Paper 4: Fundamentals of Business Mathematics & Statistic

FUNDAMENTALS OF BUSINESS MATHEMATICS AND STATISTICS I 9.11

Solution:

P No.offavourablecases

=TotalNo.ofequallylikelycases

(i) Probability of Red = 156 or 0.40

(ii) Probability of white = 154 or 0.267

(iii) Probability of Blue= 155 A or 0.333

(iv) Probability of not Red = 159 or 0.60

(v) Probability of Red and White =^1015 or 0.667

9.5 BAYES’ THEOREM

This theorem is associated with the name of Reverend Thomas Bayes. It is also known as the inverse
probability. Probabilities can be revised when new information pertaining to a random experiment is
obtained. One of the important applications of the conditional probability is in the computation of unknown
probabilities, on.the basis of the information supplied by the experiment or past records. That is, the
applications of the results of probability theory involves estimating unknown probabilities and making
decisions on the basis of new sample information. This concept is referred to as Bayes’ Theorem. Quite often
the businessman has the extra information on a particular event, either through a personal belief or from
the past history of the events. Revision of probability arises from a need to make better use of experimental
information. Probabilities assigned on the basis of personal experience, before observing the outcomes of
the experiment are called prior probabilities. For example, probabilities assigned to past sales records, to
past number of defectives produced by a machine, are examples of prior probabilities. When the probabilities
are revised with the use of Bayes’ rule, they are called posterior probabilities. Bayes’ theorem is useful in
solving practical business problems in the light of additional information. Thus popularity of the theorem
has been mainly because of its usefulness in revising a set of old probability (Prior Probability) in the light of
additional information made available and to derive a set of new probabilily (i.e. Posterior Probability)

Bayes’ Theorem: An event A can occurre only if one of the mutually exclusive and exhaustive set of events
B 1 , B 2 , ..... Bn occurs. Suppse that the unconditional probabilities

P(B 1 ), P(B 2 ), .... P(Bn)
and the conditional probabilities
P(A/B 1 ), P(A/B 2 ), .... P(A/Bn)

are known. Then the conditional probability P(Bi/A) of a specific event Bi, when A is stated to have actually
accquared, is given by

i n i i

i i i i

P(B / A) P(B ).P(A /B )

P(B ). P(A/B )

=

=