1.
A) Given group of 6 computer engineer,
7 mechanical engineer and 8 civil engineer
B) Four married couples sit in a row. In how
many ways can this be done if
a. How many teams of 5 members can be
formed?
Ans:
There are 6+7+8=21 engineers and so
 21
   20349
5
a. There are no restrictions.
Ans:
8! ways
b. Each married couple sits together.
Ans:
4! ways for married couples
2! way for each couple among themselves
b. How many with exactly 3 Computer
Engineers?
Therefore 4! 2! 2! 2! 2! ways
Ans:
If there must be exactly 3 Com. Eng., there
2. A) Suppose that P is a probability measure
6
defined on the sample space S. Prove that
are   ways to choose them, and
3
 
P  Ac  B c   1  P  A   P  B   P  A  B  .
 15 
Ans:
  ways to choose Mec. Eng. and Civil
c
2
 
P  Ac  B c   P  A  B  
Eng. Thus by the multiplication rule the
number is
 1  P  A  B   1   P  A  P  B   P  A  B  

 6 15 
    20 105  2100
 3  2 
c. How many 10-member teams with 3
computer engineer,2 mechanical engineer
and 5 civil engineer?
Ans:
For 10 member team, there are
6
  ways of choosing Com. Eng.
3
7
  ways of choosing Mec Eng.
 2
8
  ways of choosing Mec Eng.
5
The total number is, by the multipipcation rule,
then
 6  7  8
      =20.21.22=23520
3   2 5

B) Let A and B be events with
P( A)  0.3, P  A  B   0.5 and P ( B )  p.
Find p if:
Ans:
P( A) , P  B  and P  A  B  are always related via
the formula P  A  B   P  A  P  B   P  A  B  .
If we know the other three numbers in formula,
we can solve for P  B  . We are given P( A)
and P  A  B  , so it remains for us to figure out
P  A  B .
b. A and B are disjoint.
We are told that A and B are disjoint, i.e.,
P  A  B    , so P  A  B   0
P  A  B   P  A  P  B   P  A  B 
0.5  0.3 + p - 0
0.2=p
Alternatively, we could have started this part with the
equation
P  A  B   P  A   P  B  because A and B are disjoint
0.5  0.3 + p
0.2=p
c. A and B are independent.
We are told that A and B are independent, so
P  A  B   P  A P  B 
P  A  B   P  A  P  B   P  A  B 
0

1
0.5  0.3 + p - 0.3p
1
0
2 2
x2
x
2
0
 2x 1 
2 0
2
2
p
7
3. Let X be a continuous random variable with
density function,
0  x 1
kx,

 2  x, 1  x  2
0
otherwise

1
2
3
0
1
2
f ( x)dx   f ( x)dx   f ( x)dx   f ( x )dx   f ( x )dx 
1
0.2=0.7p
0
1
4. A random variable K represents
2
“  the number of head  minus  the number of Tails  ”
in three flips of a coin
a. List the sample points of the sample space for
this experiment.
S  HHH , HHT , HTH , HTT , THH , THT , TTH , TTT 
a. Evaluate k
1
2
 kxdx   (2  x)dx 
0
b. What is the set of values assigned by the
random variable K (i.e. K(S)), to the sample
points?
1
1
2
x2
x2
2
k
 2x 1 

2 0
2 1
K  1, 3,3,9
k
1
 4  2  2  1 k 1
2
2
c. Find the probability distribution of K, assuming
that the coin is biased so that “ the head is
twice as likely to occur as a tail.”
3
b. Find P ( X  )
2
1
c. Find P  1  X  3
3
2
1
3
x2
x2
2 
xdx

(2

x
)
dx


2
x
0
1
1
2 0
2
1
9 1
3 2 
2
4 2
3
2
1

P ( H )  2a
P(T )  a  a 
P ( X  3)  P (TTT ) 
1
3
1
27
6
27
12
P ( X  3)  P ( HHT )  P ( HTH )  P (THH ) 
27
8
P ( X  9)  P ( HHH ) 
27
P ( X  1)  P(TTH )  P (THT )  P ( HTT ) 
k
f(k)
-3
1/27
-1
6/27
b. If the selected balls are of the same color,
what is the probability that they are
selected from box 2?
3
9
12/27 8/27
5. Pembe has three boxes. Their contents are
Box 1: 2 red balls and 4 white balls
Box 2: 4 red balls and 2 white balls
Box 3: 3 red balls and 3 white balls
First, a box is selected, and then two balls are
drawn at random from that box (without
replacement).
6. An urn contains 4 red and 6 white chips, 5 chips
are drawn at random, without replacement.
a. What is the probability that the first three chip
flow exactly the sequence RWR?
P( R) P(W / R) P( R / RW ) 
4 6 3 1

10 9 8 10
a. What is the probability that the selected
both balls are of the same color.
Sc: Same color
B1 : 1st Box
B2 : 2nd Box
B3 : 3rd Box
P( SC )  P( B1 ) P( RR | B1 )  P( B1 ) P(WW | B1 ) 
P( B2 ) P( RR | B2 )  P( B2 ) P(WW | B2 ) 
P( B3 ) P( RR | B3 )  P( B3 ) P(WW | B3 )
b. What is the probability that the second chip is
red?
Marginalize (sum) over possible 1st chips
P(2nd R)  P(2nd R |1st W ) P(1st W )  P(2nd R |1st R) P(1st R)
=
4 6
3 4
4
+

9 10
9 10 10
c. What is probability of drawing a total of 2
red chips out of the 5?
 4  6 
4! 6!
  
2
3
6 20 10
P(2R in total)=     2!2! 3!3! 

 0.476...
10!
4 9 7 21
10 
 
5!5!
5 
d. Answer the previous question if they are
drawn with replacement
Now each draw is independent coin toss with p 
(chance of R)
=0.3456
4
10
7. Given the following frequency distributions table:
Distribution of the scores
range
# of records
0-19
20-29
30-39
40-49
50-59
60-69
70-79
80-89
90-99
3
5
9
18
9
6
5
3
1
a. Find the average (mean) of the
data.
b. Find variance and standard
deviation of data
c. Sketch the histogram and polygon
1. Let A and B be events such that P( A)  0.3, P( B)  0.6 and P( A | B)  0.25
a. Find P( A | B).
b. Let C  A  B and D  A  B. Are the events C and D independent?
2. Suppose you roll two four-faced dice, with faces labeled 1,2,3,4, and
each equally likely to appear on top. Let X denote the smaller of the two
numbers that appear. ( If both dice show the same number, then X is
equal to that common number.) Find the probability distribution of X.
3. Three are 3 bags on the shell. The first bag has 2 red balls and 4 white
balls, the second bag has 4 red balls and 2 white balls and third bag has 3
red balls and 3 white balls.
First, a bag is selected, and then two balls are drawn at random from that
bag (without replacement).
a. What is the probability that the selected both balls are of the same
color.
B1 : 1st Bag
B2 : 2nd Bag
B3 : 3rd Bag
P( SC )  P( B1 ) P( RR | B1 )  P( B1 ) P(WW | B1 ) 
P( B2 ) P( RR | B2 )  P( B2 ) P(WW | B2 ) 
P( B3 ) P( RR | B3 )  P( B3 ) P(WW | B3 )
b. If the selected balls are of the same color, what is the probability that
they are selected from bag 2?
4. Given 10 people P1 , , P10
a. In how many ways can the people be lined up in a row?
10!
b. How many lineups are there if P2 , P6 , and P9 want to stand together? (
in any order)
8!
G
P1 P2 P6 P9 P3 P4 P5 P7 P8 P10 =8!.3!
3!
c. How many lineups are there in which P2 , P6 , and P9 do not stand
together?
10!-8!3!
5. Random variables X and Y are samples from the joint probability density
function
c(2 x  y ), 0  x  1, 0  y  1
f ( x, x)  
otherwise
0,
a. Find c.
1 1
1 1
1    c(2 x  y )dxdy  c   (2 x  y )dxdy 
0 0
0 0
1 1
1 1
 c   2 xdxdy  c   ydxdy  x 2 |10 
0 0
0 0
1 2
y
2
1
0
 1
1 3
 c
2 2
3
2
1 c  c 
2
3
b. Find the marginal density of X.

g ( x) 


y 1
1
2
2
y2 
4x 1
f ( x, y )dy   (2 x  y )dy   2 xy   

30
3
2  y 0 3 3
c. Find the marginal density of Y.

h( y )  f ( x, y )dx 

1
2
2
2
(2 x  y ) dx  y  x 2

30
3
3
x 1
x 0

2y 2

3 3
d. Are X and Y independent? (Explain)
No, since f ( x, y )  g ( x)h( y )
e. Find P(Y  X ).
1 y
1
2
2
22 31 4
P Y  X     (2 x  y )dx dy   2 y 2 dy 
y 
300
30
33 0 9
y
( x 2  xy  y 2  y 2  2 y 2 )
0

6. A random variable R represents “ the number of head

2
minus
 the number of Tails  ” in three flips of a coin
2
a. List the sample points of the sample space for this experiment.
S  HHH , HHT , HTH , HTT , THH , THT , TTH , TTT 
b. What is the set of values assigned by the random variable R (i.e.
R(S)), to the sample points?
R(S )  9, 3,3,9
c. Find the probability distribution of R, assuming that the coin is biased
so that “the head is twice as likely to occur as a tail.”
P ( H )  2a
P(T )  a  a 
P ( X  9)  P (TTT ) 
1
3
1
27
6
27
12
P ( X  3)  P ( HHT )  P ( HTH )  P (THH ) 
27
8
P ( X  9)  P ( HHH ) 
27
P ( X  3)  P (TTH )  P (THT )  P ( HTT ) 
7. Following frequency distributions table shows the daily emission (in tons)
of sulfur oxides from an industrial plant:
Daily
Frequency CM
Emission
5.0 - 8.9
3
6,95
20,85
424,83
9.0 - 12.9
10
10,95
109,5
624,1
13.0 - 16.9
14
14,95
209,3
212,94
17.0 - 20.9
25
18,95
473,75
0,25
21.0 - 24.9
17
22,95
390,15
285,77
25.0 - 28.9
9
26,95
242,55
590,49
29.0 - 32.9
2
30,95
61,9
292,82
TOTAL
80
1508
2431,2
a. What is the Range of the daily emission?
Range= xmax  xmin  25.6
b. Find the class marks and class size?
=(5+8,9)/2=6.95…….30.95
c. Construct the cumulative frequency distribution
Less than 4.95
Less than 8.95
Less than 12.95
Less than 16.95
Less than 20.95
Less than 24.95
Less than 28.95
Less than 32.95
0
3
13
27
52
69
78
80
0
0,0375
0,1625
0,3375
0,65
0,8625
0,975
1
d. Estimate the number of emission of sulfur oxide in a day that is
below 22.9
22.9  20.95
17  60.28 The number of emission is 60
4
e. Compute the sample mean X and the sample variance
fi X i 1508
fi ( X i  X )2 2431.2


2
X

 18.85 S 

 30.77
N
80
N 1
79
3  10  14 
f. Draw the percentage ogive and histogram