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Unit 26: Theory of Probability: Introduction and Uses

From the above definition, it is clear that the sum of the probability of happening of an event Notes
called success (p) and the probability of non-happening of an event called failure (q) is always
one (1), i. e., p + q = 1. If p is known, we can find q and if q is known, then we can find p. In
p
practice, the value of p lies between 0 and 1, i.e., 0 ≤≤ 1 . To quote Prof. Morrison, “If an
event can happen in m ways and fail to happen in n ways, then probability of happening is
m n
m + n and that of its failure to happen is mn ”.
+
Limitations of Classical Definition
Following are the main limitations of classical definition of probability:
(1) If the various outcomes of the random experiment are not equally-likely, then we cannot
find the probability of the event using classical definition.
(2) The classical definition also fails when the total number of cases are infinite.
(3) If the actual value of N is not known, then the classical definition fails.
(2) Empirical or Relative Frequency Definition
This definition of probability is not based on logic but past experience and experiments and
present conditions. If vital statistics gives the data that out of 100 newly born babies, 55 of them
are girls, then the probability of the girl birth will be 55/100 or 55%. To quote Kenny and
Keeping, “If event has occured r times in a series of n independent trials, all are made under
the same identical conditions, the ratio r/n is the relative frequency of the event. The limit of
r/n as n tends to infinity is the probability of the occurence of the event”.

Probability is the limit of the relative frquency of success in infinite sequences of
trials.

Symbolically,
r
P(A) =Limit
n →∞ n
For example, if a coin is tossed 100 times and the heads turn up 55 times, then the relative
55
frequency of head will be = 0.55. Similarly, if a coin is tossed 1000 times and if the head
100
495
turns up 495 times, then the relative frequency will be = 0.495. In 10,000 tosses, the head
1000
turns up 5085 then the relative frequency will be 0.5085. Thus as we go on increasing the number
of trials, there is a tendency mat the relative frequency of head would approach to 0.50. The
following figure illustrate the idea:

Probability 0.5

0 50 100 150 200 250 300 350 400 450
Number of Trials ( )
n

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