1. Introduction to Probabilistic Analysis

2. 2 2.07 Introduction to Probabilistic Analysis The third phase of the cycle incorporates uncertainty into the analysis of the decision. Strategy Table Decision Structure Deterministic Analysis Probabilistic Analysis Appraisal Initial Situation Iteration Influence Diagram 1 2 3 4 5 Deterministic Model ABC Deterministic Sensitivity Decision Tree Probability Distributions Value of Information Decision Quality

3. 3 2.07 Introduction to Probabilistic Analysis We will review terminology and probability calculations used in probabilistic analysis. EV Cumulative Probability Distributions Probability Trees Decision Trees & Expected Values 25 0 100.5 50

4. 4 2.07 Introduction to Probabilistic Analysis We will work with probabilities associated with discrete events and continuous variables. * Probability of cost less than or equal to any given value. Rain Probability = p Probability = 1 – p Discrete Event No Rain Cost ($ millions) 0.2.4.6.8 1.0 050100150200250 Continuous Variable Cumulative Probability*

5. 5 2.07 Introduction to Probabilistic Analysis Probability nodes represent discrete, uncertain events in probability and decision trees. Anatomy of a Single Probability Node.25.50 Higher No Change Lower Outcome Branch (one for each outcome) Probability (sum to 1.0) Outcomes are mutually exclusive Outcomes are collectively exhaustive Uncertainty associated with continuous variables can be represented in a tree using a discrete approximation. Price Next Month

6. 6 2.07 Introduction to Probabilistic Analysis Two events may be probabilistically independent or dependent. Independent 500 200 500 200 100.5.6.4.6.4 Market Price ($/ton) Sales Volume (thousand tons) Dependent 500* 200 500* 200 100.5.4.6.8.2 *Outcomes could change along with or instead of probabilities. Market Price ($/ton) Sales Volume (thousand tons) Conditional Probability Marginal Probability The order of adjacent probability nodes can be reversed.

7. 7 2.07 Introduction to Probabilistic Analysis A “joint probability distribution” can be computed from data in the probability tree. Market Price ($/ton) Sales Volume (thousand tons) 500 200 500 200 100.5.4.6.8.2 Revenues ($ millions) Joint Probability 100 40 50 20.2* *.5 x.4 =.2.3.4.1

8. 8 2.07 Introduction to Probabilistic Analysis Sometimes it is necessary to switch the conditioning variable. The information is available in this order. Test Result Actual Event Not Sick “Negative”.98.02.99.001.999 “Positive” “Negative”.01 “Positive” Sick But we want to use the information in this order. Actual Event Test Result Sick “Positive” “Negative” Not Sick What probability would you assign to being sick, given a positive test result?

9. 9 2.07 Introduction to Probabilistic Analysis We “flip” the tree using a process called “Bayesian Revision” of probabilities. 1) Begin by computing joint probabilities Test Result Actual Event 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 “Positive” “Negative” 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?

10. 10 2.07 Introduction to Probabilistic Analysis We will review terminology and probability calculations used in probabilistic analysis. EV Cumulative Probability Distributions Probability Trees Decision Trees & Expected Values 25 0 100.5 50

11. 11 2.07 Introduction to Probabilistic Analysis A cumulative probability distribution shows the probability that a variable will be less than or equal to any given value. Cumulative Probability* Cost ($ millions) 0.2.4.6.8 1.0 050100150200250 *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.

12. 12 2.07 Introduction to Probabilistic Analysis The cumulative probability distribution displays information decision-makers need. *Probability that cost is less than or equal to a given value. Cumulative Probability* Cost ($ millions) 0.2.4.6.8 1.0 050100150200250 One chance in 10 that cost will be greater than $180 million “Median” cost is $14 million (equal chance above or below) One chance in 10 that cost will be $110 million or less 80% chance that cost will be $110 million to $180 million

13. 13 2.07 Introduction to Probabilistic Analysis Cumulative probability distributions can be plotted for discrete and continuous variables. Cost ($ millions) Cumulative Probability 0.2.4.6.8 1.0 050100150200250 Continuous Variable Cumulative Probability Days of Rain Next Week 0.2.4.6.8 1.0 012354 Discrete Variable 67

14. 14 2.07 Introduction to Probabilistic Analysis Let’s review how to construct a cumulative probability distribution in discrete form. Market Price ($/ton) Sales Volume (thousand tons) 500 200 500 200 100.5.4.6.8.2 Revenues ($ millions) 100 40 50 20 Cumulative Probability Revenues ($ millions) 0.2.4.6.8 1.0 020406080 100 Discrete Cumulative Probability Distribution Why is this a step function?

15. 15 2.07 Introduction to Probabilistic Analysis Begin by computing the value (revenues) and joint probability for each endpoint..2* *. 5 x.4 =.2 Market Price ($/ton) Sales Volume (thousand tons) 500 200 500 200 100.5.4.6.8.2 Revenues ($ millions) 100 40 50 20 Joint Probability.3.4.1

16. 16 2.07 Introduction to Probabilistic Analysis Next, list and rank unique profit outcomes, joint probabilities, and cumulative probabilities. Tree Endpoints Revenues ($ millions) Joint Probability 100.2 40 50 20.3.4.1 Probability Distribution Revenues ($ millions) 20 40 50 100 Joint Probability.1.3.4.2 Cumulative* Probability.1.4.8 1.0 *Probability that revenues are less than or equal to _____.

17. 17 2.07 Introduction to Probabilistic Analysis Plotting the cumulative distribution shows the range of outcomes and associated probabilities. Discrete Cumulative Probability Distribution Cumulative Probability Revenues ($ millions).2.4.6.8 1.0 0 020406080100120

18. 18 2.07 Introduction to Probabilistic Analysis 0.2.4.6.8 1.0 050100150200250 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 ____.

19. 19 2.07 Introduction to Probabilistic Analysis 0.2.4.6.8 1.0 050100150200250 Cumulative Probability Cumulative Probability Distribution Cost ($ millions) Continuous variables also can be plotted as “probability density functions.” Probability Density Cost ($ millions) 050100150200250 Probability Density Function

20. 20 2.07 Introduction to Probabilistic Analysis Cumulative Probability Cost ($ millions) 0.2.4.6.8 1.0 050100150200 Cumulative Probability Distribution Probability Density 050100150200250 Probability Density Function The cumulative form is easier to use for assessing and making calculations with probabilities. 250 Probability that cost is less than or equal to $120 million Cost ($ millions)

21. 21 2.07 Introduction to Probabilistic Analysis “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 –200–150–100–50050100150200250300350400 Net Present Value ($ millions) * * *** * 1 st 10 th Percentiles 90 th 99 th * Expected Value Legend

22. 22 2.07 Introduction to Probabilistic Analysis The mean, median, and mode all can be used to describe distributions, depending on which characteristics are important. Median Mode Mean Mode 0.2.4.6.8 1.0 Mean Median Probability Density Function Cumulative Probability Distribution ParameterMeaning MeanExpected value; probability-weighted average Median50 th percentile ModeMost likely value

23. 23 2.07 Introduction to Probabilistic Analysis We will review terminology and probability calculations used in probabilistic analysis. EV Cumulative Probability Distributions Probability Trees Decision Trees & Expected Values 25 0 100.5 50

24. 24 2.07 Introduction to Probabilistic Analysis The expected value (EV) is a single number that can represent an entire probability distribution. Discrete Variable Sales Volume (thousand tons) 500 200.6.4 EV = 380 thousand tons EV = 380 thousand tons 0.2.4.6.8 1.0 050100150200250 Cumulative Probability Cumulative Probability Distribution Cost ($ millions) EV = $141 million The expected value is a “probability-weighted average.” “Mean” is synonymous with expected value.

25. 25 2.07 Introduction to Probabilistic Analysis Use a right-to-left rollback procedure to compute expected values for probability trees. Market Price ($/ton) Sales Volume (thousand tons) 500 200 500 200 100.5.4.6.8.2 Revenues ($ millions) 100 40 50 20 $64 $44 EV of Revenue = $54 million EV of Revenue = $54 million The rollback proceeds right to left, one node at a time: e.g., $64 =.4 x $100 +.6 x $40. Box Indicates Expected Value

26. 26 2.07 Introduction to Probabilistic Analysis Use the same rollback procedure for decision trees, choosing the best expected value at decisions..5.2.8.3.7.3.7.5 50 10 100 –20 50 20 60 20 30 44 Indicates expected value. 4 Indicates preferred alternative for an expected value decision-maker. 30 50 44 48 60 20 40 44 DecisionUncertainty Net Value of Outcomes Decision Plan A Plan B

27. 27 2.07 Introduction to Probabilistic Analysis “Inside a complicated problem there may be a simple problem waiting to emerge!”.5.2.8.3.7.3.7.5 50 10 100 –20 50 20 60 20 30 44 Indicates expected value. 4 30 50 44 48 60 20 40 44 DecisionUncertainty Net Value of Outcomes Decision Is the initial choice between alternatives clearer now, once the inferior choices are removed? Plan A Plan B

28. 28 2.07 Introduction to Probabilistic Analysis The expected value of a cumulative distribution is the point where two areas are equal. Cumulative Probability* Cost ($ millions) 0.2.4.6.8 1.0 050100150200250 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 0.2.4.6.8 1.0 012354 Discrete Variable 6 EV = 3.1 days Area A = Area B A B

29. 29 2.07 Introduction to Probabilistic Analysis We will review terminology and probability calculations used in probabilistic analysis. EV Cumulative Probability Distributions Probability Trees Decision Trees & Expected Values 25 0 100.5 50

30. 30 2.07 Introduction to Probabilistic Analysis Change Log VersionDateChanges 0108/06/27First version for DAF - SDG LS (modified by D Wolter from DCW IPA v2.15) 0209/01/20Updated dates to >2005 and increased values in examples by 10x. (I Garrido) 0310/01/09Updated to IMS Template (MC) 407/21/10 Updated with minor revisions 506/05/12Updated to IMSCG Template (LJ)