Tanaka, Ichiro operates a garage near Sannomiya. He has just been offered the
opportunity to install a car wash at the highly discounted cost of Yen 2,000,000. This
exceptional offer, however, is only available if he accepts it within one week. If he does,
he will charge Yen1300 per car washed. Labor and materials will cost Yen300 per car
washed. Ichiro expects that monthly demand for car wash will be about 100 cars. A major
uncertainty is whether or not his competitor, Kirei Garage and Auto Repair, will also
install their own car wash. If Kirei Garage does, then Ichiro expects the demand for his
car wash will be halved. Ichiro figures that there is a 20% chance that Kirei Garage will
install their car wash very soon, in which case Ichiro and Kirei Garage would split
demand for the next 12 months; a 30% chance that Kirei will install their car wash in
about 6 months, in which case Ichiro would have the full demand for 6 months and split it
with Kirei afterwards; and a 50% chance that Kirei will not install a car wash at all.
Furthermore, Ichiro’s accountant estimates that the value of the car wash after the first 12
months is Yen 2,500,000 in the absence of competition, and Yen 1,000,000 if Kirei has
installed their own car wash.
(a) (12 points)
Construct a decision tree to represent Ichiro’s problem. Be sure to properly label all
branches, and to include all relevant probabilities and payoffs.
(b) (5 points)
Find the best strategy for Ichiro, assuming the Expected Monetary Value (EMV) criterion.
What is the EMV of this strategy?
(c) (5 points)
Is EMV an appropriate decision criterion for Ichiro in such a situation? Why, or why not?
(Please limit your answer to no more than 4 lines of text.)
(d) (8 points)
If the monthly demand was smaller or bigger than 100, would your recommendation
change? For which values of monthly demand should Ichiro choose to have the car wash
installed? Assume that he uses the EMV criterion and that all other data are as originally
Suppose that, at last, Tanaka, Ichiro opened the car wash referenced in Problem 1 and
Kirie Garage did not. Now, Ichiro has to decide the price p that he will charge his
customers for washing their cars because he is not sure if Yen1300 is the right one. In
order to do that, he improved the model he had before (described in Problem 1). Namely,
for a certain price p, he assumes that the monthly demand is a normal random variable
with mean 100 – p and constant standard deviation equal to 10. To provide a good quality
of service he needs to hire one employee if the expected monthly demand is greater than
80, but otherwise he can manage by himself. The monthly salary that the employee
makes is a random variable that is uniformly distributed between $800 and $2,000 (that is
in addition to the Yen300 labor and materials mentioned before).
How would you set up a simulation to determine how much Ichiro should charge (i.e.,
Hint: you do not need to do any math in this problem, just describe briefly how to use
simulation to solve the problem.
Problem 3 (20 points)
The J-Mart department store has sampled 225 sales records on July 14. The average sale
was Yen4200, with an observed sample standard deviation of Yen3000 per customer.
(a) (10 points)
Construct a 90% confidence interval for the mean sale value.
(b) (10 points)
How many sales records would need to be sampled for the 90% confidence interval to be
within Yen100 of the sample mean?
Problem 4 (30 points)
GuardWare Inc. has installed intrusion-detection devices on the campus of a large
university. The devices are very sensitive and, on any given weekend, each one has a
10% chance of being mistakenly activated when no intruder is present. Assume that the
mistaken activation of different devices are independent events. Hint: remember the
(a) (18 Points)
There are six devices in the Administration building. Assume there will be no intruder in
that building next weekend.
1. (6 points) What is the probability that no device will be activated next weekend?
2. (6 points) If two or more of the six devices in the Administration building are
activated during the same weekend, the system automatically signals the police. What
is the probability that this system will signal the police next weekend?
3. (6 points) What are the expected value and the standard deviation of the number of
devices (mistakenly) activated in the Administration building next weekend?
(b) (6 points)
There are a total of 200 such devices on campus. Assume there will be no intruder on any
of these locations next weekend. What is the probability that at least 22 of these devices
will be (mistakenly) activated? (Hint: use an appropriate approximation, and explain why
you may do so.)
(c) (6 points)
An intruder has a 5% chance of not activating the device located in the President’s office.
Campus police has obtained information about a student prank is in preparation and, as a
result, estimates that there is a 20% chance that an intruder will visit the President’s
office during next Holiday Day weekend. If this device is activated during that weekend,
what is the probability that there is an intruder in the President’s office
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