Paper 4: Fundamentals of Business Mathematics & Statistic

9.6 I FUNDAMENTALS OF BUSINESS MATHEMATICS AND STATISTICS

Probability

repeated a large number of times under essentially identical conditions, the limiting value of the ratio of

the number of times the event A happens to the total, number of trials.of the experiments as the number of

trials increases indefinitely, is called the probability of the occurrence of A”.

Thus, P(A ) lim=n→ ∞mn

The happening of an event is determined on the basis of past experience or on the basis of relative frequency

of success in the past. For instance, a machine produces 10% unacceptable articles of the total output. On

the basis of such experience or experiments, we may arrive at that (i) the relative frequency obtained on

the basis of past experience can be shown to come very close to the classical probability. For example, as

said earlier, a coin is tossed for 6 times, we may not get exactly 3 heads and 3 tails. But, the coin is tossed for

larger number of times, say 10,000 times, we can expect heads and tails very close to 50% (ii) There are

certain laws, according to which the ‘occurrence’ or ‘non-occurrence of the events take place. Posterior

probabilities, also called Empirical Probabilities are based on experiences of the past and on experiments

conducted. Thus, relative frequency can be termed as a measure of probability and it is calculated on the

basis of empirical or statistical findings. For instance if a machine produces 100 articles in the past, 2 particles

were found to be defective, then the probability of the defective articles is 2/100 or 2%.

9.3.2.1. Limitations of Relative Frequency Theory of Probability:

1. The experimental conditions may not remain essentially homogeneous and identical in a large number
   of repetitions of the experiment.


2. The relative frequency —, may not attain a unique value no matter however large N may be.


3. Probability P(A) defined can never be obtained in practice. We can only attempt at a close estimate
   of P(A) by making N sufficiently large.
   Example 6:
   An urn contains 8 white and 3 red balls. If two balls are drawn at random, find the probability that (a) both
   are white, (b) both are red and (c) one is of each colour.

Solution :

Total number of balls in the urn = 8 + 3 =11

Two balls can be drawn out of 11 balls in^11 C 2 ways.

Exhaustive number of cases =

(^11) C 2 11 10 55
2

= × =.

(a) Two white balls to be drawn out of 8 white, can be done in

(^8) C 2 8 7 28
2

= × =

ways.

The probability that both are white = =^2855

(b) Two red balls to be drawn out of 3 red balls can be done in^3 C 2 =3 ways.

Hence, the probability that both are red = 553

(c) The number of favourable cases for drawing one white ball and one red ball is

(^8) C 1 x (^3) C 1 = 8 x 3 = 24.