1) When sampling n objects from a population of N objects without replacement and when ordering of
objects is considered, the number of samples that can be obtained is
a)
N
Cn
b) n ! N C n
c) N n
B
d) N n
2) A sample of 500 students needs to be selected from the Faculty of Engineering at Cairo University that
represents all departments, the most appropriate sampling method is
a) Simple Random Sampling
b) Systematic Random Sampling
c) Stratified Random Sampling
d) Cluster Sampling
C
3) The flow data for a river (the discharge) for 50 years are analyzed using the following frequency table
Class No. j
1
2
3
4
5
6
Total
Class Interval I j
(m3/s)
(1200, 1800)
(1800, 2400)
(2400, 3000)
(3000, 3600)
(3600, 4200)
(4200, 4800)
Frequency f j
16
18
8
5
2
1
50
the area under the frequency polygon equals
a) 30000
b) 180000
c) 2244
d) 15000
A
c) 2244
d) 3000
B
5) In problem 3, the median of the data equals
a) 2100
b) 1900
c) 2244
d) 3000
A
4) In the previous problem, the mode of the data equals
a) 2100
b) 1900
6) If three objects are to be randomly selected from six objects, the number of unordered samples that can
be obtained is
a) 720
b) 216
c) 20
d) 120
C
7) If A and B are two events with marginal probabilities greater than zero and less than 1.0, then
a) P (A | B ) > P (A and B )
c) P (A | B )  P (A and B )
b) P (A | B ) < P (A and B )
d) P (A | B )  P (A and B )
A
8) If A and B are two independent events with marginal probabilities greater than zero and less than 1.0, then
a) P (A and B ) < P (A )
b) P (A and B ) < P (B )
c) Answers a and b together
d) None of these answers
9) For a continuous random variable with mean  and standard deviation , the area under the probability
density function between  -  and  +  equals
a) 0.95
b) 0.67
c) 0.81
d) None of these answers
10) Assuming x to be normally distributed with mean µ and standard deviation σ, 1 
a) P (x  x 1)
b) P (x 1  x  x 2)
c) P (x  x 1) + P (x x 2)

x2
x1
f ( x )dx  ...
d) Zero
C
D
C
11) Suppose A and B are mutually exclusive events. Construct a Venn diagram that contains the three
events A , B , and C such that P (A | C ) = 1 and P (B | C ) = 0
C
A
B
12) A batch of 500 containers for frozen orange juice contains 5 that are defective. Two are selected, at
random, without replacement.
(a) What is the probability that the second one selected is defective given that the first one was defective?
P (second=D│first=D) =
P( first  D) xP(sec ond  D) (5 / 500) x(4 / 499)

x100  0.8%
P( first  D)
(5 / 500)
(b) What is the probability that both are defective?
P (second=D and first=D) = P (second=D) x P (first=D) = (5 / 500) x(4 / 499) x100  0.008%
(c) What is the probability that both are acceptable?
P (second=A and first=A) = P (second=A) x P (first=A) = (495 / 500) x(494 / 499) x100  98%
13) The probability that a roadway will be flooded X times in any one year is given by the mass function
shown:
a) Find the value of k that makes this a proper probability mass function.
k
k
k
k
k
+
+
+
+
=1
1
2
3
4
5
197k
60
=1

k=
60
197
F(x) = ∑f(x) = k +
k

P( X )  k / X
 0

for X  0
for X  1, 2, ..., 5
otherwise
b) Find P (2 floods) and P (2  X < 4)
k
30
=
2 197
k
kare defective.
30
20A client50purchased seven units; what
Ten
percent
(10%)
of
the
units produced by a factory
14)
P (2 ≤ x < 4) = P (x = 2) + P (x = 3) = + =
+
=
is the probability of finding two or more defective
of these
2 units
3 out
197
197seven?
197
P (2 floods) = P (x = 2) =
14) Ten percent (10%) of the units produced by a factory are defective. A client purchased seven units; what
is the probability of finding two or more defective units out of these seven?
p = 0.1
n=7
P (x ≥ 2) = 1 – P (x ≤ 1) = 1 – [P (x = 0) + P (x = 1)]
= 1 – ( 7C0 *0.10 * 0.97 + 7C1 *0.12 * 0.96 ) = 1 – ( 0.4783 + 0.372 )
= 0.1497
15) A random variable X has a binomial distribution with mean 6 and variance 3.6. Find P (X = 4).
Binomial function
  n p  6 ..........................(1)
 2  n p (1  p)  3.6 ...........(2)
Substitution of Equation 1 into Equation 2:3.6  6 (1  p)   p  0.4
Sub. Into Equation 1  n  15
15!
P ( X  4) 
(0.4) 4 (1  0.4) (15 4)  0.1268
4! (15  4)!
p
g
pp
y
g
, , ,
16) probability of any pipe to be of type A is 0.2, of type B is 0.3, of type C is 0.4, and of type D is 0.1. For
quality control purposes, a sample is chosen randomly from this factory’s production. What is the
probability that in the sample there would be 3 pipes of type A, 4 pipes of type B, 6 pipes of type C, and
2 pipes of type D?
Multinomial function
PA  0.2 & PB  0.3 &
XA  3
& XB 4
&
PC  0.4
XC 6
&
&
PD  0.1
XD  2
n  X A  X B  X C  X D  3  4  6  2  15
The probability 
15!
(0.2) 3 (0.3) 4 (0.4) 6 (0.1) 2  0.0167
3! 4 ! 6! 2 !
17) A normally distributed random variable, x , has a mean of 4.0 and a standard deviation of 2.0. If the
probability density function of this random variable is denoted as f (x ), determine the value of

 f ( x) dx
8

 f ( x) dx = P ( x >8) = P ( Z > 2 ) = 0.5 – 0.4772 = 0.0228.
8
18) Let x represent the net profit of a certain company in millions of dollars. Suppose that a reasonable
probability model for x is given as shown in the figure.
a) Find the value of B
The Area under the curve must equal 1
0.5 B + B + B = 1
2.5 B = 1
B = 0.4
b) Find P (2.0 ≤ x ≤ 6.0)
= the area between 2 & 6
= the area between 2 &3
= 0.5 × 0.5 B = 0.25 B = 0.1
c) Find the probability that the net profit in a given year is greater than 2 million dollars
P ( X > 2 ) = the area between 2 &3 = 0.1.