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EMSE 269 - Elements of Problem Solving and Decision Making
4/29/17
7. Probability Basics
A = an event with possible outcomes A1,, An ;

= Total Event
Venn Diagrams:
Ratio of area of the event and the area of the total
rectangle can be interpreted as the probability of the event
 Probabilities must lie between 0 and 1:
0  Pr( A1 )  1, A1  
A1

 Probabilities must add up:
A1  A2    Pr( A1  A2 )  Pr( A1 )  Pr( A2 )
A2
A1

 Total Probability Must Equal 1:
A  A
i
j
  , i  j  3i 1 Ai    Pr( 3i 1 Ai )  1
A1
A2
A3

Instructor: Dr. J. Rene van Dorp
Session 3 - Page 32
Source: Making Hard Decisions, An Introduction to Decision Analysis by R.T. Clemen
EMSE 269 - Elements of Problem Solving and Decision Making
A1
A1
4/29/17
A1
A2

Note:


Pr( A1 )  1  Pr( A1 )
Pr( A1  A2 )  Pr( A1 )  Pr( A2 )  Pr( A1  A2 )
Conditional Probability:

Dow Jones Up
Stock Price Up
{
New Total Event based on condition
that we know that Dow Jones went up
Pr( Stock | Dow ) 
Pr( Stock   Dow )
Pr( Dow )
Informally: Conditioning on an event coincides with
reducing the total event to the conditioning event
Instructor: Dr. J. Rene van Dorp
Session 3 - Page 33
Source: Making Hard Decisions, An Introduction to Decision Analysis by R.T. Clemen
EMSE 269 - Elements of Problem Solving and Decision Making
4/29/17
Pr( A1  B1 )
Pr( B1 )
Pr( A1  B1 )
Pr( B1 | A1 ) 
Pr( A1 )
Pr( A1 | B1 ) 
Thus:
Also:
Note: Pr( A1  B1 )  Pr( B1 | A1 )  Pr( A1 )  Pr( A1 | B1 )  Pr( B1 )
 Independence
A with possible outcomes A1 ,, An ;
B ,, Bm
2. Event B with possible outcomes 1
1. Event
Event
A and Event B
are independent:
1.  Pr( Ai | B j )  Pr( Ai ), Ai , B j
or
2.  Pr( B j | Ai )  Pr( B j ), Ai , B j
or
3.  Pr( Ai  B j )  Pr( Ai )  Pr( B j ), Ai , B j .
Informally: Information about A does not
tell me anything about B and vice versa
Independence in Influence Diagrams:
 No arrow between two chance nodes implies
independence between the uncertain events
Instructor: Dr. J. Rene van Dorp
Session 3 - Page 34
Source: Making Hard Decisions, An Introduction to Decision Analysis by R.T. Clemen
EMSE 269 - Elements of Problem Solving and Decision Making
4/29/17
 An arrow from a chance event A to a chance event B does
not mean that "A causes B". It indicates that information
about A helps in determining the likeliness of outcomes of
B.
 Conditional Independence
A and Event B are conditionally independent given
C , , C p :
event C with possible outcomes 1
Event
1.  Pr( Ai | B j , Ck )  Pr( Ai | Ck ), Ai , B j , Ck
or
2.  Pr( B j | Ai , Ck )  Pr( B j | Ck ), Ai , B j , Ck
or
3.  Pr( Ai  B j | Ck )  Pr( Ai | Ck )  Pr( B j | Ck ), Ai , B j , Ck
Informally: If I already know C, information about A
does not tell me anything about B and vice versa
Conditional Independence in Influence Diagrams:
C
C
A
B
A
B
Instructor: Dr. J. Rene van Dorp
Session 3 - Page 35
Source: Making Hard Decisions, An Introduction to Decision Analysis by R.T. Clemen
EMSE 269 - Elements of Problem Solving and Decision Making
4/29/17
 Law of Total Probability:
B1 ,, B3 mutually exclusive, collectively exhaustive:
Pr( A1 )  Pr( A1  B1 )  Pr( A1  B2 )  Pr( A1  B3 ) 
Pr( A1 )  Pr( A1 | B1 ) Pr( B1 )  Pr( A1 | B2 ) Pr( B2 )  Pr( A1 | B3 ) Pr( B3 )
A1
B1
B3
B2
Example Law of Total Probability:
SYSTEM: X, X=failure , X= No Failure
B
A
C
1. Pr( X )  Pr( X | A) Pr( A)  Pr( X | A ) Pr( A )  1  Pr( A)  Pr( X | A ) Pr( A )
2. Pr( X | A )  Pr( X | B, A ) Pr( B | A )  Pr( X | B , A ) Pr( B | A ) 
Pr( X | B, A ) Pr( B)  0  Pr( B )  Pr( X | B, A ) Pr( B)
Instructor: Dr. J. Rene van Dorp
Session 3 - Page 36
Source: Making Hard Decisions, An Introduction to Decision Analysis by R.T. Clemen
EMSE 269 - Elements of Problem Solving and Decision Making
4/29/17
3. Pr( X )  Pr( A)  Pr( X | B, A ) Pr( B) Pr( A )
4. Pr( X | B, A )  Pr( X | C, B, A ) Pr(C | B, A )  Pr( X | C , B, A ) Pr(C | B, A ) 
Pr( X | B, A )  1  Pr(C )  0  Pr(C )  Pr(C )
5. Pr( X )  Pr( A)  Pr(C ) Pr( B) Pr( A ) 
Pr( A)  Pr(C ) Pr( B )  Pr(C ) Pr( B ) Pr( A)
 Bayes Theorem
B1 ,, B3 mutually exclusive, collectively exhaustive:
A1
B1
B3
B2
1. Pr( A1  B j )  Pr( B j | A1 ) Pr( A1 )  Pr( A1 | B j ) Pr( B j )
2. Pr( B j | A1 ) 
Pr( A1 | B j ) Pr( B j )
Pr( A1 )
3. Pr( A1 )  Pr( A1 | B1 ) Pr( B1 )  Pr( A1 | B2 ) Pr( B2 )  Pr( A1 | B3 ) Pr( B3 )
4. Pr( B j | A1 ) 
Pr( A1 | B j ) Pr( B j )
Pr( A1 | B1 ) Pr( B1 )  Pr( A1 | B2 ) Pr( B2 )  Pr( A1 | B3 ) Pr( B3 )
Instructor: Dr. J. Rene van Dorp
Session 3 - Page 37
Source: Making Hard Decisions, An Introduction to Decision Analysis by R.T. Clemen
EMSE 269 - Elements of Problem Solving and Decision Making
4/29/17
Example: Oil Wildcatter Problem
Max Profit
Dry (?)
Drill at Site 1
-100K
Low (?)
High (?)
Dry (0.2)
150K
500K
-200K
Drill at Site 2
Low (0.8)
50K
Payoff at site 1 is uncertain. Dominating factor in eventual
payoff is the presence of a dome or not.
Pr(Dome)
0.600
Outcome
Dry
Low
High
Pr(Outcome|Dome)
0.600
0.250
0.150
Pr(No Dome)
0.400
Outcome
Dry
Low
High
Pr(Outcome|No Dome)
0.850
0.125
0.025
 Law of Total Probability
Pr( Dry )  Pr( Dry | Dome) Pr( Dome)  Pr( Dry | NoDome) Pr( NoDome)
Pr( Dry )  0.600  0.600  0.850 * 0.400  0.700
Pr( Low)  Pr( Low | Dome) Pr( Dome)  Pr( Low | NoDome) Pr( NoDome)
Pr( Low)  0.250  0.600  0.125 * 0.400  0.200
Pr( High )  Pr( High | Dome) Pr( Dome)  Pr( High | NoDome) Pr( NoDome)
Pr( Low)  0.150  0.600  0.025 * 0.400  0.100
Instructor: Dr. J. Rene van Dorp
Session 3 - Page 38
Source: Making Hard Decisions, An Introduction to Decision Analysis by R.T. Clemen
EMSE 269 - Elements of Problem Solving and Decision Making
4/29/17
Dry (0.600)
Dome (0.600)
-100K
Low (0.250)
150K
High (0.150)
Dry (0.850)
No Dome (0.400)
500K
-100K
Low (0.125)
150K
High (0.025)
500K
LAW OF TOTAL PROBABILITY
Dry (0.600 0.600 + 0.850 0.400 = 0.70)
Low (0.250 0.600 + 0.125 0.400 = 0.20)
High (0.150 0.600 + 0.025 0.400 = 0.10)
-100K
150K
500K
 Bayes Theorem
We drilled at site 1 and the well is a high producer. Given
this new information what are the chances the a dome
exists?
Pr( Dome | High ) 
Pr(High|Dome) Pr(Dome)
Pr(High)
3.
Pr( Dome | High ) 
Pr(High|Dome) Pr( Dome)
Pr(High|Dome) Pr( Dome)  Pr( High|NoDome) Pr( NoDome)
4.
.150*0.600
Pr( Dome | High )  0.150*00.600
 0.0250*0.400  0.90
1.
2. Pr( High )  Pr( High | Dome) Pr( Dome)  Pr( High | NoDome) Pr( NoDome)
Pr(Dome) - Prior Probability
Pr(Dome|Data) - Posterior Probability
Instructor: Dr. J. Rene van Dorp
Session 3 - Page 39
Source: Making Hard Decisions, An Introduction to Decision Analysis by R.T. Clemen
EMSE 269 - Elements of Problem Solving and Decision Making
Dry (0.600)
Dome (0.600)
4/29/17
-100K
Low (0.250)
150K
High (0.150)
Dry (0.850)
No Dome (0.400)
Low (0.125)
High (0.025)
BAYES THEOREM
Dome (?)
-100K
150K
500K
-100K
Dry (0.7)
No Dome (?)
-100K
Dome (?)
150K
Low (0.2)
No Dome (?)
150K
Dome (0.90)
500K
High (0.10)
No Dome (?)
500K
When we reverse the order of chance nodes in a decision
tree we need to apply Bayes Theorem
 Calculating posterior probabilities using a Table
Pr(Dome)
Pr(No Dome)
0.600
0.400
X
Pr(X|Dome)
Pr(X|No Dome)
Dry
0.600
0.850
Low
0.250
0.125
High
0.150
0.025
Check
1.000
1.000
Pr(X  Dome) Pr(X  No Dome)
Pr(X)
Pr(Dome|X)
Pr(No Dome|X) Check
Instructor: Dr. J. Rene van Dorp
Session 3 - Page 40
Source: Making Hard Decisions, An Introduction to Decision Analysis by R.T. Clemen
EMSE 269 - Elements of Problem Solving and Decision Making
4/29/17
Example: Game Show
Suppose we have a game show host and you. There are
three doors and one of them contains a prize. The game
show host knows the door containing the prize but of course
does not convey this information to you. He asks you to pick
a door. You picked door 1 and are walking up to door 1 to
open it when the game show host screams: STOP. You stop
and the game show host shows door 3 which appears to be
empty. Next, the game show asks.
"DO YOU WANT TO SWITCH TO DOOR 2?"
WHAT SHOULD YOU DO?
Assumption 1: The game show host will never show the
door with the prize.
Assumption 2: The game show will never show the door
that you picked.
 Di ={Prize is behind door i }, i=1,…,3
 Hi ={Host shows door i containing no prize after you
selected Door 1}, i=1,…,3
Initially: Pr( Di ) 
3
1
3
1 1
2 3
1
3
1
3
1. Pr( H 3 )   Pr( H 3 | Di ) Pr( Di )  *  1 *  0 * 
i 1
2.
3.
1
2
1 1
*
Pr( H 3 | D1 ) Pr( D1 ) 2 3 1
Pr( D1 | H 3 ) 


1
Pr( H 3 )
3
2
1 2
Pr( D2 | H 3 )  1  Pr( D1 | H 3 )  1   .
3 3
So Yes, you should switch!
Instructor: Dr. J. Rene van Dorp
Session 3 - Page 41
Source: Making Hard Decisions, An Introduction to Decision Analysis by R.T. Clemen
EMSE 269 - Elements of Problem Solving and Decision Making
4/29/17
Uncertain Quantities & Random Variables
 Event
A with possible outcomes A1,, An

A
: {Number of Raisins in an oatmeal cookie}

Ai
: {i Raisins in an oatmeal cookie}, i=1,2, …., n
n
   i1 Ai : Total Event or Sample Space
A Random Variable Y {=Uncertain Quantity} is
a function from
R
Define:
Y :=# Raisins in a oatmeal cookie
Then:
Y ( Ai )  yi  i
often abbreviated to
Y  yi .
 When number of outcomes of the event A is finite, Y is a
discrete random variable.
Discrete Probability Distribution:
 The collection of probabilities associated with each
possible outcome of Y is called the discrete probability
distribution. Thus, if we denote Pr(Y  yi )  pi
Discrete probability distribution of Y : { p1 , pn }
Instructor: Dr. J. Rene van Dorp
Session 3 - Page 42
Source: Making Hard Decisions, An Introduction to Decision Analysis by R.T. Clemen
EMSE 269 - Elements of Problem Solving and Decision Making
n
 Note:
p
i 1
i
4/29/17
 1, pi  0. Other common notation:
 fY ( yi )  pi , i  1,, n

 fY ( y )  0, y  yi , i  1,, n
D iscrete Probability D istribution
0.4
0.3
Pr(Y= y)
0.2
0.1
0
Pr(Y = y)
1
2
3
4
5
0.1
0.15
0.3
0.35
0.1
Y
Cumulative Probability Distribution (CDF):
FY ( y)  Pr(Y  y)
C um ula tive D is tribution F unc tion
1.00
0.95
0.90
0.85
0.80
0.75
0.70
0.65
0.60
0.55
P r(Y <=y )
0.50
0.45
0.40
0.35
0.30
0.25
0.20
0.15
0.10
0.05
0.00
0
1
2
3
4
5
6
7
Y
Instructor: Dr. J. Rene van Dorp
Session 3 - Page 43
Source: Making Hard Decisions, An Introduction to Decision Analysis by R.T. Clemen
EMSE 269 - Elements of Problem Solving and Decision Making
4/29/17
In Decision Analysis a CDF is referred to as a
CUMMULATIVE RISK PROFILE
Expected Value of Y:
n
n
i 1
i 1
EY [Y ]   yi  Pr(Y  yi )   yi  pi
1 Raisin (0.10)
3.20
2 Raisins (0.15)
3 Raisins (0.30)
1
2
3
4 Raisins (0.35)
4
5 Raisins (0.10)
5
#Raisins #Raisins*Pr(Y=#Raisins)
1
1*0.10=0.10
2
2*0.15=0.30
3
3*0.30=0.90
4
4*0.35=1.40
5
5*0.10=0.50
3.20
Interpretation:
 E[Y] is “best guess” of Y
 If you were able to observe many outcomes of Y the
calculated average of all the outcomes would be close to
E[Y].
Calculation Rules:
1. Let Z be a function of Y, i.e. Z=g(Y). As Y is a random
variable, Z is a random variable and:
n
n
i 1
i 1
EZ [ Z ]   g ( yi )  Pr(Y  yi )  g ( yi )  pi
2.
g (Y )  a  Y  b  EZ [Z ]  a  EY [Y ]  b
Instructor: Dr. J. Rene van Dorp
Session 3 - Page 44
Source: Making Hard Decisions, An Introduction to Decision Analysis by R.T. Clemen
EMSE 269 - Elements of Problem Solving and Decision Making
4/29/17
3. Let X, Y be two random variables and Z=X+Y, then:
EZ [Z ]  EX [ X ]  EY [Y ]
Variance and Standard Deviation of Y:
Variance:
Var(Y )   Y2  E[Y  E[Y ] ]
2
 E[Y 2  2  Y  E[Y ]  E 2 [Y ]]
 E[Y 2 ]  2  E[Y ]  E[Y ]  E 2 [Y ]
 E[Y 2 ]  E 2 [Y ]
Standard Deviation :
 Y2   Y
Interpretation:
 Standard deviation is the best guess distance from the
mean for an arbritrary outcome
Calculation Rules:
1. Let Z, Y be random variables such that: Z=g(Y)
g (Y )  a  Y  b  Var( Z )  a 2 Var(Y )
2. Let Xi , i=1,…,n be a collection of independent random
variables.
n
n
i 1
i 1

Y   ai  X i  bi   Var(Y )   ai  Var( X i )
2

Instructor: Dr. J. Rene van Dorp
Session 3 - Page 45
Source: Making Hard Decisions, An Introduction to Decision Analysis by R.T. Clemen
EMSE 269 - Elements of Problem Solving and Decision Making
4/29/17
Example:
Max Profit
(0.24)
$35.75
A
(0.47)
$20
$35
(0.29)
$50
(0.25)
$35.75
B
(0.35)
(0.40)
-$9
$0
$95
Alternative A
Prob
Profit
0.24
0.47
0.29
Profit^2
20
35
50
Prob*Profit
400
1225
2500
4.80
16.45
14.50
 E [Y ]
Prob*(Profit^2) Variance St. Dev
96.00
575.75
725.00
35.75
1278.0625
 E [Y ]
2
1396.75
 E [Y ]
2
118.69 10.89438
 E [Y 2 ]  E 2 [Y ]
Y
Alternative B
Prob
Profit
0.25
0.35
0.4
Profit^2
-9
0
95
81
0
9025
Prob*Profit
-2.25
0.00
38.00
Prob*(Profit^2) Variance St. Dev
20.25
0.00
3610.00
35.75
1278.0625
3630.25
2352.19 48.49936
Notes:
 B has high possible yield, but also high risk
 Pr( Profit  0 | A)  0; Pr( Profit  0 | B )  0.6
Instructor: Dr. J. Rene van Dorp
Session 3 - Page 46
Source: Making Hard Decisions, An Introduction to Decision Analysis by R.T. Clemen
EMSE 269 - Elements of Problem Solving and Decision Making
4/29/17
Example: Oil Wildcatter Problem
Max Profit
Dry (0.7)
-100K
10K
Low (0.2)
Drill at Site 2
150K
High (0.1)
500K
Dry (0.2)
0K
-200K
Drill at Site 2
Low (0.8)
50K
D r ill at S ite 1
Pr o fit
Prob
-10 0
15 0
50 0
0.7
0.2
0.1
P r o b * (P r o fit ^ 2) V ar ianc e S t. De v
P r o b * Pr o fit
P r o fit^ 2
-70 .0 0
30 .0 0
50 .0 0
1 000 0
2 250 0
25 000 0
7000 .0 0
4500 .0 0
25000 .0 0
10 .0 0
10 0
36500 .0 0 36400 .0 0 190.787 8
D r ill at S ite 2
P r o fit
Prob
-200
50
0.2
0.8
P r o b * (P r o fit^ 2) V ar ian ce S t. D ev
P r o b * P r o fit
P r o fit^ 2
40000
2500
-40.00
40.00
8000.00
2000.00
0.00
0
10000.00 10000.00
100
Max Profit
Dry (0.60)
-100K
EMV=52.50K
Low (0.25)
Dome (0.6)
150K
EMV=10K
Drill at Site 1
High (0.15)
Prob
Pro fit
Prob *Pro fit
0.6 00
52. 50
31. 50
0.4 00
-53. 75
-21. 50
10. 00
Dry (0.850)
500K
-100K
EMV=-53.75K
Dome (0.4)
Low (0.125)
150K
High (0.025)
Prob
Profit
Prob*Profit
0.600 -100.00
-60.00
0.250
150.00
37.50
0.150
500.00
75.00
52.50
Prob
Profit
Prob*Profit
0.850
-100.00
-85.00
0.125
150.00
18.75
0.025
500.00
12.50
-53.75
500K
Dry (0.2)
-200K
EMV=0K
Prob
Profit
Prob*Profit
0.200
-200.00
-40.00
0.800
50.00
40.00
0.00
Drill at Site 2
Low (0.8)
50K
Instructor: Dr. J. Rene van Dorp
Session 3 - Page 47
Source: Making Hard Decisions, An Introduction to Decision Analysis by R.T. Clemen
EMSE 269 - Elements of Problem Solving and Decision Making
4/29/17
Continuous Random Variables
 Event A with possible outcomes [0, )

A
: {A components failure}
 At
   [0, )
: {component fails at time t}
: Total Event or Sample Space
A Random Variable Y {=Uncertain Quantity} is
a function from
R
Define:
Y :=# failure time of the component
Then:
Y ( At )  t
often abbreviated to
Y t.
 When the number of outcomes of the event A is infinite
and uncountable, Y is a continuous random variable.
Continuous Probability Density function:
 Pr(Y  y )  0 for any value of y in the range of Y.
 Pr(Y  [a, b])  0 , if a  b, and [a, b] falls within the range of
Y.
 Probability Density Function fY ( y )  0

  f Y ( y )dy  1
0
b
 f ( y )dy  0
a Y
Instructor: Dr. J. Rene van Dorp
Session 3 - Page 48
Source: Making Hard Decisions, An Introduction to Decision Analysis by R.T. Clemen
EMSE 269 - Elements of Problem Solving and Decision Making
4/29/17
Informally:
 dy is a very small imnterval at y

fY ( y)dy  Pr( y  Y  y  dy)

f
Y
( y )dy   Pr( y  Y  y  dy )
y
Cumulative Probability Distribution (CDF):
y
FY ( y )  Pr(Y  y )   fY (u)du
0
1.00
1.00E+00
0.80
8.00E-01
0.60
6.00E-01
0.40
4.00E-01
0.20
2.00E-01
0.00
0.00E+00
1
11 21 31 41 51 61 71 81 91
f(y)
Pr(Y<=y)
Note: CDF is always always a non-decreasing function.
Examples:
fY ( y)    exp(   y)
 1

 Weibull Distribution : fY ( y )      y exp(   y )
 Exponential Distribution:
 Beta Distribution: f Y ( y ) 
(   )  1
y (1  y )  1
( )  (  )
Instructor: Dr. J. Rene van Dorp
Session 3 - Page 49
Source: Making Hard Decisions, An Introduction to Decision Analysis by R.T. Clemen
EMSE 269 - Elements of Problem Solving and Decision Making
4/29/17
All formulas for Expectation and Variance carry
over from discrete case to the continuous case
 Dominance and making decisions
DETERMINISTIC DOMINANCE
Assume random Variable X Uniformly Distributed on [A,B]
Assume random Variable Y Uniformly Distributed on [C,D]
PDF
X
Y
0
CDF
A
B
C
D
1
X
Y
0
A
B
C
D
STOCHASTIC DOMINANCE
Assume random Variable X Uniformly Distributed on [A,B]
Assume random Variable Y Uniformly Distributed on [C,D]
PDF
X
Note:
Pr(Y<z) < Pr(X< z)
for all z
Y
0
CDF
A
C
B
D
1
Y
X
0
A
C
B
D
Instructor: Dr. J. Rene van Dorp
Session 3 - Page 50
Source: Making Hard Decisions, An Introduction to Decision Analysis by R.T. Clemen
EMSE 269 - Elements of Problem Solving and Decision Making
4/29/17
CHOOSE ALTERNATIVE WITH BEST EMV
Assume random Variable X Uniformly Distributed on [A,B]
PDF
Assume random Variable Y Uniformly Distributed on [C,D]
X E(X) E(Y)
Y
0
CDF
CA
1
0
B D
X
Y
CA
B D
MAKING DECISIONS & RISK LEVEL
DETERMINISTIC DOMINANCE PRESENT
STOCHASTIC DOMINANCE PRESENT
Chances
of unlucky
outcome
Increases
CHOOSE ALTERNATIVE WITH BEST EMV
Instructor: Dr. J. Rene van Dorp
Session 3 - Page 51
Source: Making Hard Decisions, An Introduction to Decision Analysis by R.T. Clemen