mark M. Upp has just been fired as the university book store manager
for setting prices too low (only 20% above suggested retail).  he is
considering opening a competing bookstore near the campus, and he has
begun an analysis of the situation.  There are two possible sites
under consideration.  One is relatively small, while the other is
large.  If he opens at Site 1 and demand is good, he will generate a
profit of $50,000.  If demand is low, he will lose $10,000.  If he
opens at Site 2 and demand is high, he will generate a profit of
$80,000, but he will lose $30,000 if demand is low.  He also has the
option of not opeing at either site.  He believes that there is a 50%
chance that demand will be high.  A market research study will cost
$5,000.  The probabilty of a good demand given a favorable study is
0.8.  The probablity of a good demand given an unfavorable study is
0.1.  There is a 60% chance that the study will be favorable.

a)Should Mark use the study and Why?
b)What is the maximum amount Mark should be willing to pay for this
study, or any study?
c)If the study is done and the results are favorable, what would
Mark's expected profit be?

You are considering adding a new food product to your store for
resale.  you are certain that, in a month, minimum demand for the
product will be 6 units, while maximum demand will be 8 units. 
(Unfortunately, the  new product has a one-month shelf life and is
considered to be waste at the end of the month).  You will pay
$60/unit for this new product while you plan to sell the product at a
$40/unit profit.  The estimated demand for this new product in any
given month is 6 units (p=0.1), 7 units (p=0.4),  and 8 units (p=0.5).
 Using EMV analysis, how many units of the new product should be
purchased for resale?


6)Bakery Products is considering the introduction of a new line of
products.  IN order to produce the new line, the bakery is considering
either a major or minor renovation of the current plant.  The
following conditional values table has been developed by the bakery.

Alternatives           Favorable Market ($)      Unfavorable Market ($)
----------------------------------------- ------------------------------
Major Renovation         $100,000                  -$90,000
Minor Renovation          $40,000                  -$20,000
Do Nothing                1,000                       0

Under the assumption that the probabilty of a favorable market is
equal to the probabilty of an unfavorable market, determine:
a)the EMV of a major renovation
b)The EMV of a minor renovation
c)The EMV of the do nothing option
d)The best alternative using EMV


7)The following probabilites provide information about the accuracy of
a market research firm.

Prob. of a favorable study given a favorable market = 0.7
Prob. of an unfavorable study given a favorable market = 0.3
Prob. of a favorable study given an unfavorable market = 0.05
Prob. of an unfavorable study given an unfavorable market = 0.95
If there is a 0.60 prior probability of a favorable market, find the
following probabilities:
a)Prob. of a favorable market given a favorable study
b)Prob. of an unfavorable market given an unfavorable study.

Hi k9queen!
These are the answers:

1a) I'll assume here that Mark only cares abuout the expected value of
his project and doesn't mind risk considerations. In order to answer
this question, we must then compute the expected value of all possible
alternatives. Let'sa go through them one by one.

- Mark doesn't use the study
In this case, the probability of high demand is 0.5. Therefore,

Expected value of opening at:
   Small site: 0.5*50000 + 0.5*(-10000) = $20,000
   Large site: 0.5*80000 + 0.5*(-30000) = $25,000

So clearly, if Mark doesn't use the study, then he must choose to open
the large site, because of the greater expected value of it.

- Mark uses the study, and he gets a "favorable" one.
In this case, the probability of high demand is 0.8. Therefore,

Expected value of opening at:
   Small site: 0.8*50000 + 0.2*(-10000) = $38,000
   Large site: 0.8*80000 + 0.2*(-30000) = $58,000

Again, in this case, Mark should build the large site. Finally, the
last case would be

- Mark uses the study and he gets an "unfavorable" one.
In this case, the probability of high demand is 0.1. Therefore,

Expected value of opening at:
   Small site: 0.1*50000 + 0.9*(-10000) = -$4,000
   Large site: 0.1*80000 + 0.9*(-30000) = -$19,000

Therefore, in this case, Mark should do nothing. If he does nothing,
the expected value of the "nothing" project is $0, which is greater
than $-4,000 and $-19,000.


Now, we know that there is a 60% chance that the study will be
favorable. We also know that if the study is favorable, Mark will
build the large site (EV: $58,000) and if he the study is not
favorable, he will do nothing (EV: $0). Therefore, the expected value
of performing the study is:

EV of study: 0.6*58000 + 0.4*0 - 5000 = $34,800 - $5,000 = $29,800

Notice that I substracted the $5,000 which is the cost of the study.
Obviuosly this cost must be paid whatever the result of the study is.

Finally, in order to determine wether to use the study or not, we must
compare the EV of using the study and the EV of not using it. We
concluded that if he didn't use the study, he would build the large
site, for an expected value of $25,000. We've also seen that the
expected value of the study is $29,800. Therefore, since the latter is
greater, he must choose to do the study.

1b) As you can see in the previous equation, if the study were free of
charge, the expected value of using it would have been $34,800,
compared to $25,000 of not using it. This tells us that Mark should be
willing to pay up to (34800-25000)=$9,800 for the study. Let's see
why. If the study costs a bit more (say, $9,801), the EV of the study
becomes $24,999, which is less than $25,000, so he shouldn't use the
study; and obviously he shouldn't use it if it is even more expensive.
If the study costs a bit less (say, $9,799), then the EV of the study
becomes $25,001, which is greater than $25,000 so he should use it;
and clearly he should use it if it were cheaper than that. So the
limit is at $9,800. In general, one should be willing to pay up to the
excess expect value of using a study over the expected value of not
using it.

1c) We've seen above that in this case, Mark should build the large
site for an EV of $58,000. So the project's EV "from now on" is
$58,000 if we substract the $5,000 he paid for the study, then the
value is $53,000.


2) As before, we must consider all the possible scenarios here. The
demand will be either 6, 7 or 8 units, so we must choose wether to buy
6, 7 or 8 units.

- Scenario 1: Buy 6 units
We know that the probability of a demand of 6 units is 0.1, the prob.
of a demand of 7 units is 0.4 and the prob. of a demand of 8 units is
0.5. So in any case, if we buy 6 units, we'll sell all of them. Since
each unit produces $40 in profits, then:

EMV of buying 6 units = 6*$40 = $240

- Scenario 2: Buy 7 units
If demand is either 7 or 8, we'll sell all of the 7 units, but if it's
6 we will have wasted one. Therefore:

EMV of buying 7 units = .1*(6*$40-$60) + .4*(7*$40) + .5*(7*$40)
                      = $270

Notice the *(6*$40-$60) term. It implies that we're selling 6 units
for a profit of $40 each, but we're loosing $60 because we couldn't
sell the 7th unit.

- Scenario 3: Buy 8 units
Here if demand is 6 we waste 2 units, if it's 7 we waste 1 and if it's
8 we sell all of them.

EMV of buying 8 units = .1*(6*$40-$120) + .4*(7*$40-$60) + .5*(8*$40)
                      = $260

Since the maximum EMV is attained when we buy 7 units, then we should
buy 7 units for resale.


6) Here, if the probability of a favorable market is the same as the
probability of an unfavorable market, then we can assume that the
probability of each is 0.5. So now we can compute the required values:

a) EMV Major: 0.5*100000 + 0.5*(-90000) = $5,000
b) EMV Minor: 0.5*40000 + 0.5*(-20000) = $10,000
c) EMV Nothing: 0.5*1000 + 0.5*0 = $500
d) Clearly the best alternative is a minor renovation


7) In order to answer this question, we must make use of Bayes' Rule.
Let A and B be two events. Then

Prob(A given B) = Prob(A and B)/Prob(B)

a) Let's first compute all the cases of "Prob(A and B)". Let's call
the following events:

FM: favorable market
UM: unfavorable market
FS: favorable study
US: unfavorable study

What we know:

P(FS|FM)=0.7
P(US|FM)=0.3
P(FS|UM)=0.05
P(US|UM)=0.95
P(FM) = 0.6
P(UM) = 0.4

Using this information:

P(FS|FM)=0.7=P(FS and FM)/P(FM)
Then,
P(FS and FM) = 0.7*P(FM) = 0.7*0.6 = 0.42

P(US|FM)=0.3=P(US and FM)/P(FM)
Then,
P(US and FM) = 0.3*P(FM) = 0.3*0.6 = 0.18

P(FS|UM)=0.05=P(FS and UM)/P(UM)
Then,
P(FS and UM) = 0.05*P(UM) = 0.05*0.4 = 0.02

P(US|UM)=0.95=P(US and UM)/P(UM)
Then,
P(US and UM) = 0.95*P(UM) = 0.95*0.4 = 0.38

Now, what we must compute is:

P(FM|FS) = P(FM and FS)/P(FS)

We already know P(FM and FS). In order to get P(FS), we simply do the
following calculation:

P(FS) = P(FM and FS) + P(UM and FS) = 0.42 + 0.02 = 0.44

and with this we also know P(US) = 1 - P(FS) = 0.56.

So we had to compute:

P(FM|FS) = P(FM and FS)/P(FS) = 0.42/0.44 = 0.9545

Thus the probability of a favorable market given a favorable study is 0.9545.

b) In this case, we have to find

P(UM|US) = P(US and UM)/P(US) = 0.38/0.56 = 0.6786

Thus the probability of an unfavorable market given an unfavorable study is 0.6786


I hope this helps! If you have any doubt regarding my answer, please
request a clarification before rating it. Otherwise I await your
rating and final comments.

Cheers!
elmarto