5- Chapter Five McGraw- © 2005 The McGraw-Hill Companies, Inc., All Chapter Five 5- A Survey of Probability Concepts GOALS When you have completed this chapter, you will be able to: ONE Define probability TWO Describe the classical, empirical, and subjective approaches to probability THREE Understand the terms: experiment, event, outcome, permutations, and combinations Goals 5- Chapter Five continued A Survey of Probability GOALS Concepts When you have completed this chapter, you will be able to: FOUR Define the terms: conditional probability and joint probability FIVE Calculate probabilities applying the rules of addition and the rules of multiplication SIX Use a tree diagram to organize and compute probabilities Goals Chapter Five 5- continued A Survey of Probability Concepts SEVEN Calculate a probability using Bayes’ theorem Goals 5- Movie 5- There are three definitions of probability: classical, empirical, and subjective The Classical definition applies when there are n equally likely outcomes The Empirical definition applies when the number of times the event happens is divided by the number of observations Subjective probability is based on whatever information is available Definitions continued 5- Movie 5- An Outcome is the particular result of an experiment An Event is the collection of one or more outcomes of an experiment Experiment: A fair die is cast Possible outcomes: The numbers 1, 2, 3, 4, 5, One possible event: The occurrence of an even number That is, we collect the outcomes 2, 4, and Definitions continued 5- Events are Mutually Exclusive if the occurrence of any one event means that none of the others can occur at the same time Mutually exclusive: Rolling a precludes rolling a 1, 3, 4, 5, on the same roll Events are Independent if the occurrence of one event does not affect the occurrence of another Independence: Rolling a on the first throw does not influence the probability of a on the next throw It is still a one in chance Mutually Exclusive Events 5- 10 Events are Collectively Exhaustive if at least one of the events must occur when an experiment is conducted Collectively Exhaustive Events 5- 29 The General Rule of Multiplication is used to find the joint probability that two events will occur It states that for two events A and B, the joint probability that both events will happen is found by multiplying the probability that event A will happen by the conditional probability of B given that A has occurred General Multiplication Rule 5- 30 The joint probability, P(A and B), is given by the following formula: P(A and B) = P(A)P(B/A) or P(A and B) = P(B)P(A/B) General Multiplication Rule 5- 31 The Dean of the School of Business at Owens University collected the following information about undergraduate students in her college: Example If a student is selected at random, what is the probability that the student is a female (F) accounting major (A)? 5- 32 P(A and F) = 110/1000 Given that the student is a female, what is the probability that she is an accounting major? P(A|F) = P(A and F)/P(F) = [110/1000]/[400/1000] = 275 Example continued 5- 33 A Tree Diagram is useful for portraying conditional and joint probabilities It is particularly useful for analyzing business decisions involving several stages Example 8: In a bag containing red chips and blue chips you select chips one after the other without replacement Construct a tree diagram showing this information Tree Diagrams 5- 34 6/11 7/12 5/12 R1 R2 5/11 B2 7/11 R2 B1 4/11 B2 Example continued 5- 35 Bayes’ Theorem is a method for revising a probability given additional information It is computed using the following formula: P( A1 ) P( B / A1 ) P( A1 | B) = P( A1 ) P( B / A1 ) + P( A2 ) P( B / A2 ) Bayes’ Theorem 5- 36 Duff Cola Company recently received several complaints that their bottles are under-filled A complaint was received today but the production manager is unable to identify which of the two Springfield plants (A or B) filled this bottle What is the probability that the under-filled bottle came from plant A? Example 5- 37 The following table summarizes the Duff production experience % of underfilled bottle % of total production A 5.5 3.0 B 4.5 4.0 Example continued 5- 38 P ( A) P (U / A) P( A / U ) = P ( A) P (U / A) + P ( B ) P (U / B ) 55(.03) = = 4783 55(.03) + 45(.04) The likelihood the bottle was filled in Plant A is reduced from 55 to 4783 Example continued 5- 39 The Multiplication Formula indicates that if there are m ways of doing one thing and n ways of doing another thing, there are m x n ways of doing both Example 10: Dr Delong has 10 shirts and ties How many shirt and tie outfits does he have? (10)(8) = 80 Some Principles of Counting 5- 40 A Permutation is any arrangement of r objects selected from n possible objects Note: The order of arrangement is important in permutations n n! Pr = ( n − r )! Some Principles of Counting 5- 41 A Combination is the number of ways to choose r objects from a group of n objects without regard to order n! nCr = r! (n − r )! Some Principles of Counting 5- 42 There are 12 players on the Carolina Forest High School basketball team Coach Thompson must pick five players among the twelve on the team to comprise the starting lineup How many different groups are possible? (Order does not matter.) 12! 12C = = 792 5! (12 − 5)! Example 11 5- 43 Suppose that in addition to selecting the group, he must also rank each of the players in that starting lineup according to their ability (order matters) 12! = 95,040 12 P = (12 − 5)! Example 11 continued ... permutations, and combinations Goals 5- Chapter Five continued A Survey of Probability GOALS Concepts When you have completed this chapter, you will be able to: FOUR Define the terms: conditional... + 075 =.175 Example continued 5- 16 The Complement Rule is used to determine the probability of an event occurring by subtracting the probability of the event not occurring from If P(A) is the... probability that IBM stock will increase in value next year is and the probability that GE stock will increase in value next year is Assume the two stocks are independent What is the probability