Uncertainty continued ' ' ' from lab'

1
Uncertainty continued . . . from lab.

• Probability revisions
• Continue decision trees
• Value of information

2
Todays agenda

• Probability revision computation
• Work through P4-17 and P4-18.
• Group problem solving Hunk diamond

3
Objectives understand

• Conditional probabilities
• Likelihoods
• Posteriors
• Decision tree development
• The expected value of perfect information
• The expected value of imperfect information

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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 Probability Tree
Answer
Prob.
P(favor) .8 P(knight) .6
In favor
Yes
.48
Knight
.8
Pickinhabitant
.6
.2
Not
.12
No
In favor
No (lie)
.32
.8
Knave
.2
.4
Not
.08
Yes (lie)
P(yes ? 6 n10, p.56)
1 - P(no lt 5 n10, p.44)
1 - .4696 .5304
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Joint and marginal probabilitiesfrom a frequency
table
What is
.6
P(Knight)
1000
P(Yes)
P(Knight,Favor) P(Knave,Favor)
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Marginals, joints, likelihoods, posteriors
Yes
No


336
264
600




Knight


176
224

Knave

400





560
440


Yes

Posteriors The probability of a state of nature
given a specific report P(FavorYes) 336/560
.6 and P(FavorYes) 224/560 .4 P(FavorNo)
176/440 .4 and P(FavorNo) 264/440 .6
Likelihoods The probability of a specific
report given an state of nature P(YesFavor)
480/800 .6 and P(YesFavor) 80/200
.4 P(NoFavor) 320/800 .4 and P(NoFavor)
120/200 .6
Marginals P(favor) .8 and P(favor)
.2P(knight) .6 and P(knave) .4P(yes) .56
and P(no) .44
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Revision Information received
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 (possibly) helpful visualization
We know we chose a box with a gold coin.
We want P(gold2 gold1)
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Probability revision
Likelihood
Given the prior P(addicted) .001
and
P(yes addicted) .95 and P(no not
addicted) .95
We want P(addicted yes) and P(not addicted
no)
This is the conditional probability called
arevised or posterior probability.
11
Test for drug addiction
We know this formula
P(addicted yes)
and
P(yes addicted)
.95
P(addicted) .001
We can solve for the joint.
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Test for drug addiction
P(addicted, yes)
What is P(yes)?
It consists of two joint probabilities.
The test can say yes and the subject is
addicted OR The test can say yes and the
subject is not addicted
P(addicted, yes) P(not addicted, yes) P(yes)
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Probability of yes
Answer Prob
Yes
.95
Yes .00095
Addict
.001
No .00005
No
.05
Yes
.05
Yes .04995
Not
.999
No .94905
No
.95
P(Yes) .00095 .04995 .0509
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Test for drug addiction
P(not addicted, yes)
Therefore, P(yes) .00095 .04995 .0509
Given a positive test result, the probability
that a person chosen at random from an urban
population is a drug addict is
P(addict yes) .00095/.0509 .0187
This is a posterior probability.
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Drug test joint probability table
.95
.95
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Another revision example
Priors P(disease) .01 P(disease) .99
Test accuracy P(positive disease)
.97 P(positive disease) .05
(likelihoods)
P(negative disease) .03 P(negative
disease) .95
Note that false positives gt 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?
P(positive, disease) .97 .01 .0097
P(positive) (.97 .01) (.05 .99) .0592
P(disease positive) .0097/.0592 .1639
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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
Then, P(disease positive)
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Buying information
A real estate investor has the opportunity to
purchase land currently zoned residential. If
the county board approves aa request to rezone
the property as commercial within the next year,
the investor will be able to lease the land to a
large discount firm that wants to open a new
store on the property. However, if the zoning
change is not approved, the investor will have to
sell the property at a loss. Profits are show in
the payoff table below
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Part a.
Rezoned
600,000
.5
Purchase
Not rezoned
.5
-200,000
200,000
Dont
0
200,000
Recommendation Buy the property because the
expected value of purchasing is 200,000 greater
than the expected value of not purchasing.
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Part b.
Rezoned
600,000
.18
Purchase
-200,000
Not rezoned
.82
-56,000
High
Dont
0
0
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Part b.
Rezoned
600,000
.89
Purchase
-200,000
Not rezoned
.11
512,000
Low
Dont
0
512,000
What is the probability of getting low resistance?
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Part b.
Low
512,000
.45
Buy option
High
0
.55
230,400
No option
200,000
230,400
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Part c.
EVSI 230,400 - 200,000 30,400
Recommendations First, buy the option because
it costs only 10,000, but has a value of up to
30,400. Then, if resistance is high, dont
purchase the land, but if resistance is low,
purchase the land.
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Low
400,000
.2
Medium
..35
Small condo
400,000
400,000
..45
High
400,000
Low
100,000
.2
No survey
Medium condo
Medium
.35
600,000
.45
500,000
High
600,000
Low
-300,000
500,000
.2
Large condo
Medium
.35
300,000
450,000
.45
High
900,000
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Recommendation If no survey is purchased, build
a medium development because it has the highest
expected value at 500,000.
Part c EVwPI .2(400) .35(600) .45(900)
695
Therefore, EVPI 695,000 - 500,000 195,000
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Low
400,000
Part d. Buy a survey
.39
Medium
.46
Small condo
400,000
400,000
.15
High
400,000
Low
100,000
.39
Weak
Medium condo
Medium
.46
600,000
.15
405,000
High
600,000
Low
-300,000
405,000
.39
Large condo
Medium
.46
300,000
156,000
.15
High
900,000
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No survey
500,000
Weak
405,000
.3
Average
.38
Survey
520,000
.32
541,800
Strong
696,000
Recommendation First, purchase the survey if
the cost is less than ?
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Recommendations
EVSI 541,800 - 500,000 41,800
First, buy the survey if its cost is less than
41,800. Then, if the survey says weak or
average, build medium if the survey says
strong, build a large development.
Efficiency
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Likelihoods?
P(LowWeak) P(L,W)/.3 .39, so P(L,W) .117,
so P(WeakLow) .117/.2 .585 P(MediumWeak)
P(M,W)/.3 .46, so P(M,W) .138, so
P(WeakMedium) .394 P(WeakHigh) 1 - .585 -
.394 .021
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Likelihoods
P(LowAverage) P(L,A)/.38 .16, so P(L,W)
.0608, so P(AverageLow) .0608/.2
.304 P(MediumAverage) P(M,A)/.38 .37, so
P(M,A) .1406, so P(AverageMedium)
.4017 P(AverageHigh) 1 - .304 - .4017 .2943
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Likelihoods
P(LowStrong) P(L,S)/.32 .06, so P(L,S)
.0192, so P(StrongLow) .0192/.2
.096 P(MediumStrong) P(M,A)/.32 .22, so
P(M,S) .0704, so P(StrongMedium)
.2011 P(StrongHigh) 1 - .096 - .2011 .7029
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Freemark Abbey Winery

• Based on a real decision
• No memo
• Draw the tree calculate or use the appropriate
  probabilities
• Write out recommendations based on the tree
  (e.g., First, do that. Then, if that occurs, do
  this.)
• Answer the assignment questions EVPI, but two
  kinds
• Ill be in on Saturday if you want to discuss it.