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Probability and Risk Analysis Part 1 of 2

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Overview Class Exercise Math review –Diagrams Tree diagrams, chance nodes, decision nodes, etc. –Concepts Probability, Mean, Variance, Expectation, etc. Sample problems

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Reading Assignment This Week, Read: 13.1, 13.2, 13.3, 13.4 (basics of probability and decision making under risk) Skip: 13.5, 13.6 (Monte Carlo Simulation) Next Week, Read: 13.7 (decision trees, value of information)

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Class Exercise For real money (as much as $1000) –There are three investments: A, B, and C –There are two risk factors: the economy and the number of competitors producing similar product. Procedure 1.You may vote for A,B, or C and also submit your name for the lucky draw to see who gets the money 2.The votes are totaled to see which investment the class has chosen: A, B, or C. (Ties are broken by 2 nd vote) 3.Lucky draw to see who receives the money 4.Instructor will flip coins to provide the random outcomes 5.The real money payoff is determined

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Investment A Random Factors EconomyCompetitorsProbabilityProfit StrongNone0.25$1000 StrongMany0.25$0 WeakNone0.25$0 WeakMany0.25$0

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Investment B Random Factors EconomyCompetitorsProbabilityPayoff StrongNone0.25$250 StrongMany0.25$250 WeakNone0.25$150 WeakMany0.25$150 Note: Investment B is not sensitive to competition

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Investment C EconomyCompetitorsProbabilityPayoff StrongNone0.25$450 StrongMany0.25$300 WeakNone0.25$150 WeakMany0.25$0

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Comparison EconomyCompetitorsprobabilityProfit[A]Profit[B]Profit[C] StrongNone0.25$1000$250$450 StrongMany0.25$0$250$300 WeakNone0.25$0$150 WeakMany0.25$0$150$0

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More comparison (fill in on your own) FactorInvestment A Investment B Investment C Probability of failure ($0) ??? Mean Variance ??? Maximum Minimum ??? What is this worth to you? ???

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Thoughts for Next Week Information is valuable. If you knew the outcome was strong economy, and no competitors you would choose A. In this state, A has the most profit, $1000. But A is very risky if you do not know the state. It could pay $0. Buying information can reduce the risk. How much should you pay to know: 1.The state of the economy 2.The amount of competition 3.Both 1 and 2.

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Math Review For many of you, this week will be a review of material you learned in other math or engineering courses. –Diagrams (may be new) Tree diagrams, chance nodes, decision nodes, etc. –Concepts (probably a review) Probability, Independence, Mean, Variance, Expectation, etc.

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Tree Diagram Decision Economy (chance) A B C Strong S S Weak W W Number of Competitors ManyNone Other Outcomes ………….. (try filling this in at home)$1000 $0

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Tree Diagrams These diagrams are useful for describing any process involving risk: –Risky investment performance –Risky decision making process –Alternative choices with risk More information can reduce risk: Next week we will analyze the value of information using tree diagrams. Both practice exams include a problem with a tree diagram.

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Random Variables: Definition A random variable X can take on a random value from a set of possible values. We will call a particular value a realization x. We will call the set of possible values the state space S x Example 1: Coin Flip. X can be heads or tails. S X ={heads,tails}. A particular flip x was x=tails. Example 2: X can be any number between 3.0 and 7.0. S X =[x: 3.0 x 7.0 ]. A particular x was x=4.39.

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Random Variables: Discrete vs. Continuous If the state space S x is countable (finite or infinite countable) we say X is a discrete random variable. If state space S x is uncountable we say X is a continuous random variable. This mostly affects how we treat probability – is probability and means a sum or an integral?

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Probability & discrete r.v. For a discrete random variable, the probability p(x) is a function satisfying 0 p(x i ) 1 for all x i in S x Sx p(x i ) = 1 Or, in words, probability is a number from 0 to 1. If you add up the probability of all the states, you get 1.

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Probability and continuous r.v. For a continuous random variable, the probability is given as a density function. The probability that X is between x and x+dx is p(x)dx We require that p(x)>0 everywhere, and that the density integrates to 1: x x+dx

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Probability of Sets We can extend our probability function to sets of outcomes. Suppose X* is a subset of S x. The probability of some state in X* is Discrete: p(X*) = x in X* p(x) Continuous: p(X*) = Obvious special cases: p(S x ) = 1 probability of some state in S x is 1. p( ) = 0 probability of no state is 0. (some state will occur)

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Independence Two random variables X and Y are independent if p(X=x and Y=y) = p(X=x) * p(Y=y) Independent: Two coin flips Coin1,Coin2 P(Coin1=Heads)=0.5 P(Coin2=Heads)=0.5 P(Coin1=Heads and Coin2=Heads)=0.25 Not Independent: Sky is blue, Today is rainy. suppose P(Sky=blue)=0.50 P(Today=Rainy)=0.50 but the sky is not blue when it is raining P(Sky=blue and Today=Rainy)=0

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Expectations are means or averages of functions of a random variable. E[f(X)] = x in S* p(x)*f(x) Important Note: E[f(X)] is a number, not a random variable. some familiar uses… E[X] = mean or average = x in S* p(x)*x V[X] = variance of X = E[(X-E[X]) 2 ] Expectations

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Simple Properties of Expectations Reduction: Multiplication by Constants (c is a constant, i.e. 3) E[c] = x in Sx p(x)*c = 1*c = c E[cf(X)] = x in Sx p(x)*c*f(x)= c*( x in Sx p(x)*f(x)) = c E[f(X)] Reduction: Addition E[f(X)+g(X)] = x in Sx p(x)*(f(x)+g(x)) = x in Sx p(x)*f(x)+ x in Sx p(x)*g(x)= E[f(X)]+E[g(X)] No Reduction trick for Function Multiplication E[f(X)g(X)] is NOT usually = E[f(X)]*E[g(X)] Exception: Independence If X, Y are independent R.V. E[f(X)g(Y)] = E[f(X)]*E[g(Y)]

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Variance: Definition V[X] = E[(X-E[X]) 2 ] The variance of a random variable measures how far values deviate from the mean.

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Calculating Variance From the definition, it is a 3 step process. Later, we will learn a shortcut. 1.Calculate E[X] 2.Calculate (X-E[X]) 2 for each X 3.Calculate V[X]= E[(X-E[X]) 2 ]

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Calculating Variance: part 1 Compare variance of Investment A and Investment B. Step One: Calculate the means ¼ ¼¼¼ A$1000$0$0$0 E[A] = ¼*$1000 + ¼*$0 + ¼*$0 + ¼ * $0 = $250 B$250$250$150$150 E[B]= ¼*$250+¼*$250+¼*$150+¼*$150 = $200

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Calculating Variance: part 2 Step Two: Calculate (X-E[X]) 2 for each X ¼ ¼¼¼ A$1000$0$0$0 E[A]=$250, so (A-E[A]) $750-$250-$250-$250 (A-E[A]) 2 562500625006250062500 B$250$250$150$150 E[B]=$200, so (B-E[B]) $50$50-$50-$50 (B-E[B]) 2 2500250025002500

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Calculating Variance: part 2 Step Two: Calculate (X-E[X]) 2 for each X ¼ ¼¼¼ (A-E[A]) 2 562500625006250062500 (B-E[B]) 2 2500250025002500 Here we have just cleaned up the previous slide. This slide shows only the answer to part 2. Now we are ready for part 3.

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Calculating Variance: part 3 Step Three: Calculate V[X] = E[(X-E[X]) 2 ] ¼ ¼¼¼ (A-E[A]) 2 562500625006250062500 V[A] = E[(A-E[A]) 2 ] = ¼*562500+¼*62500+¼*62500+¼*62500 = 140625+15625+15625+15625 = 187500 (B-E[B]) 2 2500250025002500 V[B] = E[(B-E[B]) 2 ] = ¼*2500+¼*2500+¼*2500+¼*2500 =2500

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The Variance Trick: A Shortcut V[X] = E[(X-E[X]) 2 ] =E[X 2 -2X*[E[X]]+[E[X]*E[X]]] But remember, E[X] is just a number, like a constant, so we can simplify further… =E[X 2 ]-E[2X*[E(X)]]+E[[E(X)] 2 ] =E[X 2 ]-2*E(X)*E(X)+[E(X)] 2 = E[X 2 ]-[E(X)] 2 =(mean of squares)-(square of mean)

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Calculating V[A] with the variance trick ¼ ¼¼¼ A$1000$0$0$0 A 2 1000000000 E[A 2 ] = ¼*1000000 + ¼*0 + ¼*0 + ¼ * 0 = 250 000 E[A] = ¼*$1000 + ¼*$0 + ¼*$0 + ¼ * $0 = $250 (E[A]) 2 = E[A]*E[A]= 250*250 = 62500 V[A]= E[A 2 ]- (E[A]) 2 = 250 000 – 62500 = 187500

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Calculating V[B] with the variance trick ¼ ¼¼¼ B$250$250$150$150 B 2 62500625002250022500 E[B 2 ] = ¼*62500 + ¼*62500 + ¼*22500 + ¼ * 22500 = 15625+15625+5625+5625 = 42500 E[B] = ¼*$250 + ¼*$250 + ¼*$150 + ¼ * $150 = $200 (E[B]) 2 = E[B]*E[B]= 200*200 = 40000 V[B]= E[B 2 ]- (E[B]) 2 = 42500 – 40000 = 2500

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Analysis of Risky Engineering Business Decisions Ideally, you should choose based on the risk neutral criterion: maximum of expected profit E[profit] –when you have both revenue and costs And/or minimum of expected cost E[cost] –for cost-only decisions Notice that V[profit] or V[cost] is not a part of this criterion.

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Risk attitudes Decisions can be swayed by risk attitudes A Risk-loving decision maker will give up a bit of expectation E[profit] to increase variance V[profit]. They like to gamble. A Risk-averse decision maker will give up a bit of expectation E[profit] to reduce variance V[profit]. They are afraid of risk. A Risk-neutral decision maker will ignore the variance and make the decision solely on E[profit] A wealthy company should be risk neutral in its decision making to maximize expected profit. However, the managers who run the company may follow their own desires and be risk averse or risk loving.

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Analysis of Class exercise Lets go back and re-examine the choice made by the class.

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Comparison EconomyCompetitionprobabilityProfit[A]Profit[B]Profit[C] StrongNone0.25$1000$250$400 StrongMany0.25$0$250$300 WeakNone0.25$0$150 WeakMany0.25$0$150$0

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More comparison FactorInvestment A Investment B Investment C Probability of failure ($0).750.25 Mean Variance $250 187500 $200 2500 $225 28125 Maximum Minimum $1000 $0 $250 $150 $400 $0 What is this worth to you? ???

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Risk neutral analysis Only mean or expected profit is important So we would recommend A FactorInvestment A Investment B Investment C Mean$250 = ($1000+$0+$0+$0)/4 $200 = ($250+$250+$150+$150) /4 $225 = ($400+$300+$150+$0) /4

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Engineering and Business Examples The criteria is the same: maximize expected profit or minimize expected cost 1.Calculate PW or AW many times to create the scenarios 2.Multiply probability times PW or AW to get contribution to E[PW] or E[AW] 3.Make decision based on overall E[PW] or E[AW]

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Accident or loss of life problems There is a probability of an accident per year. Call this probability p There is a damage $D from loss of life, loss of property, etc. when the accident occurs. D is valued in dollars. D should include all damage, and may therefore include damage that is controversial in value (human suffering, loss to environment, value of clean air/safe water, etc). The Expected Annual Cost if nothing is done to fix the problem is E[cost]=pD (AW) This number is compared to the AW of the costs of methods for fixing the problem. This method can be extended to multiple types of accidents with different ps and Ds for each type. The goal is still to minimize E[cost].

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Well look at Problem 13-22 The analysis is on the spreadsheet posted to the web.

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Summary Class Exercise –Forced you to think about risky decision making –Next week: similar scenario to learn about calculating the value of information Math review: random variables, probability, expectations, mean & variance Risk attitudes. Over time, risk neutral decision making maximizes profit Applications –Investment Goal: Maximize E[Profit] –Engineering costs Goal: Minimize E[Costs] –Danger/Safety Usually treated as Engineering Cost problem, with damage as a result of accidents factored into costs. Cost of doing nothing = E[accident cost]

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