2 chance w statistical methods for decision irwin inc

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2) Chance, W., ‘Statistical methods for Decision’ Irwin Inc., Homewood. 3) Gopikuttam, G., ‘Quantitative Methods and Operations Research’, Himalaya Publishing House, Bombay. 4) Gupta, S.P., ’Statistics Methods’, Sultan Chand and Co., New Delhi 5) Levin, R., ‘Statistics for Management’, Prentice Hall of India, New Delhi, 1984. 6) Reddy, C.R., ‘Quantitative Methods for Management Decision’, Himalaya Publishing House, Bombay, 1990. 2.11. LEARNING ACTIVITIES Explain the importance of probability in research and practice. 2.12 KEY WORDS 1) Random Experiment. 2) Trial and Event. 3) Independent and Dependent Events. 4) Mutually Exclusive Events. 5) Equally Likely Events. 6) Exhaustive Events. 7) Simple and compound Events.
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10 LESSON 3 PROBABILITY THEOREMS 3.1. INTRODUCTION The computation of probabilities can become easy and be facilitated to a great extent by the two fundamental theorems of probability the additional theorem and the multiplication theorem which is discussed in foregoing pages. 3.2. OBJECTIVES The main aim of this Lesson is to study the Basics of probability theorems. After going through this Lesson you should be able to: 1. Understand the Addition Theorems of Probability 2. Understand the Multiplication Theorem of Probability 3.3. CONTENTS 3.3.1. Addition Theorem Independent Events 3.3.2. Addition Theorem Dependent Events 3.3.3. Multiplication Theorem 3.3.1. ADDITION THEOREM INDEPENDENT EVENTS The probability of occurring, either event A or event B which are mutually exclusive events is the sum of the individual probability of A and B Symbolically. P (A or B) = P (A) + P (B) Proof If an event A can be happen in a 1 way and B in a 2 ways, then the number of ways in which either event can happen is a 1 + a 2 . If the total number of possibilities is n, then by definition the probability of either A or B even happening is: n a n a n a a 2 1 2 1 But, n a P(A) 1 and n a P(B) 2 Hence P (A or B) = P (A) + P(B) (or) P (A B) = P(A) + P (B) The theorem can be extended to three or more mutually exclusive events. Thus, P (A or B or C) = P (A) +P (B) + P (C) (or) P (A B C) = P (A) + P (B) + P (C) 3.3.2. ADDITION THEOREM DEPENDENT EVENTS If events are not mutually exclusive, the above procedure discussed is no longer holds. For example, if the probability of buying a pen is 0.6 and that of
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11 pencil is 0.3, we cannot calculate the probability of buying either pen or pencil by addicting the two probabilities because the events are not mutually exclusive. When the events are not mutually exclusive, the above said theorem is to be modified. The probability of occurring of at least one of the two events A and B which are not mutually exclusive is given by: P (A or B) = P (A) + P (B) P (A and B) (or) P (A B) = P (A) + P(B) P (A B) By subtracting P (A and B) i.e., the proportion of events as counted twice in P (A) + P (B), the addition theorem is, thus, reconstructed in such a way as to render A and B mutually exclusive events. In the case of three events P (A or B or C) = P (A) + P (B) + P (C) P (A and B) P (A and C) P (B and C) + P (A and B and C) (or) P (A B
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