$1.$ Recall problem $1$ from HW $11$ A music festival organizer has decided to present a music festival

in summer $2024.$

Size

Small Medium Large

Low $-1-2-6$

Medium $5712$

High $81215$

a$.$ Draw the decision tree below, calculate the EVs for all $3$ decisions, and figure the best decisionbased on the probability method of making a decision. Use probability of low attendance $=0\mathrm{.}20,$ probability of medium attendance $=0\mathrm{.}50.$

$2.$ For the problem above,

a$.$ Calculate the EV WITH PI$.$ Show your reasoning and work

b$.$ Using your answers from $2$a$.$ above and the best decision from problem $1,$ calculate the

EV OF PI$.$ Again show your work to get full credit. You only get $1$ point for the answer,

and more points for showing the work.

$3.$ Recall the following payoff table from problem $2$ in HW $11$:

State of economy

Weak Economy Moderate

Economy

Strong Economy

Buy Bond Funds $$200$ $$200$ $$175$

Decisions

Buy Stock Funds $-$$$1000$ $$300$ $$500$

a$.$ Draw the decision tree below, and calculate the EVs for each decision to determine the

best decision in this case. Assume the probability of Weak economy is $0\mathrm{.}2$ and of

moderate economy $0\mathrm{.}5.$

b$.$ Calculate the EV WITH PI$,($show all your work$).$

c$.$ Using the result from b and your answer in a$,$ calculate the EV OF PI for this situation.

$4.$ Remember the following table from Problem $3$ of HW $11$

Courses of action

Event Market item Do not market item

Introduction successful $$50-$$$5$

Introduction not successful $-$$$40-$$$5$

a$.$ Assuming that the probability that the introduction will be successful is $0\mathrm{.}7,$ draw the

decision below and calculate the EV for each decision, and determine the best decision.

b$.$ What is the EV WITH PI in this case?

c$.$ Using your results from a$.$ and b$.,$ what is the EV OF PI$?($as usual, show your reasoning via

calculations$).$

$5.$ The following payoff table shows profit in $millions for a decision problem involving whether to

build an industrial complex on a tract of land, or to sell the land, depending on whether the

demand for the complex is high $($S$1),$ or low $($S$2)$

demand that they identify the student who uploaded it to their site.$)$

Build

Industrial

Complex

Sell Land

instead

S$1($high

demand$)$

$30002000$

S$2($low

demand$)$

$10002000$

Below is the large decision tree that was developed for the above table, assuming that P$($S$1)=0.$

You will be developing the Bayes$$ rule tree for the posterior probabilities in this tree in a later

problem.

Note: If you don$$t print this out, you MUST redraw the tree on your homework.

Profit

Payoff

$$1000$s

S$1|$ F $0\mathrm{.}773000$

Build Complex $|$F

Favorable

report issued S$2|$ F $0\mathrm{.}231000$

P$($F$)=0\mathrm{.}7$

Sell $|$F $12000$

Market

Research

S$1|$U $0\mathrm{.}203000$

Build Complex $|$ U

Unfavorable

report issued S$2|$ U $0\mathrm{.}801000$

P$($U$)=0\mathrm{.}3$

Sell $|$U $12000$

S$10\mathrm{.}63000$

Build Complex

S$20\mathrm{.}41000$

No Market

Research

Sell $12000$

$6$

$3$

$7$

$4$

$2$

$8$

$5$

$1$

Show all work to get full credit.

a$.$ What is the EV at node $6?$

b$.$ What decision do you make at node $3,$ and what is its value? Explain your answer.

c$.$ What is the EV at node $7?$ Show your work.

d$.$ What decision do you make at node $4,$ and what is its value? Explain your answer.

e$.$ What is the EV at node $2?$ Show your work.

f$.$ What is the EV at node $8?$ Show your work.

g$.$ What is the decision at node $5,$ and what is its value? Explain your answer.

h$.$ What decision do you make at node $1?$ Explain.

$6.$ a$.$ What is the optimal decision strategy for the tree in problem $5?$

b$.$ What is the EV with SI$?$ Where did you get it from?

c$.$ What is the EV without SI$?$ Where did you get it from?

d$.$ What is the EV of SI$?$ Show your calculation.

$7.$a$.$ What is the EV with PI$?($You can show your work on the table in problem $1.)$

b$.$ What is the EV without PI$?$ Where did you get it from?

c$.$ What is the EV of PI$?$ Show your calculation.

d$.$ The EFFICIENCY of SI is calculated by EVSI$/$EVPI x $100\%.$ What is the efficiency of SI for

this problem?