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,