Unit 2 QC (1).pdf

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UNIT 2
Quality Control

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Quality control
• It is an IE technique by means of which the products of
uniform acceptable quality are manufactured
• Aiming at prevention of defects at every source by setting up
freedom of the system and corrective action procedure.
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What is the goal of QC?
• To detect SIGNIFICANT
errors rapidly
• Report out good results
in a timely manner
• Be cost effective and
simple to use
• If there is an error,
identify the source of
the error
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Tools of quality control
• There are seven basic tools of QC
– Flow chart
– Brainstorming
– Fishbone diagram
– Check sheet
– Pie chart
– Histogram
– Scatter diagram, etc
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Brainstorming
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Fishbone diagram
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Fishbone diagram
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Fishbone diagram
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Check sheet
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Scatter diagram
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Statistics as a basis of quality control
qStatistical techniques are used to identify instances of
excessive variation in a process so that corrective
actions can be taken to remedy the causes of variation.

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Statistics is the science that deals with
collection, tabulation, analysis,
interpretation and presentation of
quantitative data.

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S t a t i s t i c a l q u a l i t y
control refers to the use of
statistical methods in the
monitoring and maintaining of
the quality of products and
services.

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Types of statistics
Descriptive statistics
Inferential statistics
Pictorial
Presentation of data
Calculation of
descriptive measures
Mean Mode Median Range Standard
deviation
(It is used to describe only historical or observed data)
(It is used to predict the
probability of data in terms of
control chart so that
Defects can be prevent)
QC and QA emphasise

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Variation as a basis of quality control
• Difference exist from product to
product, person to person or
• machine to machine.
• This differences among products or
the process output over a time ,
are called variation.
• Variation could be small and even
in micro, nano - meters.

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Types of variation
Random variation Assignable variation
Inherent in a system and
hard to detect and reduce
Variation in performance

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• The variability that is contained within a process that
cannot be determined.
• These fluctuations and variations are caused by erratic and
irregular actions that are the result of random chance.
• These random variations cannot be eliminated or
determined.
Random variation

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Example: If you measure your weight five times on coin
operated weight-measuring machine in a railway platform,
you will get five different reading.

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Assignable variation
Also known as “special cause”, an assignable cause is an
identifiable, specific cause of variation in a given process or
measurement.
A cause of variation that is not random and does not occur
by chance is “assignable”.

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Example: At the end of the shift, you know that you are able to
produce around 250 bulbs everyday,
but today you find that you could produce only 100 bulbs.
This could be very specific, such as

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Sources of Variations
Five sources of variation
• Process- tool wear, way of ops, Jigs, fixtures, vibrations
• Materials – Mechanical properties, defects, rusting
• Environment- Temperature, light, humidity, radiation
• Operator – operator skills
• Inspection- Incorrect application of quality standards
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Types of Variations

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There are four phases of statistical problem solving:
1) Data collection
2) Analysis
3) Experiment
4) Corrective action

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Statistical tools
Four basic statistical tools are: (Central tendency)
1) Mean
2) Mode
3) Median
4) Standard deviation
Mean
Mean is the average and is computed as the sum of all the
observed outcomes from the samples divided by the total
number of events. If X is the symbol for the mean
X =1/n∑
n
i=1
xi

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Advantages of Mean
• It is easy to understand & simple calculate.
• It is based on all the values.
• It is rigidly defined .
• It is easy to understand the arithmetic average even
if some of the details of the data are lacking.
• It is not based on the position in the series.

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• It cannot be located graphically
• It is sensitive to extreme values/outliers,
especially when the sample size is small
• It gives misleading conclusions.
• It has upward bias.
Disadvantages of Mean

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Example:
Find the mean of: 6, 8, 11, 5, 2, 9, 7, 8

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Example Find the mean of: 6, 8, 11, 5, 2, 9, 7, 8
6 + 8 + 11 + 5 + 2 + 9 + 7 + 8
8
= 56/8 = 7
=
X =1/n∑xi

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Question: 21 bricks have a mean mass of 24.2 kg, and 29 similar
bricks have a mass of 23.6 kg. Determine the mean mass of the 50
bricks.

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Mean value = (21 x 24.2 + 29 x 23.6 )
21 + 29
= 1192.6/50 = 23.85 kg

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Mode is the value that in a set of
data.
Example: 27,33,21,32,22,43,52,47,50,40,
36,32,33,22,32,36,
27,24,26,25,32
X= 21 22 24 25 26 27 32 33 36 40 43 47 50 52
f= 1 2 1 1 1 2 4 2 2 1 1 1 1 1
Ans: Mode 32
Mode

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•It is easy to understand and simple to calculate.
•It is not affected by extreme large or small values.
•It can be located only by inspection in ungrouped
data and discrete frequency distribution.
Advantages of Mode

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§ It is not based upon all the observations.
§ It is not capable of further mathematical
treatment.
§ As compared with mean, mode is affected to a
greater extent, by fluctuations of sampling.
Disadvantages of Mode

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Median
Median is middle score. If we have an even no. of events, we
take the average of the two middles. The median is better for
describing the typical value and it is often used for income and
home price.
Median is a central value of the distribution, or the value
which divides the distribution in equal parts, each part
containing equal number of items. Thus it is the central value of
the variable, when the values are arranged in order of
magnitude.
Connor has defined as “ The median is that value of the variable
which divides the group into two equal parts, one part
comprising of all values greater, and the other, all values less
than median”

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• A standardized measure of distance from the
mean.
• Very useful and something you do read about
when making predictions or other statements
about the data.

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1)
-
(n
2
)
( X
X 

S =
=square root
=sum (sigma)
X=score for each point in data
_
X=mean of scores for the variable
n=sample size (number of
observations or cases
n
2
)
( X
X 

S =
Population standard deviation or uncorrected sample standard deviation
standard deviation for sample variance or
corrected sample standard deviation

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Example: Determine the standard deviation from the mean of
the set of numbers: {35, 22, 25, 23, 28, 33, 30} correct to 3
significant figures.
Mean= 35 + 22 + 25 + 23 + 28 + 33 + 30
= 196/7 = 28
7
(35-28)2 + (22-28)2 + (25-28)2 + (23-28)2 + (28-28)2 + (33-28)2 + (30-28) 2
7
148/7 = 4.6

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Dispersion
• To extent which the data is scattered about the zone
of central tendency.
• Main measure of dispersion- ?

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Process
• A process is a unique combination of tools, materials, methods,
and people engaged in producing a measurable output
• E.g. manufacturing line for machine parts.
• All processes have inherent statistical variability which can be
evaluated by statistical methods.

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• Being in control of a manufacturing process using statistical
process control (SPC) is not enough.
• An “in-control” process can produce bad or out-of-spec product.
• Manufacturing processes must meet or be able to achieve product
specifications.
• Further, product specifications must be based on customers
requirements.

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Process capability
• Process capability is the repeatability and consistency of a
manufacturing process relative to the customer requirements in
terms of specification limits of a product parameter.
• This measure is used to objectively measure the degree to which
your process is or is not meeting the requirements.

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Process Capability
• Process capability is defined as a statistical measure of the
inherent process variability of a given characteristic.
• that predict how many parts will be produced out of
specification (OOS).

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Process Capability
Process capability refers to the uniformity of the process.
Obviously, the variability of critical-to-quality characteristics in
the process is a measure of the uniformity of output.
A critical aspect of statistical quality control is evaluating the
ability of a production process to meet or exceed preset
specifications. This is called process capability.
How the process is performing?

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Process Capability
• Process Capability (Cp) is a statistical measurement of a process’s
ability to produce parts within specified limits on a consistent
basis.
• To determine how our process is operating, we can calculate
process capability indices: Cp (Process Capability), Cpk
(Process Capability Index),
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• Capability indices have been developed to
graphically portray that measure.
• Capability indices let you place the distribution
of your process in relation to the product
specification limits.
• Capability indices should be used to determine
whether the process, given its natural variation,
is capable of meeting established specifications.

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• Process capability can be expressed with an
index.
• Assuming that the mean of the process is
centered on the target value, the process
capability index Cp can be used.
• Cp is a simple process capability index that
relates the allowable spread of the spec limits to
the measure of the actual, or natural, variation
of the process,

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• sample of a group of items periodically from a production run and
measure the desired specification parameter, we will get subgroup
sample distributions that can be compared to that parameter’s
• Two examples of this are represented below.

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• A minimum of three possible outcomes can arise when the natural
process variability is compared with the design specifications or
customer expectations:
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• Cp<1 means the process variation exceeds specification, and a
significant number of defects are being made.
• Cp=1 means that the process is just meeting specifications.
• Cp>1 means that the process variation is less than the
specification, however, defects might be made if the process is
not centered on the target value.

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A Cp value of 1 means that 99.74 percent of the products
produced will fall within the specification limits.
This also means that 0.26 percent (100% - 99.74%) of the products
will not be acceptable

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Cp Vs Cpk
• The Cp index is calculated using specification
limits and the standard deviation only.
• This index indicates, in general, whether the
process is capable of producing products
to specifications.
• No information on the ability of the process to
adhere to the target value is included in this
index.

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• While Cp relates the spread of the process
relative to the specification width, it does not
address how well the process average, X, is
centered to the target value. Cp is often referred
to as process “potential”.

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Cpk index
• This Cpk index is calculated using specification limits, the
standard deviation, and the mean.
• The index indicates whether the process is capable of producing
within specification and is also an indicator of the ability of the
process to adhere to the target specification.

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Cpk measures not only the process variation with respect to
allowable specifications, it also considers the location of the
process average.

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Cpk estimation
• Cpk is taken as the smaller of either Cpl or Cpu where
• Cpl = (X -LSL) / 3 sigma
• Cpu = (USL – X) / 3 sigma

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• Cpk is more widely used than Cp, since it takes into account
the mean and the standard deviation in its calculation.
• Please note that the difference between Cp and Cpk is an
indicator of how far the average of the process is from the
target specification.
• When the average of the process approaches the target
value, the gap between Cpk and Cp closes.
• When the average of the specification is equal to the target
value, then Cpk is equal to Cp. Cpk can never exceed Cp.

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Process is
capable
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• If Cpk is less than 1 it indicates that some
part of the distribution is outside of the
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For example, the specifications for the width of a machine
part may be specified as 15 inches ±0.3. This means that the
width of the part should be 15 inches, though it is acceptable
if it falls within the limits of 14.7 inches and 15.3 inches.

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Bagging
machine
Std. Deviation
A 0.2
B 0.3
B 0.05
Three bagging machines at the Crunchy Potato Chip Company are
being evaluated for their capability. The following data are recorded:
If specifications are set between 12.35 and 12.65 ounces, determine
which of the machines are capable of producing with in
specifications.

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Three bagging machines at the Crunchy Potato Chip Company are
being evaluated for their capability. The following data are
recorded:

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Question :
Food served at a restaurant should be between 39°C and 49°C
when it is delivered to the customer. The process used to keep the
food at the correct temperature has a process standard deviation of
2°C and the mean value for these temperatures is 40. What is the
process capability index of the process?

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Solution of part 1:
(USL – Mean)/ 3σ
Substitute the values:
= (49-40)/3 ×2
= 9/6
Solution of part 1= 1.5
Solution of part 2:
(Mean – LSL)/ 3σ
= (40-39)/3 ×2
= 1/6
Solution of part 2= 0.166
Now, substitute the solutions in the formula, we have:
Cpk = min (part 1, part 2)
Cpk = min (1.5, 0.166)
Since the mininum value is 0.166,
The process capability index, Cpk is 0.166.

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Normal Distribution
• Normal Distribution is often called a bell curve and is broadly
utilized in statistics, business settings, and government
entities.
• Normal Distribution contains the following characteristics:
– It occurs naturally in numerous situations.
– Much fewer outliers on the low and high ends of data range.
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• Normal Distribution
– Applied to single variable continuous data
e.g. heights of plants, weights of lambs,
lengths of time
– U s e d t o c a l c u l a t e t h e p r o b a b i l i t y o f
occurrences less than, more than, between
given values
e.g. “the probability that the plants will be less than
70mm”,
“the probability that the lambs will be heavier than 70kg”,
“the probability that the time taken will be between 10 and
12 minutes”
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A normal curve is the name of
the graph of the standard
n o r m a l p r o b a b i l i t y
distribution
A s t a n d a r d n o r m a l c u r v e
provides both a visual and a
numerical representation of
h o w a g i v e n v a r i a b l e i s
d i s t r i b u t e d a c r o s s a
population when the real-
life situation represented
by the function is known to
h a v e a s y m m e t r i c a l

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• Business Applications:
– Can be utilized to model risks and following the distribution of likely
outcomes for certain events, like the amount of next month’s revenue
from a specific service.
– Process variations in operations management are sometimes normally
distributed, as is employee performance in Human Resource
Management.
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•It is continuous random distribution variables
•It is characterized, i) mean, ii) standard deviation.
•In industrial environment, the normal distribution is used to
predict the probability of producing defective products
•Relative frequency histograms that are symmetrical and
bell-shaped are said to have a normal curve

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95%
99.74%
68%
mean,
standard deviation

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95%
99.74%
68%
mean,
standard deviation
34% 34%
13.5% 13.5%
2.37% 2.37%
The 68-95-99.7% Rule

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All normal density curves satisfy the following property which
is often referred to as the Empirical Rule.
68% of the observations fall within 1 standard deviation of
the mean, that is, between and
95% of the observations fall within 2 standard deviations of
the mean, that is, between and
99.7% of the observations fall within 3 standard deviations of
the mean, that is, between
Thus, for a normal distribution, almost all values lie within 3
standard deviations of the mean.
and

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Areas under portions of a normal distribution can be
computed by using by integrating the curve between
the x-coordinates of interest.
With the normal curve, you instead look up either one
or two numbers on a table called z-values and, if
needed, perform a subtraction step.

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To solve any problems of normal distribution
we use standard normal variable Z
X
A z score is the same as
a standard score; the
number of standard
deviations above the mean.

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Let us apply the Empirical Rule
Example
The distribution of heights of American women aged 18 to 24 is
approximately normally distributed with mean 65.5 inches and
standard deviation is 2.5 inches. From the above rule, it follows
that
68%
of these American women have heights between 65.5 - 2.5 and 65.5 +
2.5 inches, or between 63 and 68 inches,
95%
of these American women have heights between 65.5 - 2(2.5) and 65.5
+ 2(2.5) inches, or between 60.5 and 70.5 inches.

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Again, you can try this out with the example below.
Therefore, the tallest 2.5% of these women are taller than
70.5 inches.
(The extreme 5% fall more than two standard deviations, or
5 inches from the mean.
And since all normal distributions are symmetric about their
mean, half of these women are the tall side.)
Almost all young American women are between 58 and 73
inches in height if you use the 99.7% calculations.

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Suppose that in a test with a mean of 80 and a standard
deviation of 10, you want to know what percentage of the
students had scores between 65 and 85.
You would start by finding the upper and lower z-scores.
This is done by subtracting the mean from your upper bound
and dividing by the standard deviation: (85 - 80)/10 = 0.50.
You then find the lower bound in the same way: (65 - 80)/10 -
1.50.

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Now, you can assign area values to these z-scores by referring to
the table.
These values are 0.68916 for z = 0.5 and 0.06681 for z = -1.5. Each
of these areas represents the area under the curve from the left
"tail" to the x-value in question, so for the area between the two
points x = 65 and x = 85, you subtract the lesser value from the
greater to get 0.63135.
Thus 63.1 percent of the scores could be expected to fall within the
range of 65 to 85 given a standard deviation of 10 in a normal
distribution.

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A study was done to determine the stress levels that students have while taking
exams. The stress level was found to be normally distributed with a mean stress
level of 8.2 and a standard deviation of 1.34. What is the probability that at your
next exam, you will have a stress level between 9 and 10?

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Find the probability P (18≤ X≤25)
To solve any problems of normal distribution
we use standard normal variable Z
X
If X= 18
Z= -1.4
If X= 25; Z= 1.4

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Find the probability P (18≤ X≤25) or P (-1.4≤ X≤1.4)
0.4596 + 0.4596 = 0.9192
91.9% PROBABILITY

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Z= -1.4 Z= 1.4
0.4596
0.4596

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Binomial Distribution
• Binomial Distribution is considered the likelihood of a pass or fail
outcome in a survey or experiment that is replicated numerous
times.
• There are only two potential outcomes for this type of
distribution, like a True or False, or Heads or Tails.
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122
P (r)= n
C
r
pr q n-r
P (r)=
n
n-r r
p
r qn-r
r= 0,1,2,3,4…….n

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– Banks and other financial institutions use Binomial Distribution to
determine the likelihood of borrowers defaulting, and apply the
number towards pricing insurance, and figuring out how much money
to keep in reserve, or how much to loan
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Example
• A coin is tossed 10 times. What is the probability of
getting exactly 6 heads?

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Solution
’Formula - nCx * Px * (1 – P)n – x
The number of trials (n) is 10
The odds of success (“tossing a heads”) is 0.5 (So 1-p = 0.5)
x = 6
P(x=6) = 10C6 * 0.5^6 * 0.5^4 = 210 * 0.015625 * 0.0625 = 0.205078125

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Q: A coin is tossed five times. Find the
probability of
1) Getting exactly 3 heads
2) Getting at most 2 heads

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Poisson Distribution
• The probability of events occurring at a specific time is
Poisson Distribution.
• In other words, when you are aware of how often the
event happened, Poisson Distribution can be used to
predict how often that event will occur.
• It provides the likelihood of a given number of events
occurring in a set period.
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Poisson Distribution Characteristics:
– An event can happen any amount of times
throughout a period.
– Events occurring don’t affect the probability of
another event occurring within the same period.
– Occurrence rate is constant and doesn’t change
based on time.
– The likelihood of an occurring event corresponds to
the time length.
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– Predicting customer sales on particular days/times of the year.
– Supply and demand estimations to help with stocking products.
– Service industries can prepare for an influx of customers, hire
temporary help, order additional supplies, and make alternative plans
to reroute customers if needed
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1. It is discrete probability variable not continuous
2. If your question has an average probability of an
event happening per unit (i.e. per unit of time, cycle,
event) and you want to find probability of a certain
number of events happening in a period of time (or
number of events), then use the Poisson Distribution
np
np
P= probability of success
q = probability of failure
q = (1-p)
M or

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Q: Suppose now that a machine which is known to produce 1%
defective components is used for a production run of 40
components. We wish to calculate the probability that two
defective items are produced, that is P(X = 2). Used both Binomial
distribution and its Poisson approximation for comparison

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