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Chapter 6

Probability

Introduction

• Processes are created to deliver product.

• Output of many processes is subject to the effects of

chance, or probability.

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6.1 Description

Definition: An experiment is a process that

results in an outcome that cannot be predicted

in advance with certainty.

Examples:

rolling a die

tossing a coin

Closing force for car doors (production sampling)

Invoice payment times (business process)

How an increase in temp. affects the yield

(experimental evaluation)

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Sample Space

Definition: The set of all possible outcomes of an experiment is called the sample space for the experiment.

Examples: • For rolling a fair die, the sample space is {1, 2, 3, 4, 5, 6}.

• For a coin toss, the sample space is {heads, tails}.

• Imagine a hole punch with a diameter of 10 mm punches holes in sheet metal. Because of variation in the angle of the punch and slight movements in the sheet metal, the diameters of the holes vary between 10.0 and 10.2 mm. For this experiment of punching holes, a reasonable sample space is the interval (10.0, 10.2).

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More Terminology

Definition: A subset of a sample space is called an

event.

• A given event is said to have occurred if the

outcome of the experiment is one of the

outcomes in the event. For example, if a die

comes up 2, the events {2, 4, 6} and {1, 2, 3}

have both occurred, along with every other

event that contains the outcome “2”.

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Random Experiment

Definition: When conducting an experiment in which results will not be essentially the same even though conditions may be nearly identical, the experiment is called a random experiment.

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Complements

The complement of an event A, denoted Ac, is

the set of outcomes that do not belong to A. In

words, Ac means “not A.” Thus the event “not

A” occurs whenever A does not occur.

Example: Consider rolling a fair sided die. Let A be

the event: “rolling a six” = {6}.

What is Ac = “not rolling a six”?

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6.2 Multiple Events:

Union

The union of two events A and B, denoted

A B, is the set of outcomes that belong either

to A, to B, or to both.

In words, A B means “A or B.” So the event

“A or B” occurs whenever either A or B (or both)

occurs.

Example: Flip coin twice, Let A = {at least one head} ={HH,

HT, TH } and B = {2nd toss results in tail} = {HT, TT}

What is A B?

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Intersections

The intersection of two events A and B, denoted

by A B, is the set of outcomes that belong to A

and to B. In words, A B means “A and B.”

Thus the event “A and B” occurs whenever both

A and B occur.

Example: Example: Flip coin twice, Let A = {at least one

head} ={HH, HT, TH } and B = {2nd toss results in tail} =

{HT, TT}

What is A B?

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Negation or Opposite

The negation or opposite of events A, denoted

by A’, is the set of outcomes that do not belong to

A. In words, A’ means “not A”

Example: Example: Flip coin twice, Let A = {at least one

head} ={HH, HT, TH } and B = {2nd toss results in tail} =

{HT, TT}

What is A’?

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Difference

The difference of two events A and B, denoted

by A - B, is the set of outcomes that belong to A

But not B. In words, A - B means “A but not B.”

Example: Example: Flip coin twice, Let A = {at least one

head} ={HH, HT, TH } and B = {2nd toss results in tail} =

{HT, TT}

What is A - B?

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6.3 Multiple-Event Relationships:

Mutually Exclusive Events

Definition: The events A and B are said to be mutually exclusive if they have no outcomes in common (cannot occur simultaneously)

More generally, a collection of events A1, A2, …, An

is said to be mutually exclusive if no two of them have

any outcomes in common.

Sometimes mutually exclusive events are referred to as disjoint events.

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Probabilities

Definition: Each event in the sample space has a probability of occurring. Intuitively, the probability is a quantitative measure of how likely the event is to occur.

Given any experiment and any event A:

The expression P(A) denotes the probability that the event A occurs.

P(A) is the proportion of times that the event A would occur in the long run, if the experiment were to be repeated over and over again.

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Axioms of Probability

1. Let S be a sample space. Then P(S) = 1.

2. For any event A, 0 ≤ 𝑃(𝐴) ≤ 1.

3. If A and B are mutually exclusive events, then

𝑃 𝐴 ∪ 𝐵 = 𝑃 𝐴 + 𝑃(𝐵). More generally, if

𝐴1, 𝐴2, … are mutually exclusive events, then

𝑃 𝐴1 ∪ 𝐴2 ∪ ⋯ = 𝑃 𝐴1 + 𝑃 𝐴2 + ⋯

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A Few Useful Things

• For any event A, P(AC) = 1 – P(A).

• Let denote the empty set. Then P( ) = 0.

• If S is a sample space containing N equally likely

outcomes, and if A is an event containing k

outcomes, then P(A) = k/N.

• Addition Rule (for when A and B are not mutually

exclusive): 𝑃 𝐴 ∪ 𝐵 = 𝑃 𝐴 + 𝑃 𝐵 − 𝑃(𝐴 ∩ 𝐵)

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6.3 Multiple-Event Relationships:

Conditional Probability and Independence

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Definition: A probability that is based on part of the

sample space is called a conditional probability.

Let A and B be events with P(B) 0. The conditional

probability of A given B is

𝑃 𝐴 𝐵 =𝑃(𝐴 ∩ 𝐵)

𝑃(𝐵)

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Independence

Definition: Two events A and B are independent if the probability of each event remains the same whether or not the other occurs.

• If P(A) 0 and P(B) 0, then A and B are independent if P(B|A) = P(B) or, equivalently, P(A|B) = P(A).

• If either P(A) = 0 or P(B) = 0, then A and B are independent.

• These concepts can be extended to more than two events.

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The Multiplication Rule

• If A and B are two events and P(B) 0, then 𝑃 𝐴 ∩ 𝐵 = 𝑃 𝐵 𝑃(𝐴|𝐵)

• If A and B are two events and P(A) 0, then 𝑃 𝐴 ∩ 𝐵 = 𝑃 𝐴 𝑃(𝐵|𝐴)

• If P(A) 0, and P(B) 0, then both of the above hold.

• If A and B are two independent events, then 𝑃 𝐴 ∩ 𝐵 = 𝑃 𝐴 𝑃(𝐵)

• If A1, A2,…, An are independent events, then 𝑃 𝐴1 ∩ 𝐴2 ∩ ⋯ ∩ 𝐴𝑛 = 𝑃 𝐴1 𝑃 𝐴2 … 𝑃(𝐴𝑛)

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6.4 Bayes’ Theorem

𝑃 𝐴 𝑍 =𝑃(𝐴 ∩ 𝑍)

𝑃 𝐴 ∩ 𝑍 + 𝑃(𝐵 ∩ 𝑍)

=𝑃 𝑍 𝐴 𝑃(𝐴)

𝑃 𝑍 𝐴 𝑃 𝐴 + 𝑃 𝑍 𝐵 𝑃(𝐵)

=𝑃 𝑍 𝐴 𝑃(𝐴)

𝑃(𝑍)

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6.4 Bayes’ Theorem: Example

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• A production process has a product nonconformance rate

of 0.5%.

• There is a functional test for nonconformance. A failed

test response is supposed to indicate that a defect exists;

however, the test is not perfect.

• For a product that has a defect, the test misses the defect

2% of the time (reports a false product conformance)

• For a product without a defect, the test incorrectly

indicates 3% of the time that they have the defect (reports

a false product nonconformance)

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6.4 Bayes’ Theorem: Example

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(.005) Yes

(.995) No

Defect

(.98) Fail

(.02) Pass

(.03) Fail

(.97) Pass

Test

6.4 Bayes’ Theorem: Example

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• Let event A be the occurrence of nonconforming product,

P(A)=.005

• Let event B be the occurrence of conforming product,

P(B)=.995

• Let event Z be the occurrence of failing the test,

P(Z)=P(A)P(Z|A)+P(B)P(Z|B)=(.005)(.98)+(.995)(.03)

=.03475

• From Bayes’ theorem,

P(A|Z)=P(A)P(Z|A) / P(Z) = (.005)(.98)/.03475 = .141

• For a test failure, there is only a probability of 0.141 that

the product is a true failure!