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# Ch04 - Chapter4 LearningObjectives 1 2 3

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Chapter 4 Introduction to Probability Learning Objectives 1. Obtain an appreciation of the role probability information plays in the decision making process. 2. Understand probability as a numerical measure of the likelihood of occurrence. 3. Know the three methods commonly used for assigning probabilities and understand when they should be used. 4. Know how to use the laws that are available for computing the probabilities of events. 5. Understand how new information can be used to revise initial (prior) probability estimates using Bayes’ theorem. 4 - 1

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Chapter 4 Solutions: 1. Number of experimental Outcomes = (3)(2)(4) = 24 2. 6 6! 6 5 4 3 2 1 20 3 3!3! (3 2 1)(3 2 1) = = = ABC ACE BCD BEF ABD ACF BCE CDE ABE ADE BCF CDF ABF ADF BDE CEF ACD AEF BDF DEF 3. P 3 6 6! 6 3 6 5 4 120 = - = = ( )! ( )( )( ) BDF BFD DBF DFB FBD FDB 4. a. H T H T H T H T H T H T H T (H,H,H) (H,H,T) (H,T,H) (H,T,T) (T,H,H) (T,H,T) (T,T,H) (T,T,T) 1st Toss 2nd Toss 3rd Toss b. Let: H be head and T be tail (H,H,H) (T,H,H) (H,H,T) (T,H,T) (H,T,H) (T,T,H) (H,T,T) (T,T,T) c. The outcomes are equally likely, so the probability of each outcomes is 1/8. 5. P (E i ) = 1/5 for i = 1, 2, 3, 4, 5 P (E i ) 0 for i = 1, 2, 3, 4, 5 P (E 1 ) + P (E 2 ) + P (E 3 ) + P (E 4 ) + P (E 5 ) = 1/5 + 1/5 + 1/5 + 1/5 + 1/5 = 1 4 - 2
Introduction to Probability The classical method was used. 6. P (E 1 ) = .40, P (E 2 ) = .26, P (E 3 ) = .34 The relative frequency method was used. 7. No. Requirement (4.4) is not satisfied; the probabilities do not sum to 1. P (E 1 ) + P (E 2 ) + P (E 3 ) + P (E 4 ) = .10 + .15 + .40 + .20 = .85 8. a. There are four outcomes possible for this 2-step experiment; planning commission positive - council approves; planning commission positive - council disapproves; planning commission negative - council approves; planning commission negative - council disapproves. b. Let p = positive, n = negative, a = approves, and d = disapproves . Planning Commission Council p n a d a d (p, a) (p, d) (n, a) (n, d) 9. 50 50! 50 49 48 47 230,300 4 4!46! 4 3 2 1 = = = 10. a. Use the relative frequency approach: P (California) = 1,434/2,374 = .60 b. Number not from 4 states = 2,374 - 1,434 - 390 - 217 - 112 = 221 P (Not from 4 States) = 221/2,374 = .09 c. P (Not in Early Stages) = 1 - .22 = .78 4 - 3

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Chapter 4 d. Estimate of number of Massachusetts companies in early stage of development = (.22)390 86 e. If we assume the size of the awards did not differ by states, we can multiply the probability an award went to Colorado by the total venture funds disbursed to get an estimate. Estimate of Colorado funds = (112/2374)(\$32.4) = \$1.53 billion Authors' Note: The actual amount going to Colorado was \$1.74 billion. 11.
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Ch04 - Chapter4 LearningObjectives 1 2 3

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