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Chapter 5

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A number expressing the likelihood that a specific event will occur, expressed as the ratio of the number of actual occurrences to the number of possible occurrences

Probability

50.2

1

11

1

tailsand heads #

heads #)(

# #

#

#

#)(

of

ofheadP

failuresofsuccessesof

successesof

iespossibilitof

successesofsuccessP

P(n) = probability of n occurrencesp= proportion success (what you are looking for)q= proportion failures (what you are not looking for)

Example: If a fair coin is tossed, what is the probability of a head occurring?

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You are rolling a fair die. A defective product is a roll of a 1. What are the odds that you will find a defective product?

Example

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Since probabilities are a ratio, or expressed as percentage, then:

0 or 0% ”impossible event”1 or 100% “sure thing”

Example: You are rolling a fair six-sided die. What are the odds that you will roll a 7? Not a 7?

Probability values

1)(0 successP

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When more than one success can occur, we describe that as “or”◦ sum of the individual probabilities

(+)

When more than one success needs to occur, we describe that as “and”◦ product of the individual

probabilities (x)

Compound probability

)( )( )( and )() and (

)( )( )( )() (

BPAPBPAPBAP

BPAPBPorAPBorAP

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You are rolling a fair die. What are the odds that you will roll a 2 or a 4? A 2 and a 4?

Compound Example

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This equation is used when the events are “mutually exclusive” – meaning they do not occur at the same time

If the successful event can occur more than once, you must use an adjusted equation

“Or” Compound probability

waysofnumberBPAPBorAP

BPAPBPAPBorAP

)( adj )() (

)( )()(or )() (

Example: Draw a King or a Queen

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You have a standard deck of cards. What are the odds that you draw a 3 or club?

Non-Mutually Exclusive Example

9999999

This equation is used when the events are “independent” – meaning they do not affect each other

If the successful event can affect each other, then you must use an adjusted equation

“And” Compound Probability

waysofnumberBPAPBandAP

BPAPBPAPBAP

)( adj )() (

)( )()( and )() and (

Example: Flip of a coin

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You have a standard deck of cards. What are the odds that you will draw four aces without replacement?

Non-Independent Example

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)!(!

!

!

!

rrn

nC

rn

nP

rn

rn

Number of ways is a listing of possible successes

Permutation, PN,n, P(n,r), nPr, the number of arrangements when order is a concern – “think word”

Combination, CN,n, , nCr, the number of arrangements when order is not a concern

Number of ways

r

n

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The product of a number and all counting numbers descending from it to 1

6! = 6x5x4x3x2x1=720

Note: 0!=1

Factorial (!)

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How many 3 letter arrangements can be found from the word C A T? How about 2 letter arrangements?

Three lottery numbers are drawn from a total of 50. How many arrangements can be expected?

Permutation Example

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How many 3 letter groupings can be found from the word C A T?

Three lottery numbers are drawn from a total of 50. How many combinations can be expected?

Combination Example

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Refers to the probability of two possible outcomes, success (s) and failure (f)

Example: Let’s look at the possibilities of flipping coins

*See table 5.1 pg.1451 flip = H(s) or T(f)2 flips = HH or HT or TH or TTEtc.

Calculated by:

Binomial Probability Distribution

P(failure) f trials#n

P(success)s successes #

) (

r

fsCtrialsninsuccessesrP rnrrn

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A single die is tossed five times. Find the probability of rolling a four, three times.

Binomial Probability Distribution example

rnrrn fsCtrialsninsuccessesrP ) (

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Refers to the probability model that can be used for non-replacement sampling

Uses combinations and the basic probability formula

Hypergeometric Probability Distribution

sample thein successes ofnumber x

size sample n

lot thein success ofnumber m

sizelot N

sample thein defectivesx

exactly getting ofy probabilitP(x)

)(

nN

xmxnmN

C

CC

iespossibilitofnumber

successesofnumberxP

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A manufacturer has received 12 parts from a supplier, 10 are good. If a sample of 4 are taken, find the probability of picking 3 good parts.

Hypergeometric Probability Distribution example

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Graphical View of Variation and Six Sigma Performance

Each unit of measure is a numerical value on a continuous scale

Size Size Size Size

Pieces vary from each other

Variation common and special causes

But they form a pattern that, if stable, is called a normal distribution

Histogram or

Frequency Distribution

Normal Distribution

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Normal Distribution

There are three terms used to describe distributions

3. Location Mean

1. ShapeBell

2. Spread

Standard

Deviation

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Symmetrical, Bell-Shaped Extends from Minus Infinity to

Plus Infinity Two Parameters

◦ Mean or Average ( )◦ Standard Deviation ( )

Space under the entire curve is 100% of the data

Mean, median and mode are the same

Normal Distribution

Basics

x,s,

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Normal Distribution

50%50%

-1s-2s-3s +1s +2s +3s0

s± ≈68%1

s± 99.73%3

s± ≈95%2

z value = distance from the mean measured in standard deviations

z value = distance from the mean measured in standard deviations

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Normal Curve theory tells us that the probability of a defect is smallest if you

◦ stabilize the process (control)◦ make sigma as small as possible

(reduce variation)◦ get Xbar as close to target as

possible (center)

Normal Curve Theory

So… For SPC we first want to stabilize the process, second we will reduce variation and last thing is to center the process.

So… For SPC we first want to stabilize the process, second we will reduce variation and last thing is to center the process.

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Specifies the areas under the normal curve

Represents the distance from the center measured in standard deviations

Values found on the normal table pg.580-585

Population Sample

z value

Remember when we talked about 3? The 3 is the z value.

Remember when we talked about 3? The 3 is the z value.

s

xxz

xz

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The known average human height is 5’8” tall with a standard deviation of 5 inches. What are the z values for 6’2” and 4’8”?

z value example

A positive value indicates a z value to the right of the mean and a negative indicates a z value to the left of the mean.

A positive value indicates a z value to the right of the mean and a negative indicates a z value to the left of the mean.

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From our answers from the last exercise, what is the values for:◦ P(Area > 6’2”)?

◦ P(Area < 4’8”)?

◦ P(4’8”< Area < 6’2”)?

◦ Prove area under the normal curve at 1s, 2s, 3s?

z table exercise

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Suppose the HR department decided to only hire people between the heights of 6’8” and 4’9” tall. What percentage of the population, based on our sample, would we not be able to hire?

Height exercise analysis

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States that any distribution of sample means from a large population approaches the normal distribution as n increases to infinity

If you chart the values, the values will have less variation than the individual measurements

Standard deviation is expressed as:

Central limit theorem

x

x

nsx

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Let’s look at an example of how the central limit theorem works

Central Limit Theorem Example (Dice Exercise)