QUIZ QUESTIONS
1. Suppose we are playing a game, we draw 3 cards with replacement from an
ordinary deck. We need to pay $2 for each drawing and will earn $20 dollar if the
heart will be drawn. What is our expected value of net profit for 4 drawings?
2. Weekly sales of automobiles at S.C’s Auto Gallery is normally distributed with a
mean of 200 and a standard deviation of 50 automobiles.
a. What is the probability that sales will be less than-170 in a given week?
b. What is the probability that sales will be in excess of 260?
c. What is the probability that sales will be between 175 and 230?
3. The probability that CIMSA’s stock price will rise in a given day is estimated to
be 0.60. When CIMSA’s stock price rises, there is a 0.70 probability that the
exchange rate of TL/$ will fall in that day. When CIMSA’s stock price falls, there
is a 0.4 probability that the exchange rate of TL/$ will fall in that day. Tomorrow
what is the probability that CIMSA’s price will rise and also the exchange rate of
TL/$ will fall?
4. Suppose we have three machines for making a particular part. The first machine
produces 4% defectives, the second machine produces 6% defectives, and the
third machine produces 1% defectives. Suppose also that the first machine
supplies 30%, the second machine 20%, and the third machine 50%, of the parts.
If a part is selected at random, what is the probability that it is defective? Given
that the part is defective, what is the probability that it came from the first
machine?
5. It has been suggested that we implement a procedure for testing for child abuse.
Suppose that when a child has been abused a doctor can correctly identify the
abuse with probability 0.99, and that when a child has not been abused the doctor
can correctly identify the non-abuse with probability 0.9 Suppose also that the
probability that a child has been abused is 0.05. If the doctor says that a child has
been abused, what is the probability that the child ha actually been abused?
6. The probability that CIMSA’s stock price will rise in a given day is estimated to
be 0.70. When CIMSA’s stock price rises, there is a 0.75 probability that the
exchange rate of TL/$ will fall in that day. When CIMSA’s stock price falls, there
is a 0.5 probability that the exchange rate of TL/$ will fall in that day. Tomorrow
what is the probability that CIMSA’s price will rise and also the exchange rate of
TL/$ will fall?
7. Weekly sales of automabiles at S.C’s Auto Gallery are normally distributed with a
mean of 300 and a standard deviation of 40 automobiles.
a. What is the probability that sales will be less than 270 in a given week?
b. What is the probability that sales will be in excess of 370?
c. What is the probability that sales will be between 284 and 328?
8. Our professor has put $3 on each of the three horses running in three different
races at ISTANBUL HORSE RACING CENTER He feels that each of the three
bets he has made, has a 0.3 probability of winning. A winning ticket on any of the
three will earn our professor $15. Assuming a Bernoulli process, what is the
expected value of his net profit?
9. A child chosen at random in a community school system comes from a low-
income family 15 percent of the time. Children from low-income families in the
community graduate from college only 20 percent of the time. Children not from
low-income families have a 40 percent chance of graduating from college. As an
employer of people from this community, you are reviewing applicants and note
that the first one had a college degree. What is the probability that person comes
from a low-income family?
10. Suppose that the past annual data about Istanbul Stock Market Index suggests that
the probability of index rising in a given year is 0.60, the probability of no change
is 0.05, and the probability of index falling in a year is 0.35. Furthermore analysts
have found out that for all the years during which index have risen, 80% of the
time interest rates have fallen and 20% of the time interest rates increased. For all
the years index have not changed, 50% of the time interest rated have fallen and
50 % of the time interest rates have increased. And for all the years’ index have
fallen, 40% of the time interests rates have fallen and 60% of the time interest
rates have increased. Latest developments in Turkey show that interest rates
started rising and will continue to rise in year 2001. Given this positive upward
trend in interest rates in 2001, what are your revised estimates for the probability
of index rising, not changing, and falling in year 2001?
11. Suppose that Ford Corporation produce 3 different models for Turkish consumers,
in its manufacturing plant located in Turkey. Annual volume of sales and
production of all the 3 models combined is 700 units: However, some of the
automobiles manufactured and sold mm out to be defective and lead to consumer
complaints. Based on the past data, production manager estimated the following
distribution of defective and non-defective automobiles for each model
manufactured in year 2001:
A B C
D 10 20 70
N 140 180 280
a. What is the probability that a randomly selected consumer (who decided to buy a Ford)
will buy either model A or model C?
b. What is the probability that the automobile that he buys will be either model B or
defective?
c. What is the probability that the automobile that he buys will be either model A or non-
defective?
d. If the automobile purchased by a randomly selected consumer is found to be defective,
what is the probability that it is a Model A automobile?
e. If the automobile purchased by a randomly selected consumer is found to be non-
defective, what is the probability that it is Model C automobile?
12. Mr X has $150,000 available for one of three investment alternatives: stock,
treasury bills, $ denominated saving deposits. The investment environment can
assume any one of three states depending on the rate of inflation (which will
determine the rate of depreciation of TL against $). The payoff table of Mr. Y
looks like this:
Type of investment
Inflation Stock Treasury Bills $ Saving deposits
High $300,000 $100,000 $ 150,000
Moderate $100 $150,000 $ 150,000
Low $50,000 $200,000 $ 150,000
a). What should we suggest Mr. X to invest by using the criterion of Maximax?
b). What should we suggest Mr. X to invest by using the criterion of realism? Assume
ά = 0.6.
13. The Province of Quebec is planning to issue hunting licenses for moose through a
province-wide lottery. They have found that by harvesting some animals during
the hunting season, they can increase the number of animals which survive
through the winter months because of the limited food supply. ‘Their estimates of
moose population (in thousands) conditional upon the number of licenses issued
and the severity of the winter are as follows:
Number of licenses Issued
Severity of winter 5,000 6,000 7,000 8,000 9,000
Mild 38 36 34 33 25
Moderate 35 33 33 29 22
Severe 28 30 32 27 20
Harsh 22 26 30 25 16
During the years when records have been kept, 20% of the winters have been mild
while 30%, 40%, and 10% have been moderate, severe, and harsh, respectively. How
many licenses should be issued in order to maximize the expected moose population?
How large will the expected springtime moose population be with this decision?
14. Monthly demand for product A at company B has been as follows:
Month Demand
Jan. 59
Feb. 53
Mar. 57
Apr. 48
May 43
Jun 37
Ju1 38
a. Use Naive Model to develop forecast for August and compute MSE of
this model.
b. Use Smoothing Linear Trend Model to develop forecast for August and
obtain MSE of the model. Choose ά1= 0.2, ά2= 0.02
c. Compare above 2 models and decide which one is more accurate.
15. Linda, the managing editor of a magazine, needs to develop a forecasting
system for monthly sales in order to schedule press runs. Sales (in
thousands of copies) for the first 8 months of 1999 (the first year of
publication) were:
Month Demand
Jan. 50
Feb. 45
Mar. 60
Apr 52
May 69
Jun. 60
Jul. 47
Aug. 53
Linda does not believe there is a seasonal pattern.
a. Use 3-period Weighted Moving Average Model to obtain a forecast for
September of 1999. Wt = 0.5, Wt-l = 0.3, Wt-2 = 0.2
b. What is the MSE of this model?
16. Monthly demand for product A at company B has been as follows:
Month Demand
Jan. 59
Feb. 53
Mar. 57
Apr. 48
May 43
Jun. 37
Jul. 38
Use linear regression model to develop a forecast for August and compute
MSE.
17. Linda, the managing editor of a magazine, needs to develop a forecasting C
f10 system for monthly sales in order to schedule press runs. Sales (in
thousands of copies) for the first 8 months of 1999 (the first year of
publication) were
Month Demand
Jan. 50
Feb. 45
Mar. 60
Apr. 52
May 69
Jun. 60
Jul. 47
Aug. 53
Linda does not believe there is a seasonal pattern.
a. Use 2-period Un-weighted Moving Average Model to obtain a forecast
for September of 1999. What is the MSE of this model?
b. Use Simple Exponential Smoothing Model to forecast the sales of
September and compute the MSE of this model. Take a = 0.6.
c. Compare above-said two models, which one is more accurate?
18. Mr X has $100,000 available for one of three investment alternatives: stock,
treasury bills, $ denominated saving deposits. The investment environment
can assume any one of three states depending on the rate of inflation (which
will determine the rate of depreciation of TL against $). The payoff table of
Mr. Y looks like this:
Type of investment Inflation Stock Treasury Bills $ saving deposits.
Type of Investment
Inflation Stock T. Bills $ Saving Deposits
High $250,000 $50,000 $100,000
Moderate $0 $100,000 $100,000
Low -$100,000 $150,000 $100,000
We have learned that Mr. Y chose treasury bills to invest. According to the
criterion of realism, find a range of alpha values that characterize the level of
optimism implied by this decision maker.
19. The ACME Company is contemplating a new product that would sell for
$11 a unit, the per-unit variable cost for this product is $8, and the fixed
cost per year allocated to this product is $ 240,000. The sales manager for
ACME estimates that annual sales for this product would have a mean of
200,000 with a standard deviation of 85,000 unit& Using this information,
answer the following questions:
a). What is the probability that ACME would loose money on this product
next year?
b). If the cost of capital investment necessary to undertake this project is
estimated to be $3 million, what is the probability of earning at least 5
percent rate of return annually?
c). What are the expected annual profit and expected annual loss from the
production of this product? (Assume that unit loss equals to unit profit)
EXAM QUESTIONS
1. Monthly Demand for TV sets at TT’s Electronics has been as follows:
Months Demand
Jan. 138
Feb. 137
March 143
Apr. 148
May 157
June 153
July 159
Using Linear Regression obtain forecast for August and September and compute MSE of
the model. (15 pts)
2. Los Bichos, the managing editor of ASTROLOGY magazine, needs to develop a
forecast for the 2nd Quarter of 2002. (10 pts)
Sales in the past quarter have been as follows:
Years Qtr. Sales
2000 2nd Qtr. 50
3rd Qtr. 52
4th Qtr. 47
2001 1st Qtr. 55
2nd Qtr. 53
3rd Qtr. 50
4th Qtr. 54
st
2002 1 Qtr. 52
Los Bichos does not believe there is a seasonal pattern.
Using 2-period Moving Average model obtain a forecast for the 2’” Qtr. of 2002 and
compute MSE of the model. (15 pts)
3. Prof. Demir is planning to invest his 1 M $ in anyone of the three investment
alternatives including GOLD, REAL ESTATE and STOCK. He believes that the
ANNUAL RETURNS (pay-offs) from each one of these investment alternatives will
critically depend on the inflation rate. Furthermore he believes that, there are 3 possible
outcomes for inflation. His corresponding pay-off table is given below:
Investment Alternatives
Gold Real Estate Stocks
High Inf. $100000 $150000 $15000
States of Nature
Medium Inf. $50000 $20000 $30000
Low Inf. $10000 $-5000 $80000
A. Using the MINIMAX REGRET criterion obtain the optimal investment alternative.
(15 pts)
B. Using the MAXIMIN criterion obtain the optimal investment alternative. (5 pts)
4. Suppose that you are the sales manager of MIGROS supermarket chain in
Istanbul. You are trying to make a decision about the optimal number of bottles of
EFES beer that should be stocked for each month. The cost of each bottle to
MIGROS is 10 S. whereas the price that MIGROS charges from the customers is
15$ per bottle. For each bottle that is unsold by the end of the week and returned back to
the producer of EFES, MIGROS receives $5 only. The sales data for the last 200 months
is given below:
QUANTITES BUYERS
NUMBER OF MONTHS
BOUGHT
THIS OCCURRED
5000 30
3500. 70
3000 60
2500 40
A. Using the EXPECTED VALUE criterion obtain the optimal number of EFES bottles
that the sales manager of MIGROS should stock per month. (15 pts)
B. Using the MAXIMUM LIKELIHOOD criterion obtain the optimal number of EFES
bottles that the sales manager of MIGROS should stock per month. (5 pts)
5. Suppose that the production manager of HONDA automobiles in Japan believes that
the expected annual volume of sales of the new model that they plan to introduce to the
market in 2003 is 200000. Furthermore he believes that the odds (chances) are 4 to 3 that
the annual sales volume will lie between 100000 and 300000. What is your best estimate
for the standard deviation of annual sales volume of this model of HONDA? (10 pts)
6. The project manager of SABANCI Corporation has been analyzing the feasibility of an
investment project, which involves the production of computer chips in Turkey. They
estimated that the cost of capital investment to undertake this çroject is $ 9 M. The selling
price of each computer chip is believed to be $ 500. The variable cost per unit is
estimated to be approximately $ 200. Annual fixed costs are likely to be around $
300000. The expected annual volume of sales is 1 500 and standard deviation of sales is
estimated to be 250.
A. What is the probability of at least breaking-even for each year? (5 pts)
B. What is the probability of earning less than $75000 profits for each year? (5 pts)
C. What is probability of losing at most $ 90000 for each year? (5 pts)
D. What is the probability of earning more than 2 % annual rate of return for each dollar
invested in this project? (5 pts)
7. Prof. TT has put $2 on each of the 3 horses running m three different races in Istanbul
Horse Racing Center He feels that each of the bets he has made has a 02 probability of
winning. A winning ticket on any of the three horses will earn Prof TT $40 Assuming a
Bernoulli process, answer each of the following questions:
a. What is the probability of at least 2 of the horses(on Prof. TT has bet) winning?
b. What is the EXPECTED EARNINGS from all three races?
c. What is the probability of at most 1 out of three horses (on which Prof TT has bet)
winning?
d. What is the probability distribution of his possible earnings from all three races?