Reference Answer to
Warm-Up Problems
MGMT4102, Jinchang Wang
0. Show in class the newspaper clips:
(1) N. J. Highways least cost-effective in nation in 2006.
NJ collected 20 times (!) as much money per mile of road as the national average.
In NJ, each mile of highway cost $2,209,394, which was the highest in the nation
and four times (!) the closest state, Florida, that was the second most inefficient highway
administration state.
In 2006, administrative cost is $71,720 per mile, which was the cost for
administrators and their office supports and does not include the “general maintenance
cost”. If an administrator’s salary is $70,000 per year, then one administrator of NJDT
takes care of one mile of highway only.
(2) Bailout proposal for Big 3 collapses – and Top 10 selling new cars in NJ.
Operating problems in Big 3 – inefficiency and uncompetitiveness. Ratio of
Output to Input was low. That is, they put in more but produced fewer.
Should the Big 3 be saved? Yes. Can they be saved? No.
1. You are selecting two from ten workers to form a team to do a project. How many
alternatives of the composition of the team?
Answer:
(1) Method 1. There are 10 choices when selecting the first worker for the team. After
the first is selected, there are 9 choices left for the second worker. So, the total number of
possible teams selected in this way is 10*9 = 90. But these teams are double counted,
since team {1, 2} and {3, 8}, for example, are same as {2, 1} and {8, 3}. So, total
number of possible distinct teams is 90/2 = 45.
(2) Method 2. The above answer can be derived by making a 10-by-10 table. Those cells
above the diagonal represent all the possible teams. Total number of alternatives of
possible teams is calculated as:
100 (total number of cells) minus 10 (number of diagonal cells), then divided by 2
(to take out the cells below the diagonal). The result is (100-10)/2 = 45.
(3) Method 3. The third way to solve this problem, which is the general method, is using
the formula for combination. This problem is to find out the combination of two selected
10  9
2
 45 .
from ten workers. That is, C10 
2 1
2. As a salesperson, you are going to visit 10 cities, once for each city. Suppose you
know the airfare between each pair of the 10 cities. How many different routes of
visiting the 10 cities?
Answer:
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For the first city on the route to visit, you have 10 choices. Once you have
selected the first city to visit, you have 9 choices for the second choice. Then you have 8
choices remaining for the third city. And so on. So, number of different route is
10*9*8*7*…*2*1 = 10! = 3,628,800.
Which route is the most inexpensive route?
Note that if there are 15 cities to visit, the number of choices is 1,307,674,368,000.
If there are 20 cities to visit, the number of choices is 2,432,902,008,176,640,000.
Suppose a computer can calculate the cost of a route in 1/1,000,000,000 second. To find
the most inexpensive route, it takes 1,307 seconds for the case of 15 cities, and
2,432,902,008 seconds for the case of 20 cities. How long is 2,432,902,008 seconds?
There are 24*60*60 = 86,400 seconds in a day and 31,536,000 seconds in a year. So,
2,432,902,008 seconds = 28,158.6 days = 77.15 years.
3. You have six jobs to be assigned to six workstations. How many different ways to get
the six jobs done?
Answer:
The first job can be assigned to one of six workstations. Once the first job is
assigned, the second job has 5 remaining workstation to be assigned. And so on, until
once five jobs are assigned, the sixth job has only one workstation to choose from. So,
total number of alternative ways to get the six jobs done is 6! = 720.
4. Kim and Joe have had three children, all of them girls. Now they are expecting their
fourth child. You wonder whether the odds favor having a boy this time. What is the
best estimate of the probability of having another girl? Why?
(a) 6.25%, or 1 in 16, because the chance of getting four girls is a row is 1 out of 16.
(b) 50%, or 1 in 2, because there is roughly an equal chance of getting each gender on a
birth of a baby.
(c) Between 6.25% and 50%.
Answer: (b)
The key here is the independence of a birth from the previous ones. It is a temptation
to think that the chance to have four girls is as low as 1/16 = 0.0625. But here is a
conditional situation where there have been already three girls. So, the chance to be a
boy or a girl is same, since we assume for any couple each birth is independent of the
previous one.
Other examples of independent events:
Guessing H or T for kicking off of a game.
Chance to win on a slot machine.
Note the difference between probability and odds:
Odds(A) = P(A) / P(A) = P(A) / 1 – P(A). So,
P(A) = Odds(A) / 1+Odds(A)
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5. An advertisement says, “Four out of five dentists surveyed recommend sugarless gum
for their patients who chew gum.”
Do you think that about 80% of dentists would recommend the sugarless gum?
Answer:
This is a tricky or cheating ad, which dose not lie but failed to mention the sample
size, - how many dentists were surveyed. If only five dentists were surveyed, of which
four recommended, this results has no value statistically since the sample is too small. If
50 or more dentists were surveyed, the result would be more reliable.
This trick can be demonstrated by using tossing coin. Let’s toss a coin twice. We
got two Heads. Can we claim that “when tossing a coin, 100% would land on Head”?
Here is another example. A certain town is served by two hospitals. In the larger
hospital, about forty-five babies are born each day. In the smaller hospital, about fifteen
babies are born each day. As we know, about 50% of all babies are boys. However, the
exact percentage of boys born varies from day to day. Sometimes it may be higher than
50%, sometimes lower.
For a period of one year, each hospital recorded the days on which more than 60% of the
babies born were boys. Which hospital do you think recorded more such days?
(a) The large hospital
(b) The small hospital
(c) About the same between the two hospitals (within 5% of difference, for example)
The answer is (b), instead of (c), because the small hospital is a small sample in
observing the disparity of boys and girls.
6. Tell what each of the following subjects is about by using a short sentence.
(1) Probability Theory
A study of chances.
(2) Statistics
A scientific study of investigating a population from a piece (sample) of it.
(3) Marketing
How to get products sold.
(4) Finance
How to effectively use the money.
(5) Accounting
Keep track activities of an organization (business, government) in terms of money.
(6) Operations Management
How to manage the process of making products.
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