Copyright © 2014, 2011 Pearson Education, Inc. 1 Chapter 7 Probability

Copyright © 2014, 2011 Pearson Education, Inc 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 © 2014, 2011 Pearson Education, Inc 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 © 2014, 2011 Pearson Education, Inc 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. Notes: The pattern must converge for LLN to apply. LLN only applies in the long run.

Copyright © 2014, 2011 Pearson Education, Inc From Data to Probability The Accumulated Relative Frequency of Easy Calls Converges to 70%

Copyright © 2014, 2011 Pearson Education, Inc 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 © 2014, 2011 Pearson Education, Inc 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 © 2014, 2011 Pearson Education, Inc Rules for Probability

Copyright © 2014, 2011 Pearson Education, Inc Rules for Probability

Copyright © 2014, 2011 Pearson Education, Inc 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 © 2014, 2011 Pearson Education, Inc Rules for Probability

Copyright © 2014, 2011 Pearson Education, Inc 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 © 2014, 2011 Pearson Education, Inc 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 © 2014, 2011 Pearson Education, Inc Rules for Probability

Copyright © 2014, 2011 Pearson Education, Inc Rules for Probability Rule 5: 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 © 2014, 2011 Pearson Education, Inc Rules for Probability

Copyright © 2014, 2011 Pearson Education, Inc Rules for Probability An Example – Movie Schedule

Copyright © 2014, 2011 Pearson Education, Inc 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 © 2014, 2011 Pearson Education, Inc 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 © 2014, 2011 Pearson Education, Inc 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 © 2014, 2011 Pearson Education, Inc 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 © 2014, 2011 Pearson Education, Inc. 22 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 © 2014, 2011 Pearson Education, Inc. 23 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 © 2014, 2011 Pearson Education, Inc. 24 4M Example 7.1: MANAGING A PROCESS Mechanics Assuming days are independent, use the multiplication rule to find P (OK for 5 days) = = 0.774

Copyright © 2014, 2011 Pearson Education, Inc. 25 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) = = 0.226

Copyright © 2014, 2011 Pearson Education, Inc. 26 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 © 2014, 2011 Pearson Education, Inc 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 © 2014, 2011 Pearson Education, Inc Independent Events

Copyright © 2014, 2011 Pearson Education, Inc Independent Events Boole’s Inequality Applied to 4M Example 7.1 P (breakdown during 5 days) = P(A 1 or A 2 or A 3 or A 4 or A 5 ) ≤ ≤ 0.25 Exact answer if the events are independent is 0.226

Copyright © 2014, 2011 Pearson Education, Inc. 30 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 © 2014, 2011 Pearson Education, Inc. 31 Best Practices (Continued)  Only add probabilities of disjoint events.  Be clear about independence.  Only multiply probabilities of independent events.

Copyright © 2014, 2011 Pearson Education, Inc. 32 Pitfalls  Do not assume that events are disjoint.  Avoid assigning the same probability to every outcome.  Do not confuse independent events with disjoint events.