Probability and Risk Analysis
Part 1 of 2
Overview
• Class Exercise
• Math review– Diagrams
• Tree diagrams, chance nodes, decision nodes, etc.
– Concepts• Probability, Mean, Variance, Expectation, etc.
• Sample problems
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)
Class ExerciseFor 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. Procedure1. 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 2nd vote)
3. Lucky draw to see who receives the money4. Instructor will flip coins to provide the random
outcomes5. The real money payoff is determined
Investment A
Random Factors
Economy Competitors Probability Profit
Strong None 0.25 $1000
Strong Many 0.25 $0
Weak None 0.25 $0
Weak Many 0.25 $0
Investment B
Random Factors
Economy Competitors Probability Payoff
Strong None 0.25 $250
Strong Many 0.25 $250
Weak None 0.25 $150
Weak Many 0.25 $150
Note: Investment B is not sensitive to competition
Investment C
Economy Competitors Probability Payoff
Strong None 0.25 $450
Strong Many 0.25 $300
Weak None 0.25 $150
Weak Many 0.25 $0
Comparison
Economy
Competitors probability
Profit[A]
Profit[B]
Profit[C]
Strong None 0.25 $1000 $250 $450
Strong Many 0.25 $0 $250 $300
Weak None 0.25 $0 $150 $150
Weak Many 0.25 $0 $150 $0
More comparison(fill in on your own)
Factor Investment
A
Investment
B
Investment
C
Probability of failure ($0)
? ? ?
Mean
Variance
? ? ?
Maximum
Minimum
? ? ?
What is this worth to you?
? ? ?
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 economy2. The amount of competition3. Both 1 and 2.
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.
Tree Diagram
Decision
Economy (chance)
AB C
Strong S SWeak W W
Number
of CompetitorsManyNone
Other Outcomes ………….. (try filling this in at home)$1000 $0
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.
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
Sx
Example 1: Coin Flip. X can be heads or tails. SX={‘heads’,’tails’}. A particular flip x was x=‘tails’.
Example 2: X can be any number between 3.0 and 7.0. SX=[x: 3.0 x 7.0 ]. A particular x was x=4.39.
Random Variables: Discrete vs. Continuous
If the state space Sx is countable (finite or infinite countable) we say X is a discrete random variable.
If state space Sx 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?
Probability & discrete r.v.
For a discrete random variable, the probability p(x) is a function satisfying
0 p(xi) 1 for all xi in Sx
Sx p(xi) = 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.
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: XS
dxxp 1)(
x x+dx
Probability of Sets
We can extend our probability function to sets of outcomes. Suppose X* is a subset of Sx.The probability of some state in X* is
Discrete: p(X*) = x in X* p(x)Continuous: p(X*) =
*)(
Xdxxp
Obvious special cases:
p(Sx) = 1 probability of some state in Sx is 1.
p() = 0 probability of no state is 0. (some state will occur)
IndependenceTwo 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 rainingP(Sky=blue and Today=Rainy)=0
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
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)]
Variance: Definition
V[X] = E[(X-E[X])2]
The variance of a random variable measures how far values deviate from the mean.
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]
Calculating Variance: part 1Compare variance of Investment A and Investment
B.
Step One: Calculate the means
¼ ¼ ¼ ¼
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
Calculating Variance: part 2Step Two: Calculate (X-E[X])2 for each X
¼ ¼ ¼ ¼
A $1000 $0 $0 $0
E[A]=$250, so
(A-E[A]) $750 -$250 -$250 -$250
(A-E[A])2 562500 62500 62500 62500
B $250 $250 $150 $150
E[B]=$200, so
(B-E[B]) $50 $50 -$50 -$50
(B-E[B])2 2500 2500 2500 2500
Calculating Variance: part 2Step Two: Calculate (X-E[X])2 for each X
¼ ¼ ¼ ¼
(A-E[A])2 562500 62500 62500 62500
(B-E[B])2 2500 2500 2500 2500
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.
Calculating Variance: part 3Step Three: Calculate V[X] = E[(X-E[X])2]
¼ ¼ ¼ ¼
(A-E[A])2 562500 62500 62500 62500
V[A] = E[(A-E[A])2] = ¼*562500+¼*62500+¼*62500+¼*62500
= 140625+15625+15625+15625 = 187500
(B-E[B])2 2500 2500 2500 2500
V[B] = E[(B-E[B])2] = ¼*2500+¼*2500+¼*2500+¼*2500
=2500
The Variance Trick: A Shortcut
V[X] = E[(X-E[X])2]=E[X2-2X*[E[X]]+[E[X]*E[X]]]But remember, E[X] is just a number, like a
constant, so we can simplify further…=E[X2]-E[2X*[E(X)]]+E[[E(X)]2]=E[X2]-2*E(X)*E(X)+[E(X)]2
= E[X2]-[E(X)]2
=(mean of squares)-(square of mean)
Calculating V[A] with the variance trick
¼ ¼ ¼ ¼
A $1000 $0 $0 $0
A2 1000000 0 0 0
E[A2] = 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[A2]- (E[A])2 = 250 000 – 62500 = 187500
Calculating V[B] with the variance trick
¼ ¼ ¼ ¼
B $250 $250 $150 $150
B2 62500 62500 22500 22500
E[B2] = 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[B2]- (E[B])2 = 42500 – 40000 = 2500
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.
Risk attitudesDecisions 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.
Analysis of Class exercise
Let’s go back and re-examine the choice made by the class.
Comparison
Economy
Competition
probability
Profit[A]
Profit[B]
Profit[C]
Strong None 0.25 $1000 $250 $400
Strong Many 0.25 $0 $250 $300
Weak None 0.25 $0 $150 $150
Weak Many 0.25 $0 $150 $0
More comparison
Factor Investment
A
Investment
B
Investment
C
Probability of failure ($0)
.75 0 .25
Mean
Variance
$250
187500
$200
2500
$225
28125
Maximum
Minimum
$1000
$0
$250
$150
$400
$0
What is this worth to you?
? ? ?
Risk neutral analysis
Only mean or expected profit is important
So we would recommend A
Factor Investment
A
Investment
B
Investment
C
Mean $250=
($1000+$0+$0+$0)/4
$200=
($250+$250+$150+$150)/4
$225=
($400+$300+$150+$0)/4
Engineering and Business Examples
The criteria is the same: maximize expected profit or minimize expected cost
1. Calculate PW or AW many times to create the scenarios
2. Multiply probability times PW or AW to get contribution to E[PW] or E[AW]
3. Make decision based on overall E[PW] or E[AW]
Accident or loss of life problems• There is a probability of an accident per year. Call this
probability “p”• There is a damage “$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 p’s and D’s for each type. The goal is still to minimize E[cost].
We’ll look at Problem 13-22
The analysis is on the spreadsheet posted to the web.
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
– InvestmentGoal: Maximize E[Profit]
– Engineering costsGoal: Minimize E[Costs]
– Danger/SafetyUsually treated as Engineering Cost problem, with damage as a result of accidents factored into costs.Cost of “doing nothing” = E[accident cost]