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Unit 12: Probability and Expected Value

3. The 2 black balls can be treated as one and, therefore, this ball along with 5 red balls can be Notes
arranged in 6! ways. Further, 2 black ball can be arranged in 2! ways.

6!2! 2
The required probability = =
7! 7
12.3.5 Statistical or Empirical 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.
In addition to the above 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 sufficiently 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 A occurs is given by
P(A) = , provided the limit exists.
This definition of probability is also termed as the empirical definition because the probability
of an event is obtained by actual experimentation.
m
Although, it is seldom possible to obtain the limit of the relative frequency, the ratio can be
n
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.

m
2. The relative frequency, , may not attain a unique value no matter how large is the total
n
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.

12.3.6 Axiomatic or Modern Approach to Probability

This approach was introduced by the Russian mathematician, A. Kolmogorov in 1930s. In his
book, ‘Foundations of Probability’ published in 1933, he introduced probability as a function of
the outcomes of an experiment, under certain restrictions. These restrictions are known as
Postulates or Axioms of probability theory. Before discussing the above approach to probability,
we shall explain certain concepts that are necessary for its understanding.

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