequency ratio as the number of repetitions becomes infinitely large is called
probability of the event A.

It is sometimes known as experimental probability also.

Example: -
The following below shows the number of heads appearing when 8 coins are tossed.
x;         0          1          2          3          4          5          6          7          8
f:          1          9          26        59        72        52        26        7          1
Find the probability of getting heads
(i) less than 4  
(ii)equal to 5 
(iii) more than 6
Solution: -
Here total number of heads = 1 + 9 + 26 + 59 + 72 + 52 + 26 + 7 + 1 = 253
(i) We have to find the probability of getting heads less than 4.
There are 1 + 9 + 26 + 59 = 95 cases in which there are less than 4 heads.
Therefore P(heads less than 4) = 95/253 = 0.37
(ii) We have to find the probability of getting heads equal to 5.
There are 52 cases in which there are 5 heads.
Therefore P(heads equal to 5) = 52/253 = 0.21
(iii) We have to find the probability of getting more than 6 heads.
There are 7 + 1 = 8 cases in which there are more than 6 heads.
Therefore P(heads more than 6) = 8/253 = 0.03

Axiomatic definition of probability

Let S be the sample space of a random experiment. 
Let A be an event of a random experiment. 
According to this definition Probability of an event is a real number which
satisfies 3 axioms.
(i) Axiom 1:P(A) ≥ 0
(ii) Axiom 2 :P(S) = 1
(iii) Axiom 3: P(A∪B) = P(A) + P(B) where A and B are disjoint events
Therefore P(A) is a real valued, non negative, totally additive set function.  Therefore P(A) is a measure and it is the probability of A.
 Axiomatic definition is usually not used to work out problems


Properties of Probability

  1. The probability of an event is always greater than or equal to 0
  2. The probability of an event lies between 0 and 1
  3. Probability of an impossible event is 0.  That is P(Φ) = 0.
  4. Probability of a sure event is 1.  That is P(S) = 1.