Copyright © 2011 Pearson Education, Inc. Probability Chapter 7

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? Copyright © 2011 Pearson Education, Inc. 3 of 32

7.1 From Data to Probability Probability = Long Run Relative Frequency  Keep track of calls (1 = easy call; 0 = hard call)  Graph the accumulated relative frequency of easy calls  In the long run, the accumulated relative frequency converges to a constant (probability) Copyright © 2011 Pearson Education, Inc. 4 of 32

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. Copyright © 2011 Pearson Education, Inc. 5 of 32

7.1 From Data to Probability The Accumulated Relative Frequency of Easy Calls Converges to 70% Copyright © 2011 Pearson Education, Inc. 6 of 32

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. Copyright © 2011 Pearson Education, Inc. 7 of 32

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 Copyright © 2011 Pearson Education, Inc. 8 of 32

7.2 Rules for Probability Rule 1: Since S is the set of all possible outcomes, P(S) = 1 Copyright © 2011 Pearson Education, Inc. 9 of 32

7.2 Rules for Probability Rule 2: For any event A, 0 ≤ P(A) ≤ 1. Copyright © 2011 Pearson Education, Inc. 10 of 32

7.2 Rules for Probability Rule 3: Addition Rule for Disjoint Events  Disjoint events are mutually exclusive; i.e., they have no outcomes in common.  The union of two events is the collection of outcomes in A, in B, or in both (A or B) Copyright © 2011 Pearson Education, Inc. 11 of 32

7.2 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). Copyright © 2011 Pearson Education, Inc. 12 of 32

7.2 Rules for Probability Rule 3: Addition Rule for Disjoint Events  Extends to more than two events  P (E 1 or E 2 or … or E k ) = P(E 1 ) + P(E 2 ) + … + P(E k ) Copyright © 2011 Pearson Education, Inc. 13 of 32

7.2 Rules for Probability Rule 4: Complement Rule  The complement of event A consists of the outcomes in S but not in A  Denoted as A c Copyright © 2011 Pearson Education, Inc. 14 of 32

7.2 Rules for Probability Rule 4: Complement Rule: P(A) = 1 – P(A c ) The probability of an event is one minus the probability of its complement. Copyright © 2011 Pearson Education, Inc. 15 of 32

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” Copyright © 2011 Pearson Education, Inc. 16 of 32

7.2 Rules for Probability Rule 5: General Addition Rule P (A or B) = P(A) + P(B) – P (A and B). Copyright © 2011 Pearson Education, Inc. 17 of 32

7.2 Rules for Probability An Example – Movie Schedule Copyright © 2011 Pearson Education, Inc. 18 of 32

7.2 Rules for Probability What’s the probability that the next customer buys a ticket for a movie that starts at 9 PM or is a drama? Copyright © 2011 Pearson Education, Inc. 19 of 32

7.2 Rules for Probability What’s the probability that the next customer buys a ticket for a movie that starts at 9 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 Copyright © 2011 Pearson Education, Inc. 20 of 32

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 Copyright © 2011 Pearson Education, Inc. 21 of 32

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) Copyright © 2011 Pearson Education, Inc. 22 of 32

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? Copyright © 2011 Pearson Education, Inc. 23 of 32

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. Copyright © 2011 Pearson Education, Inc. 24 of 32

4M Example 7.1: MANAGING A PROCESS Mechanics Assuming days are independent, use the multiplication rule to find P (OK for 5 days) = = Copyright © 2011 Pearson Education, Inc. 25 of 32

4M Example 7.1: MANAGING A PROCESS Mechanics Use the complement rule to find P (breakdown during 5 days) = 1 - P(OK for 5 days) = = Copyright © 2011 Pearson Education, Inc. 26 of 32

4M Example 7.1: MANAGING A PROCESS Message The probability that a breakdown interrupts production in the next five days is It is wise to warn the customer that delivery may be delayed. Copyright © 2011 Pearson Education, Inc. 27 of 32

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. Copyright © 2011 Pearson Education, Inc. 28 of 32

7.3 Independent Events Boole’s Inequality Copyright © 2011 Pearson Education, Inc. 29 of 32

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 1. Copyright © 2011 Pearson Education, Inc. 30 of 32

Best Practices (Continued)  Only add probabilities of disjoint events.  Be clear about independence.  Only multiply probabilities of independent events. Copyright © 2011 Pearson Education, Inc. 31 of 32

Pitfalls  Do not multiply probabilities of dependent events.  Avoid assigning the same probability to every outcome.  Do not confuse independent events with disjoint events. Copyright © 2011 Pearson Education, Inc. 32 of 32