Uncertainty: continued . . . from lab.

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Uncertainty continued . . . from lab.

• Probability revisions
• Continue decision trees
• Value of information

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Todays agenda

• Probability revision computation
• Value of information
• Group problem solving Hunk diamond

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Objectives understand

• Conditional probabilities
• Likelihoods
• Posteriors
• Decision tree development
• The expected value of imperfect information
• The expected value of perfect 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
Knight
.8
Pickinhabitant
.6
.2
Not
No
In favor
No (lie)
.8
Knave
.2
.4
Not
Yes (lie)
P(yes ? 6 n10, p ?)
1 - P(no lt 5 n10, p ?)
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Binomial

• If p ___ (if p probability of a no, n
  draws, c hits, then

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Yesterdays disjunctive event

• P(x ? 1 n 7, p .1) 1 - P(x 0n 7,p
  .1)

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Heres is what we need to know

• How do we compute the probability of a yes
  report
• Its the probability of a yes and its right,
  plus the probability of a yes and its wrong, or

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Joint and marginal probabilitiesfrom a frequency
table
What is
P(Knight)
1000
P(Yes)
P(Knight,Favor) P(Knave,Not Favor)
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Marginals, joints, likelihoods, posteriors
Yes
No


480
120
600




Knight


80
320

Knave

400





560
440


Yes

Marginals P(favor) .8 and P(favor)
.2P(knight) .6 and P(knave) .4P(yes) .56
and P(no) .44
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
Posteriors The probability of a state of nature
given a specific report P(FavorYes) 480/560
.857 and P(FavorYes) 80/560
.143 P(FavorNo) 320/440 .727 and
P(FavorNo) 120/440 .273
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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 visualization
We know we chose a box with a gold coin.
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Probability revision
Likelihoodgives theaccuracy of the info.
Given the prior P(addicted) .001
P(yes addicted) .95 and P(no not
addicted) .95
and
We want P(addicted yes) and P(not addicted
no)
This is the conditional probability called
arevised or posterior probability.
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Conditional Probability

• P(addictedyes)

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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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Test for drug addiction
P(not addicted, yes)
Therefore, P(yes)
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)
This is a posterior probability.
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Probability of yes
Answer Prob
Yes
.95
Addict
.001
No
.05
Yes
.05
Not
.999
No
.95
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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?
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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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Decision Trees
The decision maker (DM) decides
Nature decides
One coming out for each of DMs or natures
options.
P
Probabilities one for each of natures options

Net benefit or cost at the end of each branch
Each of natures nodes has an expected value.
E()
Each DM decision has a chosen expected value

Rejects a branch following DM box.
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Part a.
Recommendation
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Group Work

• The first question onHunk Diamond

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Buy a study?
Should she purchase a study on whether resistance
to rezoning is high or low? Cost 10,000
P(highno rezoning) .902P(lowrezoning)
.802
P(high,no rezoning) .902.5
.451 P(low,rezoning) .802.5 .401
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Part b.
Posterior
high
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Part b.
low
What is the probability of getting low resistance?
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Part b.
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Part c.
EVII
Recommendations.
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Whats the expected value of perfect information?
No investment errors!
Dont buy info.
200,000
No rezoning
0
Buy info.
Rezone
600,000
P(No rezoning) ?
P(Rezoning) ?
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Group Work

• The second requirement onHunk Diamond

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The end!