Although there is not much objection against the in logical content of frequency approach for defining probability but there is some inherent weakness or inelegance in the mathematical formalism. In the definition, we note that the frequency ratio is thoroughly an empirical concept, whereas the limit is postulated in a rigorous analytical sense. This combination of empirical and theoretical concept is very inelegant and naturally leads to mathematical difficulties. Now this problem is not typical of probability theory only, but arises in other branches of mathematics as well. In the theory of probability also, we are ultimately forced to give up the hope of defining probabilities and take resource to an axiomatic theory in which probability is accepted as an undefined new concept, and only the salient rules for calculation of probabilities are postulated. These rules will however, be chosen from the previous theories with necessary modification for operational convenience.
The axiomatic approach to probability was proposed by A.N. Kolmogorov in the year 1933. When this approach is followed, no precise definition of probability is given, rather we give certain axioms or postulates on which probability calculations are based.
Baye’s Theorem for evaluating new information and revising our prior estimates of the probability. If correctly used, it makes it unnecessary to gather masses of data over long periods of time in order to make good decisions based on probabilities.
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