THE INTERNATIONAL UNIVERSITY (IU) Course: Probability Models in Operations Research
Department of Industrial System Engineering
MIDTERM EXAMINATION
PROBABILITY MODELS IN OPERATIONS RESEARCH
Head of ISE Department
Lecturer:
Duration:
150 minutes
Date:
Student ID:
19th March,
2013
Name:
Assoc Prof. Ho Thanh Truong Ba Huy
Phong
INSTRUCTIONS:
1. This is an open book examination. No laptors, PDA,
2.
Use of calculator is allowed; discussion and material transfer are strictly prohibited.
Total pages:
5
(including this page)
Question 1: (20 pts)
A gambler has in his pocket a fair coin and a two-headed coin. He selects one of the coins at random, and
when he flips it, it shows heads.
1/2
1/2
1/2
1/2
F.C
2H
1/2
1/2
H1
11
H1
1/2
1/2
H2
11
H2
1/2
1/2
T3
00
T3
a. (5pts) What is the probability that it is the fair coin?
PFair .C H1  
1
1 1
1
2 2

1
1
3
2  2  1 2
Probability Models in Operations Research – Midterm Exam
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THE INTERNATIONAL UNIVERSITY (IU) Course: Probability Models in Operations Research
Department of Industrial System Engineering
b. (10pts) Suppose that he flips the same coin a second time and again it shows heads. Now what
is the probability that it is the fair coin?
PFair .C H1 H 2  
1 1
1
2 3

1  1  1 2
5
2 3
3
c. (5pts) Suppose that he flips the same coin a third time and it shows tails. Now what is the
probability that it is the fair coin?
PFair .C H1 H 2T3  
1 1
2 5
1
1  1  0 4
2 5
3
Question 2: (20 pts)
A student has to sell 2 books from a collection of 6 math, 7 science, and 4 economics books. How many
choices are possible if
a. (10pts) Both books are to be on the same subject?
C26  C27  C24  42
b. (10pts) The books are to be on different subjects?
C16 C17  C17 C14  C14 C16  94
Or : C217  C26  C27  C24   94
Question 3: (10 pts)
A rat is trapped in a maze. Initially it has to choose one of two directions. If it goes to the right, then it will
wander around in the maze for three minutes and will then return to its initial position. If it goes to the left,
then with probability 1/3 it will depart the maze after two minutes of traveling, and with probability 2/3 it
will return to its initial position after five minutes of traveling. Assuming that the rat is at all times equally
likely to go to the left or the right. What is the expected number of minutes that it will be trapped in the
maze?
1/2
1/2
E[T]
1/2
1/2
R
3+E[T]
L
1/3
1/3
2
Probability Models in Operations Research – Midterm Exam
2/3
2/3
5+E[T]
2
THE INTERNATIONAL UNIVERSITY (IU) Course: Probability Models in Operations Research
Department of Industrial System Engineering
1
3  ET   1 2  1  2 ET   5
2
2 3 3

 ET   21
ET  
Question 4: (20 pts)
The joint probability density function of random variables X and Y is given:
x
  cy
f  x, y    5

0
0  x 1 ,1 y  5
otherwise
Where c is a constant.
a. (5pts) Find the value of c.
1 5
x

   5  cy dydx  1
0 1
1
25
x c

   x  c   dx  1
2
5 2
0
2
1
  12c  1  c 
5
20
b. (10pts) Are X and Y independent?
5
4
3
x y 
f X  x      dy  x 
5 20 
5
5
1
1
1
1
x y 
f Y  y      dx 
y
5 20 
20
10
0
xy 2 x 3 y
3 x y
f X x fY  y  



 
 f  x, y 
25 25 100 50 5 20
Not independen t!
c. (5pts) Find P X Y  3
Probability Models in Operations Research – Midterm Exam
3
THE INTERNATIONAL UNIVERSITY (IU) Course: Probability Models in Operations Research
Department of Industrial System Engineering
1
P X  Y  3   P Y  3  X X  x  f X  x dx
0
3
4
  PY  3  x  x  dx
5
5
0
1
5  1
1   4
3
     y  dy  x  dx
20
10   5
5
0  3 x 
1
 x 2 x 3  4
3
   
   x  dx
40 4 5  5
5
0
1
1
 x 3 37 x 2 63 x 9 
    

 dx
50 200 100 25 
0
1
37
63
9 439





 0.732
200 600 200 25 600
Question 5: (15 pts)
A machine in a factory can be failed because of two types of failures. The time (in hours) required to repair
a machine with type 1 failure is an exponentially distributed random variable with parameter λ = 1. The
time required to recover a machine from type 2 failure follows the normal distribution with mean of 1.5 and
standard deviation of 0.5. The probabilities of type 1 and type 2 failures are 0.6 and 0.4, respectively.
0.6
1
Ex(1)
e-2
RT > 2
a.
0.4
2
N (1.5;0.52)
0.159
RT > 2
(10pts) What is the probability that a repair time exceeds 2 hours?
 P( RT  2)  0.6e 2  0.159 * 0.4  0.145
b.
(5pts) What is the expected repair time of a failed machine?
0.6*1+1.5*0.4=1.2
Probability Models in Operations Research – Midterm Exam
4
THE INTERNATIONAL UNIVERSITY (IU) Course: Probability Models in Operations Research
Department of Industrial System Engineering
Question 6: (15 pts)
The number of times that a person get a cold in a given year is a Poisson random variable with parameter λ
= 5. Suppose that a new wonder drug has just been marketed that reduces the Poisson parameter to λ = 3 for
only 75% of the population. For the other 25 percent of the population, the drug has no effect.
a. (10pts) If an individual tries the drug for a year and has fewer than 3 colds in that time, how likely is
it that the drug is beneficial for him or her?
0.75
0.25
1
(E)
P(3)
0.423
<3
2
(NE)
P(5)
0.125
<3
0.423 * 0.75
 0.911
0.423 * 0.75  0.125 * 0.25
b. (5pts) Determine the expected number of colds per year that a drug usercan get.
 P( E  3) 
EC   5 * 0.25  3 * 0.75  3.5
GOOD LUCKS!
Probability Models in Operations Research – Midterm Exam
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