Uncertainty and probability

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Uncertainty and probability

• Using probabilities
• Using decision trees
• Probability revision

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Todays agenda

• Important terms
• Simple review (objective, subjective, marginal,
  joint, and conditional probabilities)
• Examples outcomes, expected values, risk
  attitudes
• Examples action choices, decision trees

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Vocabulary

• A probability is a number between zero and one
  representing the likelihood of the occurrence of
  some event.
• Probability
• objective vs. subjective
• marginal vs. joint
• joint vs. conditional
• prior vs. posterior
• likelihood vs. posterior

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Vocabulary continued

• Outcomes or payoffs (mutually exclusive)
• Action choices
• States of nature
• Decision tree
• Expected value
• Risk

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Probability
Imagine an urn containing 1500 red, pink, yellow,
blue and white marbles.
Take one ball from the urn. What is
Probabilities are all greater than or equal to
zero and lessthan or equal to one.
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Same urn
Suppose the number of balls is as
follows Red 400 Pink 100 Yellow 400 Blue
500 White 100 Total 1500
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Dependent and Independent Events
Two events are dependent if the occurrence of one
of the events affects the probability of the
occurrence of the other event. P(AB)
P(A) P(A, B) P(A)P(B)
If two events are (statistically)
independent, the occurrence of one event will not
affect the probability of the occurrence of the
other event.
P(A B) P(A) P(A, B) P(A)P(B)
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Joint probabilities and independence
Define A as the event draw a red or a pink
marble.
We know 500 marbles are either red or pink.
What are
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Joint probabilities and independence (were
getting there)
Define B as the event, draw a pink or white
marble.
We know 200 marbles are pink or white.
What are
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Joint probabilities and independence
Define A as the event draw a red or a pink
marble.
Define B as the event draw a pink or white
marble.
What is
P(A, B) P(A ? B)
This is the joint probability of A and B.
What color is the marble?
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Joint probabilities and independence
Are A and B independent?
Are A and B mutually exclusive?
What is the probability of A or B?
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Joint probabilities and independence
Suppose we draw one marble from the urn
andreplace it. Then, we draw a second marble.
What is
Are (Red, Red) independent?
Are (Red, Blue) independent?
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Joint and marginal probabilities
1500
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Conditional probabilities
The probability that a particularevent will
occur, given we alreadyknow that another event
hasoccurred.
1500
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Definition of independence
P(B A)
Events A and B are independent if
P(B A)
P(B)
1500
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Marginal and joint probability table
The joint probabilities are in the box. The
marginalsare outside.
How do you compute conditionals from this?
A
A
B
B
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Joint probability tables
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Outcomes or payoffs
Example Win 1,000 if you draw a pink
marble, win 0 otherwise.
Outcomes 1,000 or 0
Events A pink marble or a marble of another
color
Probabilities
P(Pink) 1/15
P(Pink) 14/15
The expected value of this gamble
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Example
We expect to sell 10,000 computers if the market
isgood and to sell 1,000 computers if the market
isbad. The marketing departments best estimate
ofthe probability of a good market is .5.
Outcomes 10,000 computers sold or 1,000
computers sold.
Probabilities P(good market) .5 P(bad
market) .5
Are these objective or subjective probabilities?
What is expected computer sales?
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More expected values
For discrete outcomes, an expected value is
theprobability-weighted sum of the outcomes for
thedecision of interest.
Here are some two-outcome lotteries. Compute
theexpected values.
L1 Win 1,000 with probability .5 or lose 500
with probability .5.
L2 Win 2,000 with probability .5 or lose
1,000 with probability .5.
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More lotteries
Here are some two-outcome lotteries. Compute
theexpected values.
L3 Win 1,500 with probability 1/3 or lose
750 with probability 2/3.
L4 Win 750 with probability 2/3 or lose
750 with probability 1/3.
L5 Win 300 with probability .5 or lose 200
with probability .5.
L6 Win 10,000 with probability 9/10 or
lose 85,000 with probability 1/10.
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L1 (1000, p.5, -500)
L2 (2000, p .5, -1000)
L3 (1500, p 1/3, -750)
L4 (750, p 2/3, -750)
L5 (300, p .5, -200)
L6 (10000, p .9, -85000)
L7 (1000, p .2, 350)
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Action choices
If I build a large hotel (cost 5,000,000) and
tourismis high (P(high) 2/3), I will make
15,000,000 in revenue, but if it is low, I will
make 2,000,000.
If I build a small hotel (cost 2,000,000) and
tourismis high, I will make 5,000,000, but if
tourism is low, I will make 2,000,000.
I can also choose to do nothing.
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Action choices
If I build a large hotel (cost 5,000,000) and
tourism is high (P(high) 2/3), I will make
15,000,000 in revenue, but if it is low, I will
make 2,000,000.
If I build a small hotel (cost 2,000,000) and
tourism is high, I will make 5,000,000, but if
tourism is low, Iwill make 2,000,000.
I can also choose to do nothing.
Action choices Build large, build small, do
nothing
Outcomes 15,000,000 5,000,000 2,000,000, 0
States of nature high tourism, low tourism
Probabilities P(high) 2/3 P(low) 1/3
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Decision trees
Suppose I need to decide whether to invest
10,000 in the market or leave it in the bank to
earn interest. If I invest, there is a 50
chance that the market will increase 20 over the
coming year and a 50 chancethat the market will
be stagnant (no change). If I leave the money in
the bank, there is an 80 chance that interest
rates will increase to 10 and a 20 chance that
interest rates will remain at 5. What should I
do?
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A decision by an individual is required
Nature makes these decisions
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Homework assignment
Problems 5-15 and 16
Urn 1
Urn 2
Consider two urns
Red balls
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4
Black balls
3
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P(R1) Probability of red on first draw
P(R2) Probability of red on second draw
P(B1) Probability of black on first draw
P(B2) Probability of black on second draw
a(1) Take one ball from urn 1, replace it, and
take a second ball. What is the probability of
two reds beingdrawn?
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Homework
a(2) What is the probability of a red on the
seconddraw if a red is drawn on the first draw?
a(3) What is the probability of a red on the
second draw if a black is drawn on the first draw?
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Homework
b(1) Take a ball from urn 1 replace it. Take a
ball from urn 2 if the first ball was black
otherwise, draw a ball from urn 1.
What is the probability of two reds being drawn?
b(2) What is the probability of a red on the
second draw if a red is drawn on the first draw?
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Homework
b(3) What is the probability of a red on the
second draw if a black is drawn on the first draw?
What is the unconditional probability of red on
thesecond draw?
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Homework
5-16. Draw a tree diagram for Problem 5-15a
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Homework
5-17. Draw a tree diagram for Problem 5-15b
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Homework
5-29. This is the survey problem involving
home- ownership and income levels. The results
can be summarized by the table below
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Survey
A. Suppose a reader of this magazine is selected
atrandom and you are told that the person is a
home- owner. What is the probability that the
person has income in excess of 25,000?
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Survey
b. Are home ownership and income (measured only
as above or below 25,000) independent factors
for this group?
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Homework
5-38. The president of a large electric utility
has to decide whether to purchase one large
generator (Big Jim) or four smaller generators
(Little Arnies) to attain a given amount of
electric generating capacity. On any given summer
day, the probability of a generator being in
service is 0.95 (the generators are equally
reliable). Equivalently, there is a 0.05
probability of a failure.

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Homework
5-38. a. What is the probability of Big Jims
being out of service on a given day?
Let P(out) the probability of any generator
being out of service .05
If P(BJout) the probability of Big Jims being
out of service.
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Homework
5-38. b. What is the probability of either zero
or one of the four Arnies being out? (At least
three will be running.)
Since the probability of a failure (f) for one
Arnie is .05,
We want P(f ? 1n4, p.05)
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Homework
5-38. c. If five Little Arnies are purchased,
what is the probability of at least four
operating?
P(f ? 1 n5, p.05)
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Homework
5-38. d. If six Little Arnies are purchased,
what is the probability of at least four
operating?
P(f ? 2 n6, p.05)
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Homework
5-40. Newspaper articles frequently cite the
fact that in any one year, a small percentage
(say, 10) of all drivers are responsible for all
automobile accidents. The conclusion is often
reached that if only we could single out these
accident-prone drivers and either retrain them or
remove them from the roads, we could drastically
reduce auto accidents. You are told that
of 100,000 drivers who were involved in one or
more accidents in one year, 11,000 of them were
involved in one or more accidents in the next
year. A. Given the above information, complete
the entries in the joint probability table in
Table 5-21.
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Accidents Joint probability table
A1 accident in year 1, A2 accident in year 2
11,000/100,000 .11
Given P(A2 A1)
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Accidents
B. Do you think searching for accident-prone driv
ers is an effective way to reduce auto
accidents? Why?
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Ambiguity
Assume you can choose between two
different gambles. In one gamble, you know that
50 red and 50 black balls are in a jar, and one
ball is to be chosen perfectly randomly. You
will receive a prize if you guess what the color
of the ball drawn will be.
In the second gamble, there is an
undisclosed number of red and black balls in a
jar, and again you are to guess the color of the
draw for an identical prize.
Which gamble would you choose?
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Ambiguity
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Assessing probabilities base rates
A restaurant seats 100 people. The owner
recently decided to provide a free birthday cake
to any person having a birthday on the day he or
she dined at the restaurant. Assuming p 1/365
.003 and n 100, the owner found the folowing
entry in an extensive binomial table
P(R ? 6 n100, p.01) .0005
The owner felt that, since his p was smaller than
.01, this probability should be an upper limit on
the probability of six or more birthday cakes
being requested each night. After 10 evenings,
he had run out of cakes three times? What
happened?
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Logistics
The Acme Company has two warehouses located
in different cities. Demand for the product is
indepen- dent in each warehouse district.
However, both ware- houses have identical
probability distributions for demand as follows
Assume that each warehouse normally stocks
twounits.
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Logistics
What is the probability that one or the other of
the warehouses (not both) will have more demand
than stock?
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Logistics
What is the probability that both warehouses
will be out of stock?
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Knights Knaves
Professor Smullyan describes an island, the
inhabitants of which are either knights or
knaves. Knights never lie, and knaves never tell
the truth. Suppose that you know that 80 percent
of the inhabitants (both knights and knaves) are
in favor of electing Professor Smullyan as king
of the island. The island is made up of
60 percent knights and 40 percent knaves, but you
cannot tell which is which. Suppose you take a
sample of 10 inhabitants at random and ask, Do
you favor Smullyan as king? What is the
probability that you get six or more yes
answers?
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Knights Knaves
P(favor) .8 P(knight) .6
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Uncertainty continued . . .

• Probabilitiy revisons
• Continue decision trees

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Todays agenda

• Finish the homework problems
• Work through a decision tree example that
• Uses no information
• Uses perfect information
• Uses imperfect information
• Briefly discuss Hawthorne Plastics, Inc.
• Group problem solving

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Homework
5-42. A safety commissioner for a certain city
performed a study of the pedestrian fatalities
at intersections. He noted that only 6 of the 19
fatalities were pedestrians who were crossing the
intersection against the light (i.e., in
disregard of the proper signal), whereas the
remaining 13 were crossing with the light. He
was puzzled because the figures seemed to show
that it was roughly twice as safe for a
pedestrian to cross against the light as with
it. Can you explain this apparent contradiction
to the commissioner?
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Homework
5-42.
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Homework
5-42.
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Homework
5-43.
Probability revision
Suppose a new test is available to test for
drug addiction. The test is 95 percent accurate
each way that is, if the person is an addict,
there is a 95 percent chance the test will
indicate yes if the person is not an addict,
then 95 percent of the time the test
will indicate no. Suppose it is known that the
incidence of drug addiction in urban populations
is about 1 out of 1,000. Given a positive (yes)
test result, what are the chances that the person
being tested is addicted?
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Homework Probability revision
5-43.
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Test for drug addiction
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Test for drug addiction
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Test for drug addiction
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Drug test joint probability table
.95
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Bayes Theorem and the Multiplication Rule
P(Ai B)
and
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Another revision example
Priors P(disease) .01 P(disease) .99
Test accuracy P(positive disease)
.97 P(positive disease) .05
P(negative disease) .03 P(negative
disease) .95
Note that false positives false negatives
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Disease detection continued
Priors P(disease) .01 P(disease) .99
Test accuracy P(positive disease)
.97 P(positive disease) .05
P(negative disease) .03 P(negative
disease) .95
What is the probability that an individual chosen
at random who tests positive has the disease?
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Disease detection continued
Suppose the tested individual was not chosen from
the population at random, but instead was
selected from a subset of the population with a
greater chance of getting the disease?
Prior Suppose P(disease) .2
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Homework
5-45 Revision
This is a classical probability problem. Try out
your intuition before solving it systematically.
Assume there are three boxes and each box has two
drawers. There is either a gold or silver coin
in each drawer. One box has two gold, one box
two silver, and one box one gold and one silver
coin. A box is chosen at random and one of the
two drawers is opened. A gold coin is observed.
What is the probability of opening the second
drawer in the same box and observing a gold coin?
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Coin and box problem
Here is a helpful visualization
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Buying information
As manager of a post office, you are trying to
decide whether to rearrange a production line and
facilities in order to save labor and related
costs. Assume that the only alternatives are to
do nothing or rearrange. Assume also that the
choice criterion is that the expected savings
from rearrangement must equal or exceed 11,000.
Operating costs if you do nothing will be 200,000
If you rearrange successfully, operating costs
will be 100,000.
If you rearrange unsuccessfully, operating costs
will be 260,000.
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Post Office Example
Buying information
Operating costs if you do nothing will be 200,000
If you rearrange successfully (P(success) .6),
operating costs will be 100,000.
If you rearrange unsuccessfully (P(fail) .4),
operating costs will be 260,000.
What is the expected value of each action choice?
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Post Office Decision Tree
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You can hire a consultant, Joan Zenoff, to study
the situation. She would then render a flawless
prediction of whether the rearrangement would
succeed or fail. Compute the maximum amount you
would be willing to pay for the errorless
prediction.
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You can hire a consultant, Joan Zenoff, to study
the situation. She would then render a flawless
prediction of whether the rearrangement would
succeed or fail. Compute the maximum amount you
would be willing to pay for the errorless
prediction.
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How much would you pay for Joans report?
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Suppose now that Joans reports are not
flawless. Suppose you have been provided the
following posterior probabilities
This means that P(success optimistic)
.818, not 1 P(failure optimistic) .182, not 0
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What would you be willing to pay for
Joans report?
EVII E(decision with imperfect information) -
E(decision with no information)
Recall that E(decision with no information)
164,000
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Required 1. Compute the expected cost assuming
an optimistic report.
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2. Compute the expected costs assuming a
pessimistic report.
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3. We were given the probability of an
optimistic report (P(optimistic) .55) and the
probability of a pessimistic report
(P(pessimistic) .45).
Compute the expected value of imperfect
information.
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We were not given likelihoods. We do not know
the probability that Joan will render an
optimistic report given the rearrangement is a
success.
Can we compute that probability?
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Normally we would be given likelihoods and
priors and we would expect to compute 1. The
posterior probability of the outcome given a
particular kind of information, and 2. The
marginal probability of receiving a particular
kind of information
Therefore, given the following information
about the accuracy of Joan Zenoffs forecasts,
complete a joint probability table and compute
the necessary posterior (revised) probabilities.
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P(optimistic success)
P(optimistic success) .75 P(pessimistic
failure) .75
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Which ones go on the decision tree?
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Hawthorne Plastics, Inc.

• Read the instructions for this assignment.
• If you must make assumptions to complete your
  analysis, state what assumptions you made and
  test those assumptions using some form of
  sensitivity analysis.
• Check the sensitivity of your decisions to the
  probability assessments made in the case.
• Produce a histogram of the pressure data for
  Process 2.

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Hawthorne Plastics, Inc.

• How sensitive are your recommendations to the
  probability that 150 PSI will be maintained on
  Process 2?
• How sensitive are your recommendations to the
  probability that raw material is long chain?
• Explain to Mr. Nelson the implications of any
  assumptions you made about his risk preferences.

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Remaining

• Hawthorne Plastics (Wednesday, 4/28)
• Brockway Coates (Monday, 5/3)
• Course recap (Wednesday, 5/5)
• Final Exam
• Friday, May 7, 130-430 p.m. - Section B
• Wednesday, May 12, 700-1000 p.m. - Section C