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Introduction to Probabilistic Analysis

Introduction to Probabilistic Analysis. Structure. Deterministic Analysis. Probabilistic Analysis. Appraisal. Decision. Initial Situation. Iteration. Deterministic Model. Decision Tree. Value of Information. Strategy Table. A. B. C. 1. 2. 3. 4. 5. Influence Diagram.

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Introduction to Probabilistic Analysis

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  1. Introduction to Probabilistic Analysis

  2. Structure Deterministic Analysis Probabilistic Analysis Appraisal Decision InitialSituation Iteration DeterministicModel DecisionTree Value ofInformation Strategy Table A B C 1 2 3 4 5 InfluenceDiagram DeterministicSensitivity ProbabilityDistributions Decision Quality The third phase of the cycle incorporates uncertainty into the analysis of the decision. 2.07 • Introduction to Probabilistic Analysis

  3. 100 50 .5 0 .5 EV 25 We will review terminology and probability calculations used in probabilistic analysis. • Probability Trees • Cumulative Probability Distributions • Decision Trees & Expected Values 2.07 • Introduction to Probabilistic Analysis

  4. Continuous Variable 1.0 .8 Cumulative Probability* .6 .4 .2 0 0 50 100 150 200 250 Cost ($ millions) We will work with probabilities associated with discrete events and continuous variables. * Probability of cost less than or equal to any given value. Discrete Event Rain Probability = p No Rain Probability = 1 – p 2.07 • Introduction to Probabilistic Analysis

  5. Outcome Branch (one for each outcome) Outcomesare mutuallyexclusive Outcomesare collectivelyexhaustive Probability (sum to 1.0) Probability nodes represent discrete, uncertain events in probability and decision trees. Anatomy of a Single Probability Node Uncertainty associated with continuous variables can be represented in a tree using a discrete approximation. Price Next Month Higher .25 No Change .50 Lower .25 2.07 • Introduction to Probabilistic Analysis

  6. Dependent Market Price($/ton) Sales Volume (thousand tons) 500* Conditional Probability .4 200 .5 .6 200 Marginal Probability 500* .8 100 .5 .2 200 * Outcomes could change alongwith or instead of probabilities. Two events may be probabilistically independent or dependent. Independent The order of adjacent probability nodes can be reversed. Market Price($/ton) Sales Volume (thousand tons) 500 .6 200 .5 .4 200 500 .5 .6 100 .4 200 2.07 • Introduction to Probabilistic Analysis

  7. .3 .4 .1 A “joint probability distribution” can be computed from data in the probability tree. • * .5 x .4 = .2 Market Price ($/ton) Sales Volume (thousand tons) Revenues ($ millions) Joint Probability 500 100 .2* .4 200 .6 .5 200 40 500 50 .8 .5 100 .2 200 20 2.07 • Introduction to Probabilistic Analysis

  8. But we want to use the informationin this order. Test Result Actual Event Sick “Positive” Not Sick Sick “Negative” Not Sick Sometimes it is necessary to switch the conditioning variable. The information is available in this order. ActualEvent Test Result “Positive” .99 Sick .01 .001 “Negative” “Positive” .999 .02 Not Sick .98 “Negative” What probability would you assign to being sick, given a positive test result? 2.07 • Introduction to Probabilistic Analysis

  9. 2) Transfer joint probabilities to corresponding joint events 1) Begin by computing joint probabilities ActualEvent Test Result Joint Probability Joint Probability Test Result Actual Event “Positive” Sick 3) Add joints to get marginal probs. .00099 .00099 ~.001/.021 =.047 .99 Sick “Positive” .01 .001 .02097 “Negative” Not Sick .01998 .00001 4) Divide to get conditional probabilities “Positive” Sick .00001 .01998 .999 .02 “Negative” Not Sick .98 “Negative” Not Sick .97902 .97902 Does the resulting .047 probability of sick surprise you, given the test accuracy? We “flip” the tree using a process called “Bayesian Revision” of probabilities. 2.07 • Introduction to Probabilistic Analysis

  10. 100 50 .5 0 .5 EV 25 We will review terminology and probability calculations used in probabilistic analysis. • Probability Trees • Cumulative Probability Distributions • Decision Trees & Expected Values 2.07 • Introduction to Probabilistic Analysis

  11. 1.0 .8 Cumulative Probability* .6 *Probability that cost (in this case) is less than or equal to ____. .4 .2 0 0 50 100 150 200 250 Cost ($ millions) The complementary cumulative (drawn down from the top) shows the probability of exceeding any given value. A cumulative probability distribution shows the probability that a variable will be less than or equal to any given value. 2.07 • Introduction to Probabilistic Analysis

  12. One chance in 10 that cost will be greater than $180 million 80%chancethat costwill be$110 millionto$180 million “Median” cost is $14 million(equal chance above or below) One chance in 10 thatcost will be $110 millionor less The cumulative probability distribution displays information decision-makers need. 1.0 *Probability that cost is less than or equal to a given value. .8 Cumulative Probability* .6 .4 .2 0 0 50 100 150 200 250 Cost ($ millions) 2.07 • Introduction to Probabilistic Analysis

  13. Discrete Variable 1.0 Cumulative Probability .8 .6 .4 .2 0 0 1 2 3 4 5 6 7 Continuous Variable Days of Rain Next Week 1.0 .8 .6 Cumulative Probability .4 .2 0 0 50 100 150 200 250 Cost ($ millions) Cumulative probability distributions can be plotted for discrete and continuous variables. 2.07 • Introduction to Probabilistic Analysis

  14. Discrete Cumulative Probability Distribution 1.0 Cumulative Probability .8 .6 .4 .2 0 100 0 20 40 60 80 Revenues ($ millions) Why is this a step function? Let’s review how to construct a cumulative probability distribution in discrete form. Market Price ($/ton) Sales Volume (thousand tons) Revenues ($ millions) 500 100 .4 200 .5 .6 200 40 500 50 .5 .8 100 .2 200 20 2.07 • Introduction to Probabilistic Analysis

  15. Begin by computing the value (revenues) and joint probability for each endpoint. * . 5 x .4 = .2 Market Price ($/ton) Sales Volume (thousand tons) Revenues ($ millions) Joint Probability 500 100 .2* .4 200 .5 .6 200 40 .3 500 50 .4 .5 .8 100 .2 200 20 .1 2.07 • Introduction to Probabilistic Analysis

  16. Probability Distribution Revenues($ millions) JointProbability 20 .1 40 .3 50 .4 100 .2 Next, list and rank unique profit outcomes, joint probabilities, and cumulative probabilities. Tree Endpoints Revenues($ millions) JointProbability Cumulative*Probability 100 .2 .1 40 .3 .4 50 .4 .8 20 .1 1.0 *Probability that revenues are less than or equal to _____. 2.07 • Introduction to Probabilistic Analysis

  17. Plotting the cumulative distribution shows the range of outcomes and associated probabilities. Discrete Cumulative Probability Distribution 1.0 .8 Cumulative Probability .6 .4 .2 0 0 20 40 60 80 100 120 Revenues ($ millions) 2.07 • Introduction to Probabilistic Analysis

  18. Cumulative distributions for continuous variables are constructed by connecting cumulative points. Values on the horizontal axis are called “percentiles” (e.g., $110 million and $180 million are the 10th and 90th percentiles, respectively). Assessed Cumulative Probability Cost ($ millions) Continuous Cumulative Probability Distribution 1.0 .01 60 Cumulative* Probability .8 .10 110 .6 .50 140 .4 .90 180 .2 .99 230 0 0 50 100 150 200 250 *Probability that cost is less than or equal to ____. Cost ($ millions) 2.07 • Introduction to Probabilistic Analysis

  19. Cumulative Probability Distribution 1.0 .8 Cumulative Probability .6 .4 .2 0 0 50 100 150 200 250 Cost ($ millions) Continuous variables also can be plotted as “probability density functions.” Probability Density Function Probability Density 0 50 100 150 200 250 Cost ($ millions) 2.07 • Introduction to Probabilistic Analysis

  20. Probabilitythat cost is lessthan or equal to$120 million The cumulative form is easier to use for assessing and making calculations with probabilities. Probability Density Function Probability Density Cumulative Probability Distribution 1.0 0 50 100 150 200 250 .8 Cost ($ millions) .6 Cumulative Probability .4 .2 0 0 50 100 150 200 250 Cost ($ millions) 2.07 • Introduction to Probabilistic Analysis

  21. Legend 1st 10th Percentiles 90th 99th * * *Expected Value * * * * –200 –150 –100 –50 0 50 100 150 200 250 300 350 400 “Flying bars” highlight differences in probability distributions for many alternatives. “Flying Bar” Comparison of Strategy Risks Strategy 1 Strategy 2 Strategy 3 Strategy 4 Strategy 5 Net Present Value ($ millions) 2.07 • Introduction to Probabilistic Analysis

  22. The mean, median, and mode all can be used to describe distributions, depending on which characteristics are important. Cumulative Probability Distribution Probability Density Function 1.0 .8 Mode Mean .6 Median Median .4 Mean Mode .2 0 Parameter Meaning Mean Expected value; probability-weighted average Median 50th percentile Mode Most likely value 2.07 • Introduction to Probabilistic Analysis

  23. 100 50 .5 0 .5 EV 25 We will review terminology and probability calculations used in probabilistic analysis. • Probability Trees • Cumulative Probability Distributions • Decision Trees & Expected Values 2.07 • Introduction to Probabilistic Analysis

  24. Cumulative Probability Distribution 1.0 .8 Cumulative Probability .6 EV = $141 million .4 .2 0 0 50 100 150 200 250 Cost ($ millions) The expected value (EV) is a single number that can represent an entire probability distribution. The expected value is a “probability-weighted average.” “Mean” is synonymous with expected value. Discrete Variable Sales Volume (thousand tons) EV = 380 thousand tons 500 .6 .4 200 2.07 • Introduction to Probabilistic Analysis

  25. Use a right-to-left rollback procedure to compute expected values for probability trees. The rollback proceeds right to left, one node at a time: e.g., $64 = .4 x $100 + .6 x $40. Market Price ($/ton) Sales Volume (thousand tons) Revenues ($ millions) EV of Revenue = $54 million 500 100 $64 .4 200 Box Indicates Expected Value .5 .6 200 40 500 50 $44 .8 .5 100 .2 200 20 2.07 • Introduction to Probabilistic Analysis

  26. Net Value of Outcomes Decision Uncertainty Decision Uncertainty 30 50 .5 .5 30 10 Plan A 4 .3 100 44 .2 .8 –20 50 .7 44 50 48 20 .3 60 .7 60 Plan B 60 40 .5 60 20 .5 20 Indicates expected value. 44 Indicates preferred alternative for an expected value decision-maker. Use the same rollback procedure for decision trees, choosing the best expected value at decisions. 2.07 • Introduction to Probabilistic Analysis

  27. Net Value of Outcomes Decision Uncertainty Decision Uncertainty 30 50 .5 .5 30 10 Plan A 4 .3 100 44 .2 .8 –20 50 .7 44 50 48 20 .3 60 .7 60 Plan B 60 40 .5 60 20 .5 20 Indicates expected value. 44 Is the initial choice between alternatives clearer now, once the inferior choices are removed? “Inside a complicated problem there may be a simple problem waiting to emerge!” 2.07 • Introduction to Probabilistic Analysis

  28. Continuous Variable 1.0 Area C = Area D .8 Cumulative Probability* .6 EV = $141 million .4 .2 0 0 50 100 150 200 250 Cost ($ millions) Discrete Variable 1.0 Area A = * Probability that cost is less than or equal to ____. A .8 Cumulative Probability* Area B .6 .4 EV = B 3.1 days .2 0 0 1 2 3 4 5 6 7 Days of Rain Next Week The expected value of a cumulative distribution is the point where two areas are equal. 2.07 • Introduction to Probabilistic Analysis

  29. 100 50 .5 0 .5 EV 25 We will review terminology and probability calculations used in probabilistic analysis. • Probability Trees • Cumulative Probability Distributions • Decision Trees & Expected Values 2.07 • Introduction to Probabilistic Analysis

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