# Definitions of Probability

**Classical/Mathematical definition of probability***P*(*A*) < 1

- If
*n*(*A*) = 0,*P*(*A*) = 0 - If the event has zero favourable outcomes, the event is said to be impossible event
- If
*n*(*A*) =*n*(*s*),*P*(*A*) = 1 *P*(*A*) = 1 means the event is certain

Example

A bag contains 3 red, 4 green and 3 yellow marbles. Three marbles are randomly drawn from the bag. What is the probability that they are of (i) the same colour (ii) different colours (one of each colour)?

Solution

There are

(i) The marbles should be 3 red or 4 green or 3 yellow. Therefore,

(ii) The marbles should be one of each colour. Therefore, 3

And so,

^{10}*C*_{3}equally likely, mutually exclusive and exhaustive outcomes. Let events*A*and*B*be*A*: Selected marbles are of the same colour*B*: Selected marbles are of different colours(i) The marbles should be 3 red or 4 green or 3 yellow. Therefore,

^{3}*C*_{3}+^{4}*C*_{3}+^{3}*C*_{3}outcomes are favourable to event*A*. Hence,*P*[marbles of the same colour](ii) The marbles should be one of each colour. Therefore, 3

*C*_{1}Ã— 4*C*_{1}Ã— 3*C*_{1}outcomes are favourable.And so,

*P*[marbles of different colours]# Limitations of Mathematical Formula

- If the total number of exhaustive cases
*n*is not known, then probability cannot be obtained. (*n*should be finite) - It can be used only when the events are equally likely.
- This definition has only a limited field of applications like coin tossing, dice throwing, drawing cards etc., where the possible events are known well in advance. In the field of uncertainty or where no prior knowledge is provided, this definition is inapplicable.

**Statistical/Empirical definition of probability***A*occurs to the number of times the experiment is repeated is the probability of occurrence of*A*.

**Note:** In the above expressions it is assumed that the limiting value exists and it is unique.

Example

Out of 560 babies born in a community in a year, 252 were male. Find the probability that a new born baby is male.

Solution

Here,

*n*= 560 births were observed and among them*m*= 252 have resulted in birth to male children. Therefore, probability that a new born baby being male is*P*[male baby]# Limitations of Statistical Formula

- The experimental conditions may not remain homogenous and identical in a large number of repetitions of the experiment
- The relative frequency may not obtain a unique value

**Modern definition of probability**

- Let A be an event defined on a sample space
*i.e.*,*A*âŠ†*S*. The real valued function*P*defined on*S*is called a probability measure and*P*(*A*) is called the probability of the event*A*if*P*satisfies the following axioms:**Axiom-1:***P*(*A*) â‰¥ 0 for every*A*âŠ†*S*.**Axiom-2:***P*(*S*) = 1**Axiom-3:**For any sequence of mutually exclusive event*A*_{1},*A*_{2 }...*P*(*A*_{1 }âˆª*A*_{2 }âˆª â€¦) =*P*(*A*_{1}) +*P*(*A*_{2}) + â€¦