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?
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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)
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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.
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7.1 From Data to Probability
The Accumulated Relative Frequency of Easy Calls Converges to 70%
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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.
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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
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7.2 Rules for Probability
Rule 1: Since S is the set of all possible outcomes, P(S) = 1
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7.2 Rules for Probability
Rule 2: For any event A, 0 ≤ P(A) ≤ 1.
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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)
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7.2 Rules for Probability
Rule 3: Addition Rule for Disjoint EventsIf A and B are disjoint events, then P (A or B) = P(A) + P(B).
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7.2 Rules for Probability
Rule 3: Addition Rule for Disjoint Events
Extends to more than two events
P (E1 or E2 or … or Ek) =
P(E1) + P(E2) + … + P(Ek)
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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 Ac
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7.2 Rules for Probability
Rule 4: Complement Rule: P(A) = 1 – P(Ac)The probability of an event is one minus the
probability of its complement.
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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”
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7.2 Rules for Probability
Rule 5: General Addition RuleP (A or B) = P(A) + P(B) – P (A and B).
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7.2 Rules for Probability
An Example – Movie Schedule
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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?
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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
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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
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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)
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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?
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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.
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4M Example 7.1: MANAGING A PROCESS
Mechanics
Assuming days are independent, use the multiplication rule to find
P (OK for 5 days) = 0.955 = 0.774
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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)= 1- 0.774 = 0.226
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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.
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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.
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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.
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Best Practices (Continued)
Only add probabilities of disjoint events.
Be clear about independence.
Only multiply probabilities of independent events.
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