Statistics for business decision making and analysis robert stine and foster chapter 07

Chapter Probability Copyright © 2011 Pearson Education, Inc 7.1 From Data to Probability In a call center, what is the probability that an agent answers an easy call?  An easy call can be handled by a first-tier agent; a hard call needs further assistance  Two possible outcomes: easy and hard calls  Are they equally likely? of 32 Copyright © 2011 Pearson Education, Inc 7.1 From Data to Probability Probability = Long Run Relative Frequency  Keep track of calls (1 = easy call; = hard call)  Graph the accumulated relative frequency of easy calls  In the long run, the accumulated relative frequency converges to a constant (probability) of 32 Copyright © 2011 Pearson Education, Inc 7.1 From Data to Probability The Law of Large Numbers (LLN) The relative frequency of an outcome converges to a number, the probability of the outcome, as the number of observed outcomes increases Note: The pattern must converge for LLN to apply of 32 Copyright © 2011 Pearson Education, Inc 7.1 From Data to Probability The Accumulated Relative Frequency of Easy Calls Converges to 70% of 32 Copyright © 2011 Pearson Education, Inc 7.2 Rules for Probability Sample Space  Set of all possible outcomes  Denoted by S; S = {easy, hard}  Subsets of samples spaces are events; denoted as A, B, etc of 32 Copyright © 2011 Pearson Education, Inc 7.2 Rules for Probability Venn Diagrams  The probability of an event A is denoted as P(A)  Venn diagrams are graphs for depicting the relationships among events of 32 Copyright © 2011 Pearson Education, Inc 7.2 Rules for Probability Rule 1: Since S is the set of all possible outcomes, P(S) = of 32 Copyright © 2011 Pearson Education, Inc 7.2 Rules for Probability Rule 2: For any event A, ≤ P(A) ≤ 10 of 32 Copyright © 2011 Pearson Education, Inc 7.2 Rules for Probability An Example – Movie Schedule 18 of 32 Copyright © 2011 Pearson Education, Inc 7.2 Rules for Probability What’s the probability that the next customer buys a ticket for a movie that starts at PM or is a drama? 19 of 32 Copyright © 2011 Pearson Education, Inc 7.2 Rules for Probability What’s the probability that the next customer buys a ticket for a movie that starts at PM or is a drama? Use the General Addition Rule: P(A or B) = P(9 PM or Drama) = 3/6 + 3/6 – 2/6 = 2/3 20 of 32 Copyright © 2011 Pearson Education, Inc 7.3 Independent Events Definitions  Two events are independent if the occurrence of one does not affect the chances for the occurrence of the other  Events that are not independent are called dependent 21 of 32 Copyright © 2011 Pearson Education, Inc 7.3 Independent Events Multiplication Rule Two events A and B are independent if the probability that both A and B occur is the product of the probabilities of the two events P (A and B) = P(A) X P(B) 22 of 32 Copyright © 2011 Pearson Education, Inc 4M Example 7.1: MANAGING A PROCESS Motivation What is the probability that a breakdown on an assembly line will occur in the next five days, interfering with the completion of an order? 23 of 32 Copyright © 2011 Pearson Education, Inc 4M Example 7.1: MANAGING A PROCESS Method Past data indicates a 95% chance that the assembly line runs a full day without breaking down 24 of 32 Copyright © 2011 Pearson Education, Inc 4M Example 7.1: MANAGING A PROCESS Mechanics Assuming days are independent, use the multiplication rule to find P (OK for days) = 0.955 = 0.774 25 of 32 Copyright © 2011 Pearson Education, Inc 4M Example 7.1: MANAGING A PROCESS Mechanics Use the complement rule to find P (breakdown during days) = - P(OK for days) = 1- 0.774 = 0.226 26 of 32 Copyright © 2011 Pearson Education, Inc 4M Example 7.1: MANAGING A PROCESS Message The probability that a breakdown interrupts production in the next five days is 0.226 It is wise to warn the customer that delivery may be delayed 27 of 32 Copyright © 2011 Pearson Education, Inc 7.3 Independent Events Boole’s Inequality  Also known as Bonferroni’s inequality  The probability of a union is less than or equal to the sum of the probabilities of the events 28 of 32 Copyright © 2011 Pearson Education, Inc 7.3 Independent Events Boole’s Inequality 29 of 32 Copyright © 2011 Pearson Education, Inc Best Practices  Make sure that your sample space includes all of the possibilities  Include all of the pieces when describing an event  Check that the probabilities assigned to all of the possible outcomes add up to 30 of 32 Copyright © 2011 Pearson Education, Inc Best Practices (Continued)  Only add probabilities of disjoint events  Be clear about independence  Only multiply probabilities of independent events 31 of 32 Copyright © 2011 Pearson Education, Inc Pitfalls  Do not multiply probabilities of dependent events  Avoid assigning the same probability to every outcome  Do not confuse independent events with disjoint events 32 of 32 Copyright © 2011 Pearson Education, Inc ... Rules for Probability Rule 3: Addition Rule for Disjoint Events If A and B are disjoint events, then P (A or B) = P(A) + P(B) 12 of 32 Copyright © 2011 Pearson Education, Inc 7.2 Rules for Probability... Education, Inc 7.2 Rules for Probability Rule 5: General Addition Rule  The intersection of A and B contains the outcomes in both A and B  Denoted as A ∩ B read “A and B” 16 of 32 Copyright... Education, Inc 7.2 Rules for Probability Rule 1: Since S is the set of all possible outcomes, P(S) = of 32 Copyright © 2011 Pearson Education, Inc 7.2 Rules for Probability Rule 2: For any event A,