MER301: Engineering
Reliability
LECTURE 5:
Chapter 3:
Probability Plotting,The Rules of Counting,
Binomial Distribution, Poisson Distribution
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 5
1
Summary of Topics
 Probability Plotting
 Rules of Counting
 Binomial Distribution
 Poisson Distribution
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 5
2
Probability Plotting
 Graphical method of testing a data set
to see if it conforms to some specific
distribution
 For an ordered data set x(j) the
Cumulative Distribution Function is
plotted x(j)vs (j-0.5)/n where j=1,…,n
 Probability plotting often used in failure
analysis to predict future failures….
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 5
3
Normal Probability Plot
Descriptive Statistics
Variable: C1
Anderson-Darling Normality Test
A-Squared:
P-Value:
180
190
200
210
220
95% Confidence Interval for Mu
0.257
0.636
Mean
StDev
Variance
Skewness
Kurtosis
N
195.700
14.032
196.9
0.511343
-6.2E-01
10
Minimum
1st Quartile
Median
3rd Quartile
Maximum
176.000
184.500
191.500
207.250
220.000
95% Confidence Interval for Mu
185.662
190
200
210
205.738
95% Confidence Interval for Sigma
9.652
25.617
95% Confidence Interval for Median
95% Confidence Interval for Median
L Berkley Davis
Copyright 2009
184.315
208.081
MER301: Engineering Reliability
Lecture 5
4
Normal
Probability Plot
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 5
5
Normal
Probability Plot
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 5
6
Rules of Counting
 Fundamental Rule of Counting
 Permutation Rules
 Combinations
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 5
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Rules of Counting
 Fundamental Counting Rule

The total number of ways a sequence of k events can
occur with ni denoting the number of ways the ith event
(i= 1 to k) can occur is
n  n1  n2    nk
 For buying a computer, there are choices of three
hard drives n1, two levels of RAM n2, two video cards
n3, and three monitors n4 so the total number of
options is
n  n1  n2  n3  n4  3  2  2  3  36
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 5
8
Rules of Counting-con’t

Permutation Rules

A permutation is an arrangement of n distinct objects in a specific
order. The number of arrangements of n distinct objects in a specific
order equals
n!

If the n objects are taken k at a time, then the number of
arrangements in a specific order is
n!
(n  k )!

A case where the order of selection is not important is called a
Combination. The number of ways of selecting k objects from n
objects without regard to order is
n!
k!(n  k )!
 This is the permutation divided by k!, the number of ways the k
objects can be arranged…
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 5
9
Rules of Counting-con’t
 Combinations

A case where the order of selection is not important is called a
Combination. The number of ways of selecting k objects from n
objects without regard to order is
n!
k!(n  k )!
 This is the permutation divided by k!, the number of ways the k
objects can be arranged…

For the case where the objects are taken n at a time but some are
identical, the number of possible arrangements in a specific order is
n!
r1!r2 !  r j !
r1  r2    r j  n
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 5
10
Rules of Counting-con’t


A Combination for n objects taken k at a time is equal to
the Permutation of the n objects (n!/(n-k)! )divided by the
number of ways the k objects can be arranged (k!)
For five objects taken three at a time
n!
5!
1 5!
1


  60  10
k!(n  k )! 3!(5  3)! 3! (5  3)! 6
ARRANGEMENTS OF A,B,C,D,E TAKEN THREE AT A TIME
ABC
ACB
BCA
BAC
CAB
CBA
L Berkley Davis
Copyright 2009
ABD
ADB
BDA
BAD
DAB
DBA
ABE
AEB
BEA
BAE
EAB
EBA
ACD
ADC
CDA
CAD
DAC
DCA
ACE
AEC
CEA
CAE
EAC
ECA
ADE
AED
DEA
DAE
EAD
EDA
MER301: Engineering Reliability
Lecture 5
BCD
BDC
CDB
CBD
DBC
DCB
BCE
BEC
CEB
CBE
EBC
ECB
BDE
BED
BED
DBE
EBD
EDB
CDE
CED
DEC
DCE
ECD
EDC
11
Binomial Distribution
 The Binomial distribution describes
the results of n independent identical
success-failure trials
 Constant chance of a success or
failure outcome(called probability p)
 Knowing the outcome of any one
repetition does not change chance of
any other repetition
 Must be able to count the number of
successes and failures
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 5
12
Binomial Distribution
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 5
13
Binomial Distribution Combinations and
Mean/Variance
 From counting combinations, the number
of combinations of n distinct objects
selected x at a time is given by
 n
n!
  
 x  x!(n  x)!
n!  n  ( n  1)  ( n  2)    3  2  1
 Mean and Variance of the Binomial
Distribution
E( X )  n  p
V  n  p  (1  p )
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 5
14
Binomial Distributions for n=10 and
p=0.1,0.5,0.9
X
0
1
2
3
4
5
6
7
8
9
10
n
10
10
10
10
10
10
10
10
10
10
10
n*p
1
p
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
p(X=x)
0.34867844
0.38742049
0.19371024
0.05739563
0.01116026
0.00148803
0.00013778
8.748E-06
3.645E-07
9E-09
1E-10
Variance=0.9
L Berkley Davis
Copyright 2009
x*p(X=x)
0
0.387420489
0.387420489
0.172186884
0.044641044
0.007440174
0.000826686
6.1236E-05
2.916E-06
8.1E-08
0.000000001
Mean
1
X
0
1
2
3
4
5
6
7
8
9
10
n
10
10
10
10
10
10
10
10
10
10
10
n*p
5
p
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
p(X=x)
0.0009766
0.0097656
0.0439453
0.1171875
0.2050781
0.2460938
0.2050781
0.1171875
0.0439453
0.0097656
0.0009766
Variance=2.5
MER301: Engineering Reliability
Lecture 5
x*p(X=x)
0
0.00976563
0.08789063
0.3515625
0.8203125
1.23046875
1.23046875
0.8203125
0.3515625
0.08789063
0.00976563
Mean
5
X
0
1
2
3
4
5
6
7
8
9
10
n
10
10
10
10
10
10
10
10
10
10
10
n*p
9
p
0.9
0.9
0.9
0.9
0.9
0.9
0.9
0.9
0.9
0.9
0.9
p(X=x)
1E-10
9E-09
3.64E-07
8.75E-06
0.000138
0.001488
0.01116
0.057396
0.19371
0.38742
0.348678
Variance=0.9
15
x*p(X=x)
0
9E-09
7.29E-07
2.62E-05
0.000551
0.00744
0.066962
0.401769
1.549682
3.486784
3.486784
Mean
9
Binomial Distributions for n=10 and
p=0.1,0.5,0.9
Excel formula for Binomial
=binomdist(x,n,p,f(X=x)=false)
=binomdist(x,n,p,cumulative=true)
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 5
16
Example 5.1
 An electronics manufacturer claims that
10% or less of power supply units fail
during warranty. To test this claim, an
independent laboratory purchases 20 units
and conducts accelerated life testing to
measure failure during the warranty period.
 Let p denote the probability that a power supply
unit fails during the testing period.
 The laboratory data resulting from testing will be
compared to the claim that p0.10.
 Let X denote the number among the 20 sampled
that fail. What is the expected value and
variance of X if the manufacturer’s claim is true?
 Find the probability that less than 5 of the 20
power supplies will fail if the manufacturer’s
claim is true
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 5
17
Normal Approximation to the
Binomial Distribution
 For values of np>5 and n(1-p)>5,
probabilities from the binomial
distribution can be approximated
as follows:
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 5
18
Poisson Distribution
 The Poisson Distribution gives the
predicted probability of a specific
number of events occurring in an
interval of known size, when the
mean number of events in such
intervals is known
 Number of goals scored in a game
 Errors in transmission of data
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 5
19
Poisson Distribution
 Assumptions for Poisson Distribution
 Random discrete events that occur in an
interval that can be divided into subintervals
 Probability of a single occurrence of the
event is directly proportional to the size of a
subinterval
 If the sampling subinterval is sufficiently
small, the probability of two or more
occurrences of the event is negligible
 Occurrences of the event in nonoverlapping
subintervals are independent
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 5
20
Poisson Distribution
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 5
21
Poisson Examples:lambda =1,4.5,9
lambda
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
x
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
lambda=1
f(X=x)
0.3678794
0.3678794
0.1839397
0.0613132
0.0153283
0.0030657
0.0005109
7.299E-05
9.124E-06
1.014E-06
1.014E-07
9.216E-09
7.68E-10
5.908E-11
4.22E-12
2.813E-13
Cumulative
0.36787944
0.73575888
0.9196986 lambda
4.5
0.98101184
4.5
0.99634015
4.5
0.99940582
4.5
0.99991676
4.5
0.99998975
4.5
0.99999887
4.5
0.99999989
4.5
0.99999999
4.5
1
4.5
1
4.5
1
4.5
1
4.5
1
4.5
4.5
4.5
x
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Excel Formula for Poisson
=poisson(x,lambda,f(X=x)=false)
=poisson(x,lambda,cumulative=true)
L Berkley Davis
Copyright 2009
lambda=4.5
f(X=x) Cumulative
0.011109 0.011109
lambda
0.04999 0.0610995
9
0.112479 0.1735781
9
0.168718 0.342296
9
0.189808 0.5321036
9
0.170827 0.7029304
9
0.12812 0.8310506
9
0.082363 0.9134135
9
0.046329 0.9597427
9
0.023165 0.9829073
9
0.010424 0.9933313
9
0.004264 0.9975957
9
0.001599 0.9991949
9
0.000554 0.9997484
9
0.000178 0.9999263
9
5.34E-05 0.9999797
9
9
Poisson Distribution
9
9
9
9
9
MER301: Engineering Reliability
Lecture 5
x
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
lambda=9.0
f(X=x)
0.00012341
0.001110688
0.004998097
0.014994291
0.033737155
0.060726879
0.091090319
0.117116124
0.13175564
0.13175564
0.118580076
0.097020062
0.072765047
0.050375802
0.032384444
0.019430666
0.01092975
0.005786338
0.002893169
0.001370449
0.000616702
Cumulative
0.00012341
0.001234098
0.006232195
0.021226486
0.054963641
0.115690521
0.20678084
0.323896964
0.455652604
0.587408244
0.70598832
0.803008383
0.875773429
0.926149231
0.958533675
0.977964341
0.988894091
0.994680429
0.997573598
0.998944046
0.999560748
22
Poisson Examples:lambda =1,4.5,9
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 5
23
Poisson Process
 Observing discrete events in a
continuous “interval” of time,
length or space
 The number of white blood cells in a
drop of blood
 The number of times excessive
pollutant levels are emitted from a
gas turbine power plant during a
three-month period
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 5
24
Example 5.2

Yeast is added when mash is prepared for fermentation in
the beer making process. The yeast is cultured in vats and
the exact amount of yeast added to the mash is of critical
importance. The yeast cell count in culture fluid averages
6000 yeast cells per cubic millimeter of fluid in the culture
vats. It is however necessary to know the concentration
in a specific vat to know how much fluid to add to the
mash. The distribution of yeast cells is known to follow a
Poisson Distribution.
 To establish the yeast cell concentration, a 0.001
cubic millimeter drop is taken and the number of yeast
cells X is counted
 What is the probability of getting two or less yeast
cells in a single sample?
 What is the probability of getting more than two yeast
cells in a single sample?
 How many yeast cells are expected if the sample is
from a typical culture vat?
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 5
25
Normal Approximation of
the Poisson Distribution
 Poisson developed for case of n
approaching infinity
 For mean >5 then:
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 5
26
Summary of Topics
 Probability Plotting
 Rules of Counting
 Binomial Distribution
 Poisson Distribution
L Berkley Davis
Copyright 2009
MER301: Engineering Reliability
Lecture 5
27