Definitions of Probability

1.  Definitions of Probability:

There are two types of probability. They are Mathematical probability and Statistical probability.

    Mathematical Probability (or a priori probability):

If the probability of an event can be calculated even before the actual happening of the event, that is, even before conducting the experiment, it is called Mathematical probability.

If the random experiments results in ‘n’ exhaustive, mutually exclusive and equally likely cases, out of which ‘m’ are favourable to the occurrence of an event A, then the ratio m/n is called the probability of occurrence of event A, denoted by P(A), is given by P (A) = m/n 

where;  

m= number of cases favourable to the event A

n= Total number of exhaustive cases

Mathematical probability is often called classical probability or a priori probability because if we keep using the examples of tossing of fair coin, dice etc., we can state the answer in advance (prior), without tossing of coins or without rolling the dice etc.,

The above definition of probability is widely used, but it cannot be applied under the

following situations:

(1)   If it is not possible to enumerate all the possible outcomes for an experiment.

(2)   If the sample points (outcomes) are not mutually independent.

(3) If the total number of outcomes is infinite.

(4) If each and every outcome is not equally likely.

Some of the drawbacks of classical probability are removed in another definition given

below:

1.1.2    Statistical Probability (or a posteriori probability):

If the probability of an event can be determined only after the actual happening of the event, it is called Statistical probability.

If an event occurs m times out of n, its relative frequency is m/n.

In the limiting case, when n becomes sufficiently large it corresponds to a number

which is called the probability of that event.

In symbol, P(A) = Limit (m/n)

n → ∞

The  above  definition  of  probability  involves  a  concept  which  has  a  long  term

consequence. This approach was initiated by the mathematician Von Mises .

If a coin is tossed 10 times we may get 6 heads and 4 tails or 4 heads and 6 tails or any other result. In these cases the probability of getting a head is not 0.5 as we consider in Mathematical probability.

However, if the experiment is carried out a large number of times we should expect approximately equal number of heads and tails and we can  see  that  the  probability  of getting head approaches 0.5. The Statistical probability calculated by conducting an actual experiment is also called a posteriori probability or empirical probability.