SMU
EMIS 5300/7300
NTU
SY-521-N
Systems Analysis Methods
Dr. Jerrell T. Stracener,
SAE Fellow
Decision Analysis
Decision-Making Under Risk
updated 11.06.01
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Decision Making Under Risk
• Decision Trees
• Revising Probabilities Based on Sample
Information
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Decision Trees
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Revising Probabilities Base on
Sample Information
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Bayes Theorem
Let {B1, B2, ..., Bn) be a set of events forming a partition of the
sample space S, where P(Bi)  0, for i = 1, 2, ... , n. Let A be any
event of S such that P(A)  0. Then, for k = 1, 2, ..., n,
P(B k | A) 
P(Bk  A)
n
 P(B  A)
i 1

i
P(B k )P(A | B k )
n
 P(B )P(A | B )
i 1
i
i
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In a sense, Bayes’ Rule is updating or revising the prior
probability P(B) by incorporating the observed information
contained within event A into the model.
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Example
Let us examine the revision of probabilities using Bayes’ rule
with sampling information. Determine how to best invest
$10,000 for the next year. However there are only two
decision alternatives available to the investor.
1. Bonds
2. Stocks
Only two states of the investment climate can occur:
1. No growth
2. Rapid growth
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Example
There is a 0.65 probability of no growth in the investment
climate and 0.35 probability of rapid growth. The payoffs are
$500 for a bond investment in a no-growth state, $100 for a
bond investment in a rapid-growth state, -$200 for a stock
investment in a no-growth state, and a $1100 payoff for a stock
investment in a rapid-growth state.
STATE OF NATURE
No Growth (0.65)
DECISION
ALTERNATIVE
Bonds
Stocks
$500
-$200
Rapid Growth (0.35)
$100
$1100
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Example
The expected monetary value for the bonds decision alternative is
EMV(bonds) = $500(0.65) + $100(0.35) = $360
The expected monetary value for the stocks decision alternative is
EMV(stocks) = -$200(0.65) + $1100(0.35) = $255
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Example
Now suppose that the decision-maker has a chance to obtain
some information from an economic expert regarding the future
state of the investment economy. This expert does not have a
perfect record of forecasting, but she has a track record of
predicting a no-growth economy about 0.80 of the time when
there actually is a no-growth economy. She has been slightly
less successful in predicting rapid-growth economies, with a 0.70
probability of success.
ACTUAL STATE OF ECONOMY
No Growth (s1)
Forecaster Predicts No Growth (F1)
Forecaster Predicts Rapid Growth (F2)
0.80
0.20
Rapid Growth (s2)
0.30
0.70
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Example
Using these condition probabilities, we can revise prior
probabilities of the states of the economy using Bayes’ rule.
PX i | Y  
PX i   PY | X i 
PX1   PY | X1   PX 2   PY | X 2   ...  PX n   PY | X n 
Suppose the forecaster predicts no growth (F1).
PF1   PF1  s1   PF1  s 2 
 0.520  0.105
 0.625
PF2   PF2  s1   PF2  s 2 
 0.130  0.245
 0.375
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Example
Revision Based on Forecast of No Growth (F1)
STATE OF
ECONOMY
PRIOR
PROBABILITIES
CONDITIONAL
PROBABILITIES
JOINT
PROBABILITIES
REVISED
PROBABILITIES
No Growth
(s1)
P (s1)=0.65
P (F1 |s1)=0.80
P (F1 s1)=0.520
0.520/0.625 = 0.832
Rapid
Growth
(s2)
P (s2)=0.35
P (F1 |s2)=0.30
P (F1 s2)=0.105
0.105/0.625 = 0.168
P (F1)=0.625
Revision Based on Forecast of Rapid Growth (F2)
STATE OF
ECONOMY
PRIOR
PROBABILITIES
CONDITIONAL
PROBABILITIES
JOINT
PROBABILITIES
REVISED
PROBABILITIES
No Growth
(s1)
P (s1)=0.65
P (F1 |s1)=0.20
P (F1 s1)=0.130
0.130/0.375 = 0.347
Rapid
Growth
(s2)
P (s2)=0.35
P (F1 |s2)=0.70
P (F1 s2)=0.245
0.245/0.375 = 0.653
P (F1)=0.375
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