Probability and Risk Analysis Part 1 of 2. Overview Class Exercise Math review –Diagrams Tree diagrams, chance nodes, decision nodes, etc. –Concepts Probability,

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]