Classical and Statistical definition of Probability

The scope of the classical definition was found to be very limited as it failed to determine the probabilities of certain events in the following circumstances :

  1. When n, the exhaustive outcomes of a random experiment is infinite.
  2. When actual value of n is not known.
  3. When various outcomes of a random experiment are not equally likely.
  4. This definition doesn't lead to any mathematical treatment of probability.

In view of the above shortcomings of the classical definition, an attempt was made to establish a correspondence between relative frequency and the probability of an event when the total number of trials become su1fficiently large.

Definition (R. Von Mises)

If an experiment is repeated n times, under essentially the identical conditions and, if, out of these trials, an event A occurs m times, then the probability that

Definition (R. Von Mises)

This definition of probability is also termed as the empirical definition because the probability of an event is obtained by actual experimentation.

Although, it is seldom possible to obtain the limit of the relative frequency, the ratio m/n can be regarded as a good approximation of the probability of an event for large values of n.

This definition also suffers from the following shortcomings :

  1. The conditions of the experiment may not remain identical, particularly when the number of trials is sufficiently large.
  2. The relative frequency, m/n, may not attain a unique value no matter how large is the total number of trials.
  3. It may not be possible to repeat an experiment a large number of times.
  4. Like the classical definition, this definition doesn't lead to any mathematical treatment of probability.

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