Introduction to Probabilistic Analysis. 2 2.07 Introduction to Probabilistic Analysis The third phase of the cycle incorporates uncertainty into the analysis

Introduction to Probabilistic Analysis

22.07 • Introduction to Probabilistic Analysis

The third phase of the cycle incorporates uncertainty into the analysis of the decision.

Strategy Table

DecisionStructure

DeterministicAnalysis

ProbabilisticAnalysis

AppraisalInitial

Situation

Iteration

InfluenceDiagram

12345

DeterministicModelA B C

DeterministicSensitivity

DecisionTree

ProbabilityDistributions

Value ofInformation

Decision Quality


We will review terminology and probability calculations used in probabilistic analysis.

EV

• Cumulative Probability Distributions

• Probability Trees

• Decision Trees & Expected Values25

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We will work with probabilities associated with discrete events and continuous variables.

* Probability of cost less than or equal to any given value.

RainProbability = p

Probability = 1 – p

Discrete Event

No Rain

Cost ($ millions)

0

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1.0

0 50 100 150 200 250

Continuous Variable

Cumulative Probability*


Probability nodes represent discrete, uncertain events in probability and decision trees.

Anatomy of a Single Probability Node

.25

.25

.50

Higher

No Change

Lower

Outcome

Branch(one for each outcome)

Probability(sum to 1.0)

Outcomesare

mutuallyexclusive

Outcomesare

collectivelyexhaustive

Uncertainty associated with continuous variables can be represented in a tree using a discrete approximation.

Price Next Month


Two events may be probabilistically independent or dependent.

Independent

500

200

500

200

200

100

.5

.5

.6

.4

.6

.4

Market Price($/ton)

Sales Volume(thousand tons)

Dependent

500*

200

500*

200

200

100

.5

.5

.4

.6

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.2

* Outcomes could change alongwith or instead of probabilities.

Market Price($/ton)


ConditionalProbability

MarginalProbability

The order of adjacent probability nodes can be reversed.


A “joint probability distribution” can be computed from data in the probability tree.

Market Price($/ton)


500

200

500

200

200

100

.5

.5

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.6

.8

.2

Revenues($ millions)

JointProbability

100

40

50

20

.2*

•* .5 x .4 = .2

.3

.4

.1


Sometimes it is necessary to switch the conditioning variable.

The information is available in this order.

Test Result

ActualEvent

Not Sick

“Negative”.98

.02

.99

.001

.999“Positive”

“Negative”.01

“Positive”

Sick

But we want to use the informationin this order.

Actual Event

Test Result

Sick

Sick

“Positive”

“Negative”

Not Sick

Not Sick

What probability would you assign to being sick, given a positive test result?


We “flip” the tree using a process called “Bayesian Revision” of probabilities.

1) Begin by computing joint probabilities

Test Result

ActualEvent

Not Sick

“Negative”.98

.02

.99

.001

.999“Positive”

“Negative”.01

“Positive”

Sick

.00099

.00001

.01998

.97902

Joint Probability

2) Transfer joint probabilities to

corresponding joint events

Actual Event

Test Result

Sick

Sick

“Positive”

“Negative”

Not Sick

Not Sick

.00099

.00001

.01998

.97902

Joint Probability

3) Add joints to get

marginal probs.

.02097

4) Divide to get

conditional probabilities

~.001/.021 =.047

Does the resulting .047 probability of sick surprise you, given the test accuracy?



EV




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A cumulative probability distribution shows the probability that a variable will be less than or equal to any given value.

CumulativeProbability*

Cost ($ millions)

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0 50 100 150 200 250

*Probability that cost (in this case) is less than or equal to ____.

The complementary cumulative (drawn down from the top) shows the probability of exceeding any given value.


The cumulative probability distribution displays information decision-makers need.

*Probability that cost is less than or equal to a given value.

CumulativeProbability*

Cost ($ millions)

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One chance in 10 that cost will begreater than $180 million

“Median” cost is $14 million(equal chance above or below)

One chance in 10 thatcost will be $110 millionor less

80%chancethat costwill be$110 millionto$180 million


Cumulative probability distributions can be plotted for discrete and continuous variables.

Cost ($ millions)

Cumulative Probability

0

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0 50 100 150 200 250

Continuous Variable

Cumulative Probability

Days of Rain Next Week

0

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1.0

0 1 2 3 54

Discrete Variable

6 7


Let’s review how to construct a cumulative probability distribution in discrete form.

Market Price($/ton)


500

200

500

200

200

100

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100

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20

CumulativeProbability

Revenues ($ millions)

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Discrete Cumulative Probability Distribution

Why is this a step function?


Begin by computing the value (revenues) and joint probability for each endpoint.

.2*

* . 5 x .4 = .2

Market Price($/ton)


500

200

500

200

200

100

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.5

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.2


100

40

50

20

Joint Probabilit

y

.3

.4

.1


Next, list and rank unique profit outcomes, joint probabilities, and cumulative probabilities.

Tree Endpoints


JointProbability

100 .2

40

50

20

.3

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.1

Probability Distribution


20

40

50

100

JointProbability

.1

.3

.4

.2

Cumulative*Probability

.1

.4

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1.0

*Probability that revenues are less than or equal to _____.


Plotting the cumulative distribution shows the range of outcomes and associated probabilities.

Discrete Cumulative Probability Distribution


Revenues ($ millions)

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1.0

0

0 20 40 60 80 100 120


0

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Cumulative distributions for continuous variables are constructed by connecting cumulative points.

Values on the horizontal axis are called “percentiles” (e.g., $110 million and $180 million are the 10th and 90th percentiles, respectively).

Cumulative*Probability

Cost($ millions)

Assessed Cumulative

Probability

60

110

140

180

230

.01

.10

.50

.90

.99

Continuous Cumulative Probability Distribution

Cost ($ millions)*Probability that cost is less than or equal to ____.


0

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1.0

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Cumulative

Probability

Cumulative Probability Distribution

Cost ($ millions)

Continuous variables also can be plotted as “probability density functions.”

Probability

Density

Cost ($ millions)

0 50 100 150 200 250

Probability Density Function


Cumulative

Probability

Cost ($ millions)

0

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1.0

0 50 100 150 200


Probability

Density

0 50 100 150 200 250

Probability Density Function

The cumulative form is easier to use for assessing and making calculations with probabilities.

250

Probabilitythat cost is lessthan or equal to$120 million

Cost ($ millions)


“Flying bars” highlight differences in probability distributions for many alternatives.

Strategy 5

Strategy 4

Strategy 3

Strategy 2

Strategy 1

“Flying Bar” Comparison of Strategy Risks

–200 –150 –100 –50 0 50 100 150 200 250 300 350 400

Net Present Value ($ millions)

*

*

*

*

*

*1st 10th Percentiles

90th 99th

*Expected Value

Legend


The mean, median, and mode all can be used to describe distributions, depending on which characteristics are important.

MedianMode

MeanMode

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Mean

Median

Probability Density FunctionCumulative

Probability Distribution

Parameter MeaningMean Expected value; probability-weighted averageMedian 50th percentileMode Most likely value



EV




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The expected value (EV) is a single number that can represent an entire probability distribution.

Discrete Variable


500

200

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EV = 380thousand tons

EV = 380thousand tons

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Cost ($ millions)

EV = $141 million

The expected value is a “probability-weighted average.” “Mean” is synonymous with expected value.


Use a right-to-left rollback procedure to compute expected values for probability trees.

Market Price($/ton)


500

200

500

200

200

100

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100

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20

$64

$44

EV ofRevenue =$54 million

EV ofRevenue =$54 million

The rollback proceeds right to left, one node at a time: e.g., $64 = .4 x $100 + .6 x $40.

Box IndicatesExpectedValue


Use the same rollback procedure for decision trees, choosing the best expected value at decisions.

.5

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44Indicates expected value.

4

Indicates preferred alternative for an expected value decision-maker.

30

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44

48

60

60

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44

Decision Uncertainty UncertaintyNet Value ofOutcomesDecision

PlanA

PlanB


“Inside a complicated problem there may be a simple problem waiting to emerge!”

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44Indicates expected value.

4

30

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48

60

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44

Decision Uncertainty UncertaintyNet Value ofOutcomesDecision

Is the initial choice between alternatives clearer now, once the inferior choices are removed?

PlanA

PlanB


The expected value of a cumulative distribution is the point where two areas are equal.

Cumulative

Probability*

Cost ($ millions)

0

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Continuous Variable

Area C =Area D

* Probability that cost is less than or equal to ____.

EV =$141 million

7

Cumulative

Probability*

Days of Rain Next Week

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Discrete Variable

6

EV =3.1 days

Area A =Area B

A

B



EV




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Documents

Introduction to Probabilistic Analysis. 2 2.07 Introduction to Probabilistic Analysis The third phase of the cycle incorporates uncertainty into the analysis