Theoretical Probability Models

1
Theoretical Probability Models

• Dr. Yan Liu
• Department of Biomedical, Industrial Human
  Factors Engineering
• Wright State University

2
Introduction

• Use Theoretical Probability Models When They
  Describe the Physical Model Adequately
• The results of intelligent tests Normal
  Distribution
• The length of a telephone call Exponential
  Distribution
• The number of people arriving at a bank within an
  hour Poisson Distribution
• The number of defects in a bottle production line
  Binomial Distribution
• Discrete Distributions
• Binomial and Poisson distributions
• Continuous Distributions
• Exponential, Normal, and Beta distributions

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Binomial Distribution

• Characteristics
• Totally n trials
• Dichotomous outcomes
• Each trial results in one of two possible
  outcomes (e.g. yes/no, true/false)
• Constant Probability
• Each trial has the same probability of success, p
• Independence
• Different trials are independent
• Probability Mass Function (PMF)

(See Appendix A)
X of successes in a sequence of n
independent trials, and the probability of
success in each trial is p (Success means the
occurrence of an event)
number of ways you can choose x successes from
n trials
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Binomial Distribution (Cont.)

• Cumulative Distribution Function

(See Appendix B)
Y of failures in a sequence of n independent
trials, then Y n X

• Expected Value
• Variance

5
Pretzel Example
You are planning to sell a new pretzel, and you
want to know whether it will be a success or not.
If your pretzel is a Hit, you expect to gain
30 of the market. If it is a Flop, on the
other hand, the market share is only 10.
Initially, you judged these outcomes to be
equally likely. You decided to test the market
first and found out that 5 out of 20 people
preferred your pretzel to the competing product.
Given the new data, what do you think of the
chance of your pretzel being a Hit?
Let X number of people out of 20 tasters who
preferred the new pretzel
Pr(Hit X5)?
(Bayes Theorem)
?
?
0.5
6
In conclusion, the new data suggest that the new
pretzel is very likely to be a hit
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Poisson Distribution

• Represent occurrences of events over a unit of
  measure (time or space)
• e.g. number of customers arriving, number of
  breakdowns occurring
• Assumptions
• Events can happen at any point along a continuum
• At any particular point, the probability of an
  event is small (i.e. events do not happen
  frequently)
• Events happen independently of one another
• The average number of events is constant over a
  unit of measure
• Probability Mass Function

(See Appendix C)
X of events in a unit of measure
m is the average number of events in a unit of
measure
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Poisson Distribution (Cont.)

• Cumulative Distribution Function

(See Appendix D)

• Expected Value
• Variance

Y of events in t units of measure
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Pretzel Example (Cont.)
Based on your previous market research, you
decide to invest in a pretzel stand. Now you need
to select a good location. You consider a
location to be good, bad, or dismal if you
sell 20, 10, or 6 pretzels per hour,
respectively. You have found a new stand and your
initial judgment is that the probabilities of the
location being good, bad, and dismal are 0.7,
0.2, and 0.1, respectively. After having the
stand for a week, you decided to run a test.
Within 30 minutes, you sold 7 pretzels. Now, what
are your probabilities regarding the quality of
the stand?
Let Xnumber of pretzels sold within 30 minutes
or 0.5 hour
Pr(Good X 7) ?
(Bayes Theorem)
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In light of the new data, you feel that the
chance of the current stand being a good location
has slightly increased and thus you should stay.
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Exponential Distribution

• If the number of events occurring within a unit
  of measure follows a Poisson distribution, then
  the time or space between the occurrence of two
  events follows an exponential distribution
• Exponential distribution has the same assumptions
  as Poisson distribution
• Probability Density Function

Let T Time (space) between two consecutive
events
m is the same average rate used in Poisson
distribution

• Cumulative Distribution Function

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Exponential Distribution (Cont.)

• Expected Value
• Variance
• Other Important Probabilities

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Pretzel Example (Cont.)
You wonder if you can provide fast service to
your customers. It takes 3.5 minutes to cook a
pretzel, so what is the probability that the next
customer arrives before the pretzel is
finished? As in the previous example, you assume
customers arrive according to a Poisson process,
and you consider your location being good, bad or
dismal if you sell 20, 10, 6 pretzels per hour,
respectively. Your prior belief is that
Pr(Good)0.7, Pr(Bad)0.2, and Pr(Dismal)0.1.
Let Tthe time between two consecutive customers
Pr(Tlt3.5) ?
?
?
?
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In other words, about 60 of your customers will
have to wait until the pretzel is ready.
Therefore, the fast service does not seem very
appealing.
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Normal Distribution

• Bell-Shaped Curve
• Particularly good for modeling situations in
  which the uncertain quantity is subject to many
  different sources of errors
• many measured biological phenomena (e.g. height,
  weight, length)
• Probability Density Function
• Expected Value
• Variance
• Some Handy Empirical Rules

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Normal Distribution (Cont.)

• Standard Normal Distribution
• Convert to Standard Normal Distribution

(See Appendix E for Cumulative Probability)
,
X N(µ10, s2400), then the probability X is
less than or equal to 35 is
(Appendix E)
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Normal Distribution (Cont.)

• Other Important Probabilities

Because standard normal distribution is symmetric
around zero,
X N(µ10, s2400), then
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Standard Normal Distribution
19
Quality Control Example
Your plant manufactures disk drivers for personal
computers. One of your machines produces a part
that is used in the final assembly. The width of
the part is important to the proper functioning
of the disk driver. If the width falls below
3.995 or above 4.005 mm, the disk driver will not
work properly and must be repaired at a cost of
10.40. The machine can be set to produce parts
with width of 4mm, but it is not perfectly
accurate. In fact, the width is normally
distributed with mean 4 and the variance depends
on the speed of the machine. The standard
deviation of the width is 0.0019 at the lower
speed and 0.0026 at the higher speed. Higher
speed means lower overall cost of the disk
driver. The cost of the driver is 20.45 at the
higher speed and 20.75 at the lower speed.
Should you run the machine at the higher or
lower speed?
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Let X width of a disk driver
P1Pr(Defective Low Speed)
P2Pr(Defective High Speed)
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E(CostLow Speed)0.008631.150.991420.7520.84
E(CostHigh Speed)0.054830.850.945220.4521.0
2
Conclusion Because E(CostLow Speed)ltE(CostHigh
Speed), you should run the machine at the lower
speed
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Beta Distribution

• Useful in modeling an uncertain ratio or
  proportion (ranging from 0 to 1)
• e.g the proportion of voters who will vote for
  the Republican candidate
• Probability Density Function

(See Appendix F for Cumulative Probability)
Let Qthe proportion of interest
n, r are parameters that determine the shape of
f(qn,r). n determines the tightness of the
distribution the larger n is, the tighter the
distribution is. r determines the skewness of
the distribution. In particular, When r n/2,
the distribution is symmetric around 0.5.
Otherwise, the distribution is skewed to the
right and left when r lt n/2 and r gt n/2,
respectively.
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Beta Distribution
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Beta Distribution (Cont.)

• Expected Value
• Variance

Loosely speaking, r and n can interpreted as r
successes in n trials
Suppose your guess for the preference of the
Republican candidate is that 40 people would
vote for the Republican candidate.
You can set n10, r4. This coincides with the
expected proportion of 40.
What if you set n100, r40?
This still coincides with the expected proportion
of 40. However, the variances of the two cases
are different.
When n10, r4,
When n100, r40,
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Pretzel Example (Cont.)
You want to re-evaluate your decision to invest
in a pretzel stand. At this point, you estimate
that you are 50 sure that your market share is
less than 20 and 75 sure that your market share
is less than 38.
Let Q market share, you decide to model the
uncertainty in Q as a Beta distribution
Pr(Q0.20)0.5, Pr(Q0.38)0.75
Using the table in Appendix F, you find that
You think the beta distribution is close enough
and thus should proceed with the analysis
The expected value of Q, E(Q)0.25
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Pretzel Example (Cont.)
You estimate that the total market is 100,000
pretzels. You sell a pretzel at 0.50. It costs
you 0.10 to produce a pretzel, in addition to
8,000 fixed cost for marketing, financing, and
overhead.
Net Profit Revenue Cost 100,000Q0.5
(100,000Q0.18,000) 40,000Q 8,000
E(Net Profit) 40,0000.25 8,000 2,000 gt 0
So it seems to be a good idea to start a pretzel
career.
However, as a careful person, suppose you also
want to evaluate your chances of losing money.
Net Profit lt0 gt 40,000Q-8,000lt0 gt Q0.2
(Appendix F)
Therefore, there is about 50 chance of losing
money. Are you willing to continue to take this
risk?
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Exercises

• Bottle Production
• In bottle production, bubbles that appear in the
  glass are considered defects. Any bottle that has
  more than two bubbles is classified as
  nonconforming and is sent to recycling. Suppose
  that a particular production line produces
  bottles with bubbles at a rate of 1.1 bubbles per
  bottle. Bubbles occur independently of one
  another.
• What is probability that a randomly chosen bottle
  is nonconforming?
• Bottles are packed in cases of 12. An inspector
  chooses one bottle from each case and examines it
  for defects. If it is nonconforming, she inspects
  the entire case, replacing nonconforming bottles
  with good ones. If the chosen one conforms, then
  she passes the case. In total, 20 cases are
  produced. What is the probability that at least
  18 of them pass?

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a. X of bubbles in a bottle X
Possion(m1.1) Pr(X gt 2 m 1.1) 1 - Pr(X 2
m 1.1) 1.00 - 0.90 0.1 b. Y of cases
out of 20 cases that do not pass Y Binomial
(n20, p0.1) Pr(Y2n20,p0.1) 0.677
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Exercises

• Greeting Card
• A greeting card shop makes cards that are
  supposed to fit into 6 in. envelopes. The paper
  cutter, however, is not perfect. The length of a
  cut card is normally distributed with mean 5.9
  in. and standard deviation 0.0365 in. If a card
  is longer than 5.975 in., it will not fit into a
  6 in. envelope.
• Find the probability that a card will not fit
  into a 6 in. envelope
• The cards are sold in boxes of 20. what is the
  probability that in one box there will be two or
  more cards that do not fit in 6 in. envelopes?

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a. L the uncertain length of an envelope. L
N(µ 5.9, s 0.0365) Pr (L gt 5.975 µ 5.9, s
0.0365) Pr(Z gt(5.975-5.9)/0.0365) Pr(Z gt
2.055) 1-Pr(Z2.055)1-0.980.02 b. X of
cards in one box that do not fit in the
envelopes X Binomial(n20, p0.02) Pr(X2n20,p
0.02) 1-Pr(X1n20,p0.02) 1-0.940.06