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Quality
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Chapter 8- Fundamentals
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Outline
 Definition of Probability
 Theorems of Probability
 Counting of Events
 Discrete Probability Distributions
 Continuous Probability Distribution
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© 2013, 2008 by Pearson Higher Education, Inc
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Learning Objectives
When you have completed this chapter you
should be able to:
 Define probability using the frequency
definition.
 Know the seven basic theorems of probability.
 Identify the various discrete and continuous
probability distributions.
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Dale H. Besterfield
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© 2013, 2008 by Pearson Higher Education, Inc
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Learning Objectives cont’d.
When you have completed this chapter you
should be able to:
 Calculate the probability of non-conforming
units occuring using the Hypergeometric,
Binomial and Poisson distributions.
 Know when to use the Hypergeometric,
Binomial and Poisson distributions.
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© 2013, 2008 by Pearson Higher Education, Inc
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Definition of Probability
 Likelihood, chance, tendency, and trend.
 The chance that something will happen.
 If a Nickel is tossed, the probability of a head
is 1/2 and the probability of the tail is 1/2.
When a die is tossed, the probability of one
spot is 1/6, the probability of two spots is 1/6,.....
 Drawing a card from a deck of cards. The
probability of a spade is 13/52.
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Definition of Probability
 The area of each distribution is equal to 1.
 The area under the normal distribution
curve, which is a probability distribution, is
equal to 1.
 The total probability of any situation will
be equal to 1.
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© 2013, 2008 by Pearson Higher Education, Inc
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Definition of Probability
 The probability is expressed as a decimal
(the probability of a head is 0.5).
 An event is a collection of outcomes
(six-sided die has six possible outcomes).
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Definition of Probability
When the number of outcomes is known or when the
number of outcomes is found by experimentation:
P(A) = NA/N
where:
P(A) = probability of event A ocurring to 3 decimal places
NA = number of successful outcomes of event A
N = total number of possible outcomes
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© 2013, 2008 by Pearson Higher Education, Inc
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Definition of Probability
 The probability calculated using known
outcomes is the true probability, and the
one calculated using experimental
outcomes is different due to the chance
factor.
 For an infinite situation (N = ∞), the
definition would always lead to a
probability of zero.
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© 2013, 2008 by Pearson Higher Education, Inc
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Definition of Probability
In the infinite situation the probability of
an event occurring is proportional to the
population distribution.
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Theorems of Probability
Theorem 1
Probability is expressed as a number between
1 and 0, where a value of 1 is a certainty that
an event will occur and a value of 0 is a
certainty that an event will not occur.
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Theorems of Probability
Theorem 2
If P(A) is the probability that event A will
occur, then the probability that A will not
occur is:
P(notA) = 1- P(A)
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Theorems of Probability
One Event
Out or Two
or More Events
Mutually
Exclusive
Not Mutually
Exclusive
Theorem 3
Theorem 4
Two or More Event
Out or Two
or More Events
Independent
Dependent
Theorem 6
Theorem 7
Figure 7-2 When to use Theorems 3,4,6 and 7
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© 2013, 2008 by Pearson Higher Education, Inc
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Theorems of Probability
Theorem 3
If A and B are two mutually exclusive events
(the occurrence of one event makes the
other event impossible), then the probability
that either event A or event B will occur is
the sum of their respective probabilities:
P(A or B) = P(A) +P(B)
This is the “additive law of probability”.
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© 2013, 2008 by Pearson Higher Education, Inc
Upper Saddle River, New Jersey 07458 • All Rights Reserved
Theorems of Probability
Theorem 4
If event A and event B are not mutually
exclusive, then the probability of either event A
or event B or both is given by:
P(A or B or both) = P(A) +P(B) – P(both)
Events that are not mutually exclusive have some
outcomes in common
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© 2013, 2008 by Pearson Higher Education, Inc
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Theorems of Probability
Theorem 3
If A and B are two mutually exclusive events
(the occurrence of one event makes the
other event impossible), then the probability
that either event A or event B will occur is
the sum of their respective probabilities:
P(A or B) = P(A) +P(B)
This is the “additive law of probability”.
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© 2013, 2008 by Pearson Higher Education, Inc
Upper Saddle River, New Jersey 07458 • All Rights Reserved
Theorems of Probability
Theorem 4
If event A and event B are not mutually
exclusive, then the probability of either event A
or event B or both is given by:
P(A or B or both) = P(A) +P(B) – P(both)
Events that are not mutually exclusive have some
outcomes in common
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© 2013, 2008 by Pearson Higher Education, Inc
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Theorems of Probability
Theorem 5
The sum of the probabilities of the events of
a situation is equal to 1.000
P(A) + P(B) + …..+ P(N) = 1.000
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Theorems of Probability
Theorem 6
If A and B are independent events (one
where its occurrence has no influence on the
probability of the other event or events),
then the probability of both A and B
occurring is the product of their respective
probabilities:
P(A and B) = P(A) X P(B)
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© 2013, 2008 by Pearson Higher Education, Inc
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Theorems of Probability
Theorem 7
If A and B are dependent events, the
probability of both A and B occurring is the
probability of A and the probability that if A
occurred, then B will occur also:
P(A and B) = P(A) X P(B\A)
P(B\A) is defined as the probability of event B,
provided that event A has ocurred.
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Counting of Events
1. Simple multiplication
If an event A can happen in any of a ways
or outcomes and, after it has occurred,
another event B can happened in b ways
or outcomes, the number of ways that
both events can happen is ab.
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Counting of Events
2. Permutations
A permutation is an ordered arrangement of a
set of objects.
n!
P 
(n  r )!
n
r
Example: The word “cup”…… cup, cpu, upc,
ucp, puc, and pcu.
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© 2013, 2008 by Pearson Higher Education, Inc
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Counting of Events
3. Combinations
If the way the objects are ordered is unimportant, then
we have a combination:
n!
C 
r !(n  r )!
n
r
Example: The word “cup” has 6 permutations
when the 3 objects are taken 3 at a time. There
is only one combination, since the same three
letters are in different order.
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© 2013, 2008 by Pearson Higher Education, Inc
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Discrete Probability Distributions
Binomial Probability Distribution
1. It is applicable to discrete probability problems
that have an infinite number of items or that
have a steady stream of items coming from a
work center.
2. It is applied to problems that have attributes.
n(n  1) n2 2
n
( p  q)  p  np q 
p q  .........  q
2
n
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n
n 1
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Discrete Probability Distributions
Figure 8-6 Distribution of the number of tails for an infinite number
of tosses of 11 coins
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Discrete Probability Distributions
Binomial Probability Distribution cont’d
3. See Figure 8-6. Since
p=q, the distribution is
symmetrical regardless of the value of n,
however, when p is not equal to q, the
distribution is asymmetrical.
4. In quality work
p is the portion or fraction
nonconforming and is usually less than 0.15
n!
d n d
P(d ) 
p0 q0
d !(n  d )!
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Discrete Probability Distributions
Binomial Probability Distribution cont’d.
5. As the sample size gets larger, the shape of
the curve will become symmetrical even
though p is not equal to q.
6. It requires that there be two and only two
possible outcomes (C, NC) and that the
probability of each outcome does not change.
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Discrete Probability Distributions
Binomial Probability Distribution cont’d.
7. The use of the binomial requires that the
trials be independent.
8. It can be approximated by the Poisson
when Po≤0.10 and nPo≤5.
9. The normal curve is an excellent
approximation when Po is close to 0.5
and n>̳ 10 or
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Discrete Probability Distributions
Poisson Probability Distribution
1. It is applicable to many situations that
involve observations per unit of time.
2. It is also applicable to situations involving
observations per unit amount.
3. In each of the preceding situations, there
are many equal opportunities for the
occurrence of an event.
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© 2013, 2008 by Pearson Higher Education, Inc
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Discrete Probability Distributions
Poisson Probability Distribution cont’d.
4. The Poisson is applicable when
n is quite
large and Po is small.
5. When Poisson is used as an approximation
to the binomial, the symbol c has the same
meaning as d has in the binomial and
hypergeometric formulas.
c
(np0 )  np0
P( c ) 
e
c!
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Discrete Probability Distributions
Poisson Probability Distribution cont’d.
6. When
nPo gets larger, the distribution
approaches symmetry.
7. Table C in the Appendix.
8. The Poisson probability is the basis for
attribute control charts and for acceptance
sampling.
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© 2013, 2008 by Pearson Higher Education, Inc
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Discrete Probability Distributions
Poisson Probability Distribution cont’d.
9. It is used in other industrial situations,
such as accident frequencies, computer
simulation, operations research, and
work sampling.
10. Uniform (generate a random number
table), Geometric, and Negative binomial
(reliability studies for discrete data).
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Discrete Probability Distributions
Poisson Probability Distribution cont’d.
11. The Poisson can be easily calculated
using Table C.
12. Similarity among the hypergeometric,
binomial, and Poisson distributions can
exist.
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© 2013, 2008 by Pearson Higher Education, Inc
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Continuous Probability Distributions
Normal Probability Distribution
1. When we have measurable data.
2. The normal curve is a continuous
probability distribution.
3. Under certain condition the normal
probability distribution will approximate
the binomial probability distribution.
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© 2013, 2008 by Pearson Higher Education, Inc
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Computer Program
 Microsoft EXCEL/Minitab will solve for
permutations, combinations,
hypergeometric, binomial, and Poisson
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Homework
 33, 35, 37, 41, 43 by hand and minitab
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