BUSINESS MATHEMATICS AND STATISTICS THE ADDITION AND THE MULTIPLICATION THEOREM OF PROBABILITY A PRESENTATION ON

INTRODUCTION

THEOREMS OF PROBABILITY There are three main theorems of probability are : ADDITION THEOREM MULTIPLICATION THEOREM COMBINED USE OF ADDITION AND MULTIPLICATION THEOREM BERNOULLI’S THEOREM BAYES’ THEOREM

THE ADDITION THEOREM The Addition Theorem of Probability is studied under two headings: Addition Theorem for Mutually Exclusive Events Addition Theorem for Not Mutually Exclusive Events THE ADDITION THEOREM FOR MUTUALLY EXCLUSIVE EVENTS The Addition Theorem states that if A and B are two mutually exclusive events, then the probability of occurrence of either A or B is the sum of the individual probabilities of A and B. Symbolically, P (A or B) = P (A) + P (B) OR P (A + B) = P (A) + P (B) It is also known as the ‘Theorem of Total Probability.’

PROOF OF THE THEOREM

EXAMPLES ILLUSTRATING THE APPLICATION OF THE ADDITION THEOREM

ADDITION THEOREM FOR NOT MUTUALLY EXCLUSIVE EVENTS Two or more events are known as partially overlapping if part of one event and part of another event occur together. Thus, when the events are not mutually exclusive the addition theorem has to be modified. Modified Addition Theorem states that if A and B are not mutually exclusive events, the probability of occurrence of either A or B or both is equal to the probability of that event A occurs, plus the probability that event B occurs minus the probability that events common to both A and B occur simultaneously. Symbolically, P (A or B or Both ) = P (A) + P (B) – P (AB) The following figure illustrates this point: NOT MUTUALLY EXCLUSIVE EVENTS Overlapping Events AB AB

Generalisation The theorem can be extended to three or more events. If A, B and C are not mutually exclusive events, then P (Either A or B or C) = P (A) + P (B) + P (C) – P (AB) – P (AC) – P (BC) + P (ABC) EXAMPLES ILLUSTRATING THE APPLICATION OF THE MODIFIED ADDITION THEOREM

MULTIPLICATION THEOREM MULTIPLICATION THEOREM FOR INDEPENDENT EVENTS

Generalisation The theorem can be extended to three or more independent events. If A, B and C are three independent events, then P (ABC) = P (A) x P (B) x P (C) EXAMPLES ILLUSTRATING THE APPLICATION OF THE MULTIPLE THEOREM

Probability of happening of atleast one event in case of n independent events P (happening of atleast one of the events) = 1 – P (happening of none of the events) EXAMPLES ILLUSTRATING THE APPLICATION OF THIS THEOREM

CONDITIONAL PROBABILITY

MULTIPLICATION THEOREM IN CASE OF DPENDENT VARIABLES When the events are not independent, i.e., they are dependent events, then the multiplication theorem has to be modified. The Modified Multiplication Theorem states that if A and B are two dependent events, then the probability of their simultaneous occurrence is equal to the probability of one event multiplied by the conditional probability of the other. P (AB) = P (A). P (B/A) OR P (AB) = P (B). P (A/B) Where, P (B/A) = Conditional Probability of B given A. P (A/B) = Conditional Probability of A given B.

EXAMPLES ILLUSTRATING THE APPLICATION OF THE MODIFIED MULTIPLICATION THEOREM

COMBINED USE OF ADDITION AND MULTIPLICATION THEOREM

BERNOULLI’S THEOREM IN THE THEORY OF PROBABILITY Bernoulli’s theorem is very useful in working out various probability problems. This theorem states that if the probability of happening of an event on one trial or experiment is known, then the probability of its happening exactly, 1,2,3,…r times in n trials can be determined by using the formula: P (r) = n C r p r. q n-r r = 1,2,3,…n Where, P (r) = Probability of r successes in n trials. p = Probability of success or happening of an event in one trial. q = Probability of failure or not happening of the event in one trial. n = Total number of trials.

EXAMPLES ILLUSTRATING THE APPLICATION OF BERNOULLI’S THEOREM

BAYE’S THEOREM Baye’s Theorem is named after the British Mathematician Thomas Bayes and it was published in the year With the help of Baye’s Theorem, prior probability are revised in the light of some sample information and posterior probabilities are obtained. This theorem is also called Theorem of Inverse Probability. STATEMENT OF BAYE’S THEOREM

PROOF OF THE THEOREM Since, A 1 and A 2 are mutually exclusive events and since the event B occurs with only one of them, so that B = BA 1 + BA 2 orB = A 1 B + A 2 B By the addition theorem of probability, we have P (B) = P (A 1 B) + P(A 2 B)…(i) A1A1 A2A2 B

The theorem can be expressed by means of the following figure: P (B/A 1 )P (A 1 ). P (B/A 1 ) First Branch Second Branch P (A 1 ) P (A 2 ) P (B/A 2 ) P (A 2 ). P (B/A 2 ) Prior ProbabilityConditional ProbabilityJoint Probability Of A 1 and A 2 of B given A 1 and A 2

EXAMPLES ILLUSTRATING THE APPLICATION OF BAYE’S THEOREM Events (1) Prior Probability (2) Conditional Probability (3) Joint Probability (2) X (3) AP(A) = 0.25P (D/A) = x0.05 BP (B) = 0.35P (D/B) = x0.04 CP (C) = 0.40P (D/C) = x0.02

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