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Stat 111
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STAT 111
Chapter Zero
Question1
Suppose two dice are tossed and the numbers on the
upper faces are observed. Let S denote the set of all
possible pairs that can be observed. Define the
following subsets of S.
A: the number on the second die is even.
B: the sum of the two numbers is even.
C: at least one number in the pair is odd.
List the elements in A,CC,A∩B ,A∩BC,AC∪B, and
AC∩C.
Answer1
S = { (1,1),(1,2),(1,3),(1,4),(1,5),
(1,6),(2,1),(2,2)….(6,6).
N(S) = 6X6 = 36.
A = {(1,2),(2,2),(3,2),(4,2),(5,2),(6,2),
(1,4),(2,4),(3,4),(4,4),(5,4),(6,4),(1,6),(2,6),
(3,6),(4,6),(5,6),(6,6).
B = {(2,2),(4,2),(6,2),(2,4),(4,4),(6,4),(2,6),(4,6),(6,6).
C = {(1,1),(1,2),(1,3),(1,4),(1,5),(2,1),(2,3),(2,5),(3,1),
(3,2),(3,3),(3,4),(3,5),(3,6),(4,1),(4,2),(4,5),(5,1),
(5,2),(5,3),(5,4),(5,5),(5,6),(6,1),(6,3),(6,5).
Cont..Answer1
CC =S – C ={(2,2),(2,4),(2,6),(4,2),(4,4),(4,6), (6,2),(6,4),(6,6)}
A∩B = B
A∩ BC = A – B ={(1,2),(3,2),(5,2),(1,4),(3,4),(5,4),(1,6),
(3,6),(5,6)}
AC = {(1,3),(2,3),(3,3),(4,3),(5,3),(6,3),(1,5),(2,5),(3,5),
(4,5),(5,5),(6,5),(1,1),(1,2),(1,3),(1,4),(1,5),(1,6)}
AC ∪B ={(1,1),(1,2),(1,3),(1,4),(1,5),(1,6),(4,2),(2,2),(6,2),
(1,3),(2,3),(3,3),(4,3),(5,3),(6,3),(2,4),(4,4),(6,4),(1,5),
(2,5),(3,5),(4,5),(5,5),(6,5),(2,6),(4,6),(6,6)}
AC ∩C= AC
Question2
Suppose a family contains two children of different ages, and we
are interested in the sex of these children. Let F denote that a
child is female and M that the child is male, and let a pair such as
FM denote that the older child is female and the younger male.
There are four elements in the set S of possible observation.
S= { FF ,FM,MF,MM}
Let A denote the subset of possibilities containing no males, B
the subset containing two males, and C the subset containing at
least one male. List the elements of A , B ,C ,A ∩ B ,A ∪ B ,
A ∩ C ,A∪ C ,B ∩ C and C ∩ BC
Answer2
A={FF}, B={MM}, C={ FM,MF,MM}
 A∩B=φ
 A ∪ B ={FF,MM}
A∩C=φ
A∪C=S
 B ∩ C ={MM}
 C ∩ BC = C- B ={FM,MF}
Question3
A total of 36 members of club play tennis, 28 play squash, and
18 play badminton. Furthermore, 22 of the members play tennis
and squash, 12 play both tennis and badminton, 9 play both
squash and badminton, and 4 play all three sport. How many
members of this club play at least one of these sports.
Answer3
N(T)=36
N(S)=28
N(b)=18
N(T ∩ S)= 22
N(S ∩B)=9
N(T ∩S ∩B)=4
N(T ∩B)=12
N(T ∩S ∩B)=N(T)+N(S)+N(B)-N(T ∩S)-N(T ∩B)N(S ∩B)+N(T ∩S ∩B)=36+28+18-22-9-12+4=43
Question4
In a survey carried out in a school snack shop. The following
results were obtained. Of 100 boys questioned, 78 linked sweets,
74 ice cream, 53cake, 57linked both sweets an ice cream, 46
liked both sweets and cake while only 31 boys liked all three. If
all the boys interviewed linked at least one item, draw a Venn
diagram to illustrate the results. How many boys both ice cream
and cakes?
Answer4
N(S)=100,
6
26
N(SW)=78,
y
31
N(I)=74,
15
x
N( C) =53,
N(SW ∩I)=57,
z
N(SW∩C)=4,
N(SW∩I ∩C)=31.
N(S)=N(SW ∪I ∪C)=N(SW+N(I)+N( C ) –N(SW ∩C)N(SW ∩C)-N(I ∩C)+N(SW ∩I ∩C)=100.
N(I ∩C)=33.
STAT 111
Chapter One
Question1
Find the number of ways in which 6 teacher can be
assigned to 4 section of an introductory psychology
course if no teacher is assigned to more than one
section?
Answer1
6P
4
= 6 x 5 x 4 x 3= 360
Question2
A toy manufacture makes a wooden toy in tow parts; the
top part may be colored red , white, or blue and the
bottom part brown, orange, yellow or green. How many
differently colored toys can be produces?
Answer2
3 X 4 = 12
Question3
How many different signals may be formed by
displaying 6 flags in row if there are 3 blue flags, 2 red
flags, and 1 white flags available if all flags of the same
color are identical?
Answer3
6!
 60
3!2!1!
Question4
Proteins in living cells are composed of 20 different
kinds of amino acids. Most proteins consist of several
hundred amino acids in along chin structure. How many
different proteins of length 100 can be constructed?
Answer4
2
20 20 20..... 20
 (20)
100tim e
Question5
Find the number of subsets of a set X containing n
elements?
Answer5
Either the subsets containing no element  n  ,
0
 
n
n
n




1 elements1  2 elements 2  , n elements  
 
 
n
2
 n  n
 n
      ........    (1  1)
 0  1 
 n
Question6
In how many can a person gathering data for a market
research organization interview 3 of the 20 families
living in a certain apartment house?
Answer6
 20
20
   c3  1140
3 
Question7
Suppose that someone wants to go by bus, by train, or
by place on a week's vacation to one of the five East
North states. Find the number of different ways in
which this can be done?
Answer7
3 X 5 = 15
Question8
a)How many ways can one make a true –false test
consisting of 20 questions?
In how many ways can they be marked true or false so
that
b) 7 are right and 13 are wrong?
c) at least 17 are right?
Answer8
a) 2x2x2x2Xx….x2 = 20 = 1048,576
b)  20   20   20  77,520
 7   7,13 c7
  

 20   20
20
     c10  184,756
c) 
10,10 10 
d)
c
20
17
 c18  c19  c20  1351
20
20
20
Question9
How many license plates may be formed beginning with
2 different letters of the Arabic alphabet following by 4
different digits? How many be formed if repetition of
letters and digits is allowed?
Answer9
Without repetition :
28x27x10x9x8x7= 3,810,240.
With repetition :
28x28x10x10x10x10=7,840,000.
Question10
A test has 10 true-false questions and 6 multiple-choice
questions with 5 possible choices for each. How many
possible sets of answers are there?
Answer10
2 5
10
6
 16,000 ,000
Question11
How many different 6-digits numbers may be formed
using the digits from the 4,4,5,6,6,6.
Answer11
6!
 60
2!1!3!
Question12
A telephone company in a certain area has all telephone
numbers prefixed by either 465,475, or 482, followed
by 4 digits. How many different telephone numbers are
possible in this area? How many if repetition in each
number is allowed?
Answer12
With repetition
3 104  30,000
Without repetition
3  7  6  5  4  2,520
Question13
A student may select one of 3 English classes, one of 2
mathematics classes ,and one of 2 history classes for his
program. In how many ways may he build his program?
Answer13
c c c
3
2
3
1
1
1
 3  2  2  12
Question14
A person has 8 friends, of whom 5 will be invited to a
party.
a)How many choices are there if 2 of the friends are
feuding and will not attend together?
b) How many choice if 2 of the friend will not attend
together?
Answer14
 2  6   2  6 
a)         6  2 15  36
 0  5  1  4 
 2  6   2  6 
b)         26
 2  3   0  5 
Question15
How many ways can we arrange the letters in the word
cold?
Answer15
4! = 24
Question16
How many ways may 5 people be seated in a
5-passenger vehicle if one of two people must drive?
Answer16
 2
  4!
1 
The drive seat must be filed in 2 ways, after that, the
remaining 4 can be arranged in 4!
Question17
A student is to answer 7 out of 10 questions in an
examination. How many choices have she? How many
if she must answer at least 3 of the first questions?
Answer17
a) c7  120
10
b) c3 c4 c4 c3 c5 c2  110
5
5
5
5
5
5
Question18
In how many ways can 2 oaks, 3 pines, and 2 maples be
arranged in a straight line if one does not distinguish
between trees of the same kind?
Answer18
7!
 210
2!3!2!
Question19
Ten persons have organized a club. How many different
committees consisting of 3 persons may be formed from
these 10 people? In how many ways may a president a
secretary and treasure be selected?
Answer19
a) c7  120
10
10
b) p3  720
Question20
If eight persons are having dinner together, in how
many different ways can three order chicken, four order
steak, and one order lobster?
Answer 20
8!
 280
3!4!
Question21
a)How many distinct permutations are there of the
letters in the word statistics?
b) How many of these begin and end with letters?
Answer 21
1)
10!
 50,400
3!3!2!
8!
2)
 3,360
3!2!
Question22
In how many ways can 7 books be arranged on a shelf if
a) Any arrangement is possible,
b) 3 particular books must always stand together,
c) 2 particular books must occupy the ends?
Answer 22
1) 7! 5,040
2) 3!5! 720
3) 2  5! 240
Question23
How many numbers consisting of five digits each can
be made from the digits 1,2,3,…,9 if
a) the numbers must be odd.
b) the first two of each number are even?
Answer 23
a) 5  6  7  8  5  75,600
b) 6  7  3  4  504
Question24
In how many ways can 3 women and 3 children be
seated at a round table if
a) No restriction is imposed.
b) Two particular children must not sit together.
c) Each child is to be seated between two women?
Answer 24
a) 5! 120
b) 5!2  4! 72
c) (3  1)!3! 12
Question25
From a group of 8 women and 6 children a committee
consisting of 3 women and 3 children is to be formed.
How many different committees are possible if
a)2 of the children refuse to serve together?
b) 2 of the women refuse to serve together?
c) 1 child and 1 woman reuse to serve together?
Answer 25
a)
c c c  c c c
2
4
8
2
4
8
0
3
3
1
2
3
 896
6   2  6   2  6 

b)           1000
 3   0  3  1  2 
       
 2  7  5   7  5   7  5 
c)      1    1    910
 0  3  3   3  2   2  3 
Question26
There are 3 paths leading from A to B and 2 paths
leading from B to C. In how many ways may one make
the round trip from A to C and back without retracing
any paths?
Answer 26
ABC 3x2x1x2=12
Question27
A child has 12 blocks, of which 6 are black, 4 are red, 1
is white, and 1 is blue (blocks of the same color are
identical). If the child puts the blocks in a line, how
many arrangements are possible?
Answer 27
12!
6!4!1!1!
Question28
A robot must pick up ten items from the floor. In how
many ways can the task be performed? If the items are
divided into two sub collections, the first containing six
items and the second containing four items, and if, once
an item from a sub collection is selected, the robot is
programmed to pick up the remaining items in that sub
collection before proceeding to the order sub collection,
in how many ways can the task be performed?
Answer 28
a)
b)
10! = 3628,800
6! 4! 2 = 34,560
Question29
In a class of 10 students, how many ways can the
students be seated so that there are 1 student in each of
the five rows and 5 students in the last row?
Answer 29
10 

  30,240
 .....5 
Question30
a small community consists of 10 women, each of
whom has 3 children. If one woman and one of her
children are to be chosen as mother and child of the
year, how many different choices are possible?
Answer 30
c c
10
3
1
1
 30
Question31
How many functions defined on n points are possible if
each functional value is either 0 or 1 ?
Answer 31
2  2  2  ... 2
 2n
n _ tim e
Question32
How many even three digit number can be formed from
the digits 1,2,5,6 and 9 if each digit can be used only
once.
Answer 32
3  4  2  24
Question33
How many sample points are in the sample space when
a pair of dice is thrown once?
Answer 33
6X6=36
S={(1,1),(1,2),…..(1,6)}
Question34
Calculate the number of permutations of the letters a, b,
c, d had taken two at a time?
Answer 34
p
4
2
 12
Question35
four names are drawn from the 24 members of a club,
for the offices of president, vice - president , treasurer,
and secretary. In how many differently ways can this be
done?
Answer 35
24
p
4
 255,024
Question36
In how many ways may 3 books be placed next to each
other on a shelf?
Answer 36
3!  6
Question37
Four different mathematics books, 6 different physics
books, and 2 different arrangements are possible if
a)The books in each in each particular subject must all
stand together,
b) Only the mathematics books must stand together?
Answer 37
no rest = 12!
a) 3! (4! 6! 2! ) = 207360
b) 4! 9! = 8,709,120
Question38
Four women and four children to be seated in a row of
chairs numbered 1 thought 8;
a) How many total arrangements are possible?
b) How many arrangements are possible if the women
are required to sit in alternate chairs?
c) How many arrangements are possible if the four
women are considered indistinguishable and the four
children are considered indistinguishable?
d) How many arrangements are possible if the four
women are considered indistinguishable but the four
children are considered indistinguishable
Answer 38
a)
8! = 40, 320
b)
2! (4! 4!) + 1,152
c)
8!
 70
4!4!
8!
 1680
4!
d)
Question39
From 4 chemists and 3 physicists find the number of
committees that be formed consisting of 2 chemists and
1 physicist?
Answer 39
 4  3 
    18
 2 1 
Question40
Form a group of 5 teachers and 7 students, how
different committees consisting of 2 teachers and 3
students can be formed? What if 2 of the students refuse
to serve on the committee together?
Answer 40
a) c2 c3  350
2 5 5
2 5 5
b) c0 c3 c2  c1 c2 c2  300
5
7
or
c c c  c c   300
5
2
5
2
5
2
0
3
1
2
Question41
From 5 statisticians and 6 economists a committee
consisting of 3 statisticians and 2 economists is to be
formed. How many different committees can be formed
if
a) no restrictions are imposed,
b) Two particular statisticians must be on the
committee.
c) One particular economist cannot be on the
committee.
Answer 41
a)
b)
c)
 150
5
cc
6
3
2
2
3
6
2
1
2
5
5
3
2
ccc
cc
 45
 100
Question42
A shipment of 10 television sets includes three that are
defective. In how many ways can a hotel purchase four
of these and receive at least two at least two of the
defective sets?
Answer 42
c c c c
3
7
3
7
2
2
3
1
 70
Question43
In how many ways can a set of four objects be
partitioned into three subsets containing, respectively,
2,1 and 1 of the objects?
Answer 43
4 

  12
 2,1,1
Question44
In how many ways can seven scientists be assigned to
one triple and two double hotel rooms?
Answer 44
7


  210
 3,2,2 
Question45
a)How many license plates are there if the first three
place are form the Arabic letters and the last three are
numbers?
b)If each number can be used any one time?
Answer 45
a)28 28 28101010  21,952,000
b)28 27 2610 9  8  14,152,320
Question46
If a travel agency offers special weekend trip to 12
different cities, by air, rail or bus. In how many different
ways can such a trip be arranged?
Answer 46
12  3  36
Question47
In an experiment consists of throwing a die and then
drawing a letter at random from the English alphabet,
how many points are possible?
Answer 47
6  26  156
Question48
In a medical study are patients are classified in 8 way
according to whether they have blood type AB +, AB - ,
A+ , A - ,B +,B –, O +, O - , also according to whether
their blood pressure is low, normal, or high. Find the
number of ways in which a patient can be classified?
Answer 48
8  3  24
Question49
If 4 teachers, 3 engineers, and 3 doctors are to be seated
in a row, how many seating arrangement are possible
when people of the same jobs must sit next to each
other?
Answer 49
3!(4!3!3!)  5,184
Question50
Find the number of ways in which one a, three B's, two
C's and one F can be distributed among seven students
taking a course in statistics?
Answer 50
7


  420
1,3,2,1
Question51
10 Math student, 5 chemistry students and 5 geo;ogy
students. In how many different ways we can select 6
such that
a)Any 6;
b)2 from chemistry;
c)Number of Math students range from 2 to 4?
Answer 51
a) c6  38,760
20
b) c2 c4  13,650
5
10
c ) c2
15
c c c c c
10
10
10
10
10
4
3
3
4
2
 33,300
STAT 111
Chapter Two
Question1
The following data were given in a study of a group of
1000 subscribers to a certain magazine. In reference to
sex, marital status, and education, there were 312 males,
470 married persons , 525 college graduates, 42 male
college graduates, 147 married college graduates, 86
married males, and 25 married male college graduates.
Show that the numbers reported in the study must be
incorrect.
Answer 1
M  Male
Mr  Married
G  College
P (M) 312
P (MR)  .470
P (G)  .525
P (M  G)  .042
P (Mr G)  .147
P (M  Mr  G)  .25
P (M  Mr  G)  1.057 1
Question2
The mathematics department consists of full professors,
15 associate professors, and 35 assistant professors; a
committee of 6 is selected at random from the faculty of
the department. Find the probability that all the
members of the committee are assistant professors. Find
also the probability that the committee of 6 is composed
of 2 full professors, 3 associate professors, and 1
assistant professor.
Answer 2
 75
n( s )   
6 
 35
 
6 

p ( A) 
 .00806
 75
 
6 
 2515 35
   
2  3 1 

p ( A) 
 .029
 75
 
6 
Question3
Let P be a probability measure such that
P(A)=1/3 , p(b)= 1/2 , P(A∪B)= 2/3 .
Find P(A∩B), P(A∩BC), P(AC∪ BC), P(AC∪ B).
Answer 3
P (A B)  1/6
P (A Bc )  1/6
P (Ac  Bc )  5/6
P (Ac  B)  5/6
A
AC
B
1/6
2/6
1/2
BC
1/6
2/6
1/2
1/3
2/3
1
Question4
If 3 books are picked at random from a shelf containing
5 mathematics, 3 books of statistics, and a chemistry,
what is the probability that
a) the chemistry is selected
b)2 mathematics and 1 book of static are selected
Answer 4
1 8 
  
1 2 

a)
 .33
9
 
3
 5  3 
  
2 1 

b)
 .3571
9
 
3
Question5
A system containing two components A and B is wired
in such a way that it will work if either component
works. If it is known from previous experimentation
that the probability of A working is 0.9, that of B
working is 0.8, and the probability that both work is
0.72, determine the probability that the system will
work?
Answer 5
p( A)  0.9
p( B)  0.8
p( A  B)  0.72
p (The system will work)=
p( A  B)  P( A)  P( B)  P( A  B)  .9  .8  .72  .98
Question6
Let A and B be events with P(A)=1/2 ,
P(A∪B)= 3/4,P(Bc)=5/8 . Find P(A∩B), P(Ac ∪ Bc)
c
c
c
,P(A ∩ B ),and P(B ∩A ).
A
B
Bc
Ac
Answer 6
P (A  B)  1/8
P (Ac  B c )  1/4
P (Ac  Bc )  7/8
P (B A c )  2/8
Question7
A pair of fair dice is tossed. Find the probability that the
maximum of the two numbers is greater than 4?
Answer 7
 20 
p   .5556
 30 
Question8
Of 120 students, 60 are studying French, 50 are
studying Spanish, and 20 are studying French and
Spanish. If a student is chosen at random, Find the
probability that the student?
a) is studying French or Spanish.
b)is studying neither French nor Spanish.
c)is studying exactly of them.
Answer8
n(S)  120,n(F)  60,n(SP )  50,n(F SP )  20
60
50
P (F) 
, P (SP ) 
120
120
20 1
40
C
P (F SP ) 
 , P( F  SP ) 
120 6
120
30
C
P( F  SP) 
120
a) p( F  SP)  .75
b) P( F C  SPC )  1  P( F  SP)  ..
c) P( F  SPC )  P( F C  SP)  .5833
Question9
A committee of 5 is to be selected from a group of 6
teachers and 9 students. If the selection is made
randomly, what is the probability that the committee
consists of teachers and 2 students?
Answer9
 6  9 
  
 3  2   .2398
15
 
5 
Question10
A jar contains 3 red, 2 green,4blue, and 2 white marbles.
Four marbles are selected at random without
replacement from this jar. What is the probability of
drawing 2 red, a blue, and a white marble?
Answer10
 3  2  4  2 
    
 2  0 1 1   .0727
11
 
4 
Question11
If 2 balls are randomly drawn from a bowl containing 6
white and 5 black balls, what is the probability that one
of the drawn balls is white and the other black?
Answer11
 6  5 
  
1 1   .5455
11
 
2 
Question12
If the probability that a student A will fail a certain
statistics examination is 0.5, the probability that student
B will fail the examination is 0.2, and the probability
that both student A and student B will fail the
examination 0.1, what is the probability that at least one
of these two student will fail the examination?
Answer12
P(A)  .5
P(B)  .2
P(A  B)  .1
P(A  B)  P(A) P(B)- P(A  B)  .6
Question13
An experiment consists of tossing a die and then
flipping a coin once if the number on the die is even. If
the number on the die is odd, the coin is flipped twice.
List the element of the sample space S?
Answer13
S  {(2,H),(2,T ),(4,H),(6,H)} {(1,H, H),
(1,H, T ),(1,T ,H),(1,T ,T ),.}
Question14
If the probability are, respectively, 0.09, 0.15, 0.21, and
0.23, that a person purchasing a new automobile will
choose the color green, white, red, or blue, what is the
probability that a given buyer will purchase a new
automobile that comes in one of those colors?
Answer14
P(g  w  r  b )  p(g)  p(w)  p( r )  p(b)  .68
(mutuallyexclusive)
[ or at least one of thosecolures]
Question15
A die is loaded in such a way that the probability of any
particular face's showing is directly proportional to the
number on that face. What is the probability that an
even number appears?
Answer15
P (S)  w 1  2w  3w  4w  5w  6w  1  21w  1
1
w
21
12
P (E) P (2) P (4) P (6)
21
Question16
A jar contains 12 marbles, 2 of which are red, 2 green, 4
blue, and 4 white. A marble is selected at random from
the jar. What is the probability that it is blue?
Answer16
 4
 
1   4
12 12
 
1 
Question17
It is known that a patient will respond to a treatment of
a particular disease with probability equal to 0.9. If
there patients are treated in an independent manner, find
the probability that at least one will respond?
Answer17
P(at least one )=
p( A1  A2  A3 )  1  P( A1  A3  A3 ) 
1  P( A  A2  A3 )  0.9
C
1
C
C
Question18
If A and B are independent events with P(A)=0.5, and
P(B)=0.2, find the following
a) P(A∪ B)
b)P(Ac ∩ Bc )
c)P(Ac ∪ Bc )
Answer18
a) p( A)  p( B)  p( A) p( B)  .6
b) p( AC ) p( BC )  .4( AC , BC are indep also)
Question19
A mixture of candies contains 6 mint, 4 toffees, and 3
chocolates. If a person makes a random selection of one
candies, Find the probability of getting
a)a mint.
b) a toffee or a chocolate.
Answer19
6
 
1
6

a)
 .4615 or
13
13
 
1 
Question20
A die is tossed 50 times. The following table gives the
six numbers and their frequency of occurrence
Number
1
2
3
4
5
6
Frequency
7
9
8
7
9
10
Find the relative frequency of the event
a) a 4 appears.
b) and odd numbers appears.
c)a prime number appears.
Answer20
07
a)
 .14
50
789
b)
 .48
50
989
c)
 .52
50
Question21
Three women and three children sit in a row. Find the
probability that
a) the 3 children sit together.
b)the woman and children sit in alternate seats.
Answer21
(3!4!)
a)
 .2
6!
3!3!2!
b)
 .1
6!
Question22
Let A and B be events with P(A∪ B)=7/8 ,
P(A∩ B)= 1/4 ,P(AC)=5/8 .
Find P(A),P(B), and P(A∩ BC).
Answer22
3
P (A)
8
6
P (B) 
8
A
B
BC
1
P (A B ) 
8
c
AC
2/8
4/8
6/8
1/8
1/8
1/8
3/8
5/8
1
Question23
A balanced die is tossed twice. If A is the event that an
even number comes up on the first toss, B is the event
that an even number comes up on the second toss and C
is the event that both toss result in the same number, are
the events A, B and C independents?
Answer23
A  {(1,2).(2,6),(4,1),....(4,6),(6,1),.....(6,6)}
B  {(1,2),...(6,2),(1,4)..(6,4),(1,6)....(6,6)}
C  {(1,1),(2,2),..(6,6)}
n(A)  18,n(B)  18,n(S)  36,n( C )  6
A  B  C  {(2,2),(4,4),(6,6)}
3
P (A B  C) 
 .083
36
1818 6
P( A) P( B) P(C ) 
 .041
36
P( A  B  C )  P( A) P( B) P(C )
.:not indep
Question24
Three names to be selected from a list of seven names
for use in a particular public opinion survey. Find the
probability that the first on the list is selected for the
survey?
Answer24
1 6 
  
1 2   .4286
7
 
3 
Question25
A hat contains twenty white slips of paper numbered from 1
through 20, ten red slips of paper numbered from 1 through 10,
forty yellow slips of paper numbered from 1 through 40, and ten
blue slips of paper numbered from 1 through 10. If these 80 slips
of paper are thoroughly shuffled so that each slip has the same
probability of being drawn, find the probabilities of drawing a
slip of paper which is
a)blue or white .
b) numbered 1,2,3,4 or 5.
c) red or yellow and numbered 1,2,3,or4.
d) numbered 5,15,25,or 35.
e)white and numbered higher than 12 or yellow and numbered
higher than 26.
Answer25
a)(10 20)/80 .375
b)20/80 .25
c)8/80 .1
d)8/80  .1
e).275
Question26
Three students A,B and C are in a swimming race. A and B have
the same probability of winning and each is twice as likely to win
as C. Find the probability that B or C wins.
Answer26
P(a)  2/5
P(b)  2/5
P(c)  1/5
P(c b)  p(c )  p(b) - p(c  b)  3/6  .6
Question27
Of 10 girls in a class, 3 have blue eyes. If two of the girls are
chosen at random, what is the probability that
a)both have blue eyes.
b) Neither have blue eyes.
c)at least one has blue eyes.
Answer27
 3  7 
  
2  0 

a)
 .06
10
 
2 
 3  7 
  
0  2 

b)
 .4667
10
 
2 
 3  7   3  7 
      
1 1   2  0 

c)
 .533
10
 
2 
Question28
Consider families with two children. Let E be the event that a
randomly chosen family has at most one girl, and F ,the event
that the family has children of both sexes. Show that E and F are
not independent.
Answer 28
S  {BB, GG, GB, BG}
E  {GB, BG, BB}
F  {GB, BG}
P (E  F)  2/4
P (E) P (E)  3/4  2/4  6 / 16
 3/8  2/4
Question29
Find the probability of getting three heads in three (independent)
tosses of a balanced coin.
Answer 29
p(HHH,HHT ,HT H,T HH,T T H,T HT ,HT T ,T T T }
1
P (HHH)
8
Question30
Relays used in the construction of electric circuits function
properly with probability 0.9. A assuming that the circuits operate
independently, which of the designs in Figure 0-2 yields the
higher probability that current will flow when the relays are
activated?
Answer 30
For design(1):
=.9
For design(2):
=.96639
Question31
Find the probability of getting at least one head in 5 tossed of a
balanced coin?
Answer 31
P(at least one head)=1-p(HHH)=.969
Question32
Among a shipment of 4 electrical components of types A,B,C and
D, there are 3 of type A,4 of type B, 5 of type C, and 6 of type D.
From this shipment 3 components are randomly selected. Find
the probability that
1. all are of type C.
2. one of each of the type B,C,D
3. at least 2 of type B and nothing of types A,D?
Answer 32
 5
 
3
10

a)

 .0123
18 816
 
3 
 4  5  6 
   
1 1 1 

b)
 .1471
18
 
13
 4  5   4  5 
      
2 1   3  0 

c)
 .041
18
 
3 
Question33
An urn contains M white and N black balls. If a random sample
of size r is chosen, what is the probability that it will contain
exactly K white balls?
What if M=K=1?
Answer 33
 M  N   N 
 
 

 K  r  K    r  1 or if M  K  1
M  N 
 N  1




r

r

Question34
A single die is tossed. Find the probability of a 3 or 6 turning up?
Answer 34
P(3or 6)  p(3) p(6) 1/6  1/6  2/6  1/3  .33
Question35
If A and B are mutually exclusive events and P(A)=0.3,and
P(B)=0.5, find
1. p ( AC )
2. p ( A  B)
3. p ( A C  B )
Answer 35
1. p ( AC )  0.7
2. p ( A  B)  p ( A)  p( B)  .8
3. p ( AC  B)  p ( B)  .5
Question36
If three events A ,B , and C are independent, show that
1. A and B∩C are independent.
2. A and B ∪C are independent .
3. AC and B ∩CC are independent.
1
3
B
A
2
4
1
3
A
B
2
4
Answer 36
1)p(A ∩(B ∩C))=P(A ∩B ∩C)=
P(A)x P(B)x P(C)=P(A)x P(B ∩C)
2)P(A ∩(B ∩C))=P((A ∩B) ∪ (A ∩C))=P(A).P(B ∪C)
3)P(AC ∩(B ∩CC )=P(AC )x P(B)x P(CC )
P(AC )P(B ∩CC )
If T,F independent => TC and FC independent
STAT 111
Chapter Three
Question1
Find P(A|B)if
a) B is a student of A,
b)A and B are mutually exclusive.
Answer 1
P A  B 
a) p( A B) 
1
P( B)
0
b) p ( A B ) 
0
P( B)
Question2
A box contains three coins, one coin is fair, one coin is twoheaded, and one coin is weighted so that the probability of head
appearing is 1/3 . A coin is selected at random and tossed. Find
the probability that heads appears.
Answer 2
P( H )  P( H I ) P( I )  P( H II ) P( II )  P( H III) P( III)  0.61111
Question3
An urn contains 3 red marbles and 7 white marbles. A marble is
drawn from the urn and a marble of the other color is then put
into the urn. A second marble is drawn from the urn. A) Find the
probability that the second marble b)If both marbles were of the
same color, what is the probability that they were both white?
Answer 3
p(both white both same color) =
p (both white same color)
=
p(same color)
p (both white)
p( (both red)+p(both white)
=
7 6

10 10
 0.875
p( R1 ) p( R2 R2 )  p( w1 ) p( w2 w2 )
Question4
The probability that three men hit a target are respectively , and
each shoots once at the target (independent) . a) Find the
probability that exactly one of them hits the target. b) If only one
hit the target, what is the probability that it was the first men?
Answer 4
a) p(exactly one man)=
p(E)  P( M 1 M 2 M 3 )  P( M 1 M 2M 3 )  P( M 1 M 2 M 3 )
C
C
C
C
31
  0.43
72
P( M 1  E )
C
C
p(M1 E) 
 P(M1  M 2  M 3 )  0.1935
P( E )
C
C
Question5
Only one in 1000 adults is affected with a rare disease for which
a diagnostic test has been developed. The test is such that, when
and individual actually has the disease, a positive result will
occur 99% of the time, while an individual without the disease
will show a positive test result only 2% of the time. If a randomly
selected individual is tested and the result is positive, what is the
probability that the individual has the disease?
Answer 5
P(With  ve) 
p(ve with) p( with)
p(ve)
p(ve)  p(ve with) p( with)  p(ve without) p( without)
 0.99 0.001 0.02 0.999
 0.02097
0.99 0.001
p( with  ve) 
 0.0472
0.02097
Question6
The members of a consulting firm rent cars from three agencies;
60 percent from agency 1,30 percent from agency 2, and 10
percent from agency 3. If 9 percent of the cars 1 need a tune-up,
20 percent of the cars from agency 2 need a tune-up, and 6
percent of the cars from agency 3 need a tune-up , what is the
probability that a rental car delivered to the firm will need a tuneup.
Answer 6
P(I)=0.6
P(T | I)=0.09
P(II)=0.3
P(T| II)=O.2
P(III)=0.1
P(T| II)=0.06
P(T)=0.09x0.6+0.2x0.3+0.06x0.1=0.12
Question7
Each of 2 cabinets identical in appearance has 2 drawers.
Cabinet A contains a silver coin in each drawer, and cabinet B
contains a silver coin in one of its drawers and a gold coin in the
other. A cabinet is randomly selected, one of its drawers is
opened, and a silver coin is found. What is the probability that
there is silver coin in the other drawer?
Answer7
1 1 1
P( S )  1    0.75
2 2 2
p( S A) P( A)
P (A | S ) 
0.6667
P( S )
Question8
Suppose we have 10 coin such that if the ith coin is flipped, head
will appear with probability , i=1,…,10. When one of the coin is
randomly selected and flipped, it shows a head. What is the
conditional probability that it was the fifth coin?
Answer8
P(5th| H)  0.09
P(H|)  0.55
Question9
The probability that a regularly scheduled flight departs on time
is 0.83, the probability that it arrives on time is 0.82; and the
probability that it departs and arrives on time is 0.78. Find the
probability that a plane
a)arrives on time given that it departed on time, and
b) departed on time given that it has arrived on time.
Answer9
P (D)  0.38
P (A)  0.83
P (DA) 0.78
a)P (A| D)  0.94
b)P (D| A)  0.95
Question10
An urn contains 10 white, 5 yellow, and 10 black marbles. A
marble is chosen at random from the urn, and it is noted that it is
not one of the black marbles. What is the probability that it is
yellow?
Answer10
P(Y| 5)  0.33
Question11
Suppose that a fair coin is tossed until a head appears for the first
time. Determine the probability that exactly n tosses will be
required.
Answer11
1 1 1 1
P(T , T ,....T , H )  P(T ) XP(T ) X ...P( H )  X X   
2 2 2  2
n
Question12
A die is tossed. If the number is odd, what is the probability that
it is prime?
Answer12
P( prim eodd) 
p( pr odd)
p(odd)
 0.667
Question13
A box of fuses contains 20 fuses, of which 5 are defective. If 3 of
the fuses are selected at random and removed from the box in
succession without replacement, what is the probability that all
there fuses are defective?
Answer13
p( D1  D2  D3 )  0.00877
Question14
A number is picked at random from { 1,2,3,…,100}. Given that
the chosen number is divisible by 2 what is the probability it is
divisible by 3 or 5?
Answer14
Let A1  {the numbers which divisible by 2}
A3  {the numbers which divisible by 3}
A5  {the numbers which divisible by 5}
P (A3? A5 | A2)  0.46
P (A3 A2)  0.16
P (A5 A2)  0.1
P (A2 A3  A5)  0.03
Question15
Three members of a private country club have been nominated
for the office of president. The probability that the first will be
elected is 0.3, the probability that the second member will be
elected is 0.5, the probability that the third member will be
elected is 0.2. If the first member is elected, the probability for an
increase in membership fees is 0.8, If the second or third member
be elected, the corresponding probabilities for an increase in fees
are 0.1, and 0.4. What is the probability that there will be an
increase in membership fees? If someone is considering joining
the club but delays his or her decision for several weeks only to
find out that the fees have been increased, what is the probability
that the third member was elected president of the club?
Answer15
a)0.37
b)0.2162
Question16
Suppose that 5 percent of men and 0.25 percent of women are
color blind. A color blind person is chosen at random. What is the
probability of this person's being male? Assume that there are an
equal number of males and females?
Answer16
P (m) 1/2
P (c| m)  0.05
P (c| f)  0.0025
P (c)  0.02625
P (m| c)  0.9524
Question17
In recent years much has been written about the possible link
between cigarette smoking and lung cancer. Suppose that in a
large medical centre ,,of all the smokers who were suspected of
having lung cancer,90 percent of them did, while only 5 percent
of the nonsmokers who were suspected of having lung cancer
actually did. If the proportion of smokers is 0.45, what is the
probability that a lung cancer patient who is selected by chance is
a smoker?
Answer17
P(c )  0.4325
P(s| c)  0.9364
Question18
A laboratory blood test is 95 percent effective in detecting a
certain disease . when it is, in fact, percent. However, the test also
yields a 'false positive" result for 1 percent of the healthy persons
tested. ( That is, if a healthy person is tested, then, with
probability 0.01, the test result will imply he or she the disease.)
If 0.5 person of the population actually has the disease, what is
the probability a person has the disease given that the test result is
positive?
Answer18
P( ve)  0.0147
P(D|  ve)  0.3231
Question19
In a certain town, 40% of the people have brown hair,25% have
brown eyes, and 15% have both brown hair an brown eyes. A
person is selected at random from the town
If he has brown hair, what is the probability that he also has
brown eyes;
b) if he has brown eyes, what is the probability that he does not
have brown hair
c) what is the probability that he has neither brown hair nor
brown eyes.
Answer19
a)0.375
b)0.4
c)0.5
Question20
twenty percent of the employees of a company are college
graduates. Of these, 75% are in supervisory position. Of those
who did not attend college, 20% are in supervisory position.
What is the probability that a randomly selected supervisor is a
college graduate?
Answer20
P (c1 )  0.2
P (s1 | c1 )  0.75
P (c2 )  0.8
P (s1 | c 2 )  0.2
P (s2 | c 2 )  0.8
P (c1 | s)  0.4839
STAT 111
Chapter Four
Question1
Two fair dice tossed, and let X denote the sum of the spots that
appears on the top face.
a)Obtain the probability distribution for X.
b) Construct a graph for this probability distribution.
Answer1
x
2
F(x) 1/36
3
4
5
6
2/36 3/36
4/36
5/36
7
8
9
10
6/36 5/36 4/36 3/36
11
12
2/36 1/36
Question2
A machine has been producing ball-point pens with a defective
rate of 0.02. A sample of size 5 is taken from a carton of pens
produced by the machine. Let Y represent the number of
defective pens in the sample. What is the value of f(0),f(5),f(3.5)?
Answer2
F(0)  p ( y  0)  p ( NNNNN )  (.98) 5  .904
F(5)  p ( y  4)  p ( DDDDD)  (.02) 5  3.2 10-9
F(3.5) 0
Question3
A urn holds 5 white and 3 black marbles. If two marbles are
drawn at random without replacement and Z denote the number
of white marbles
1.Find the probability distribution for Z
Graph the distribution.
Answer3
z
0
1
2
sum
3/28
15/28
10/28
28/28
Question4
For each of the following functions determine if it is a
probability function or not and sketch the distribution
(
)=
( )=
( )=
Answer4
a) Yes
b) Yes
c) No
Question5
Two dice are rolled. Let X be the difference of the face numbers
showing, the higher minus the lower, and 0 for ties. Find the
probability mass function of X.
Answer5
x
0
1
2
3
4
5
F(x)
6/36
10/36
8/36
6/36
4/36
2/36
Question6
6. The probability mass function of X is given by
F(x)=k|x-2|
x=-1,1,3,5
Find
a)K
b)The cumulative function of X and plot its graph
c)P(X
d)P(-0.4<X<4)
e)P(X>1)
Answer6
x2
1
a) k  , f ( x ) 
8
8
b)
5
c) 8
1
d) 4
1
e)
2
x
-1
1
3
5
F(x)
3/8
1/8
1/8
3/8
Question7
Let S={(I,j):I,j be the set of all subsets of S. Let ((I,j))=for all 62
pairs(I,j) in S. Define
X(I, j)  i  j
1 i j 6
Find
Find the distribution F(x)for the random variable X
Graph this distribution function.
Answer7
F(x)=
Question8
A box contains good defective items. If an item drawn is good.
We assign the number 1 to the drawing; otherwise, the number 0.
Find the p.m.f and the c.d.f?
Answer8
F(0)  p(x  0)  1 - p
F(1)  p(x  1)  p
Question9
Consider the toss of balanced dice. Let X denote the random
variable representing the sum of the two faces,find
a)The probability distribution of X.
b) Draw the graph of p.m.f
c)find P(X>7),P(5 X 9).
Answer9
Question10
Suppose that a random variable X has a discrete distribution with
the following p.m.f
(x)=
Find the value of the constant c.
Answer10
c
 1
1 1 1

1  x 1 f ( x)  x 1 x  cx 1 x  c   2  3  .....
2
2
2 2 2



c 1 
 
 c  c 1

2 1  1 
 2


Question11
A fair coin is tossed until a head or five tails occurs. Find the
p.m.f and the c.d.f
Answer11
F(1)  1/4
1
F(2)  f(T H)
4
1
F(3)  F(T T H)
8
1
F(4) 
16
1 1
2 1
F(5)  F(T T T T H) F(T T T T T T)   
32 32 32 16
Question12
Independent trials, consisting of the flipping of a coin having
probability p of coming up heads, are continually performed until
either a head occurs or a total of n flips in made. Let Y denote the
number of times the coin is flipped.
a)Find the probability distribution of Y.
b)Graph the distribution.
Answer12
F(1)  P
F(2)  (1- P )P
F(3)  (1- P ) P
.
.
.
F(N - 1)  (1- P ) P
F(N)  (1- P )
Question13
The distribution function of the random variable X is given by
F(x)=
1.Graph F(X)
2.Determine the p.m.f
3.Compute P(X<3),P(X 2)and P(1 X 3),P(X>1.4)
Answer13
x
1
2
3
4
F(x)
1/8
2/8
3/8
2/8
3
P(X  3)  ,
8
7
P(X 2)  ,
8
6
P(1 X 3)  ,
8
7
P(X  1.4) 
8
Question14
a certain school gives only three letter grades in its course:
p(pass),F(fail),and W (withdrew) . In computing grade points
p =1 point, F=-1 point, and W=0 points . A student has enrolled in
a mathematics course and a history course. Let x represent the
total of the grade points that the student may earn in the two
classes , Using the letter grade to represent the outcome in the
course ,describe a sample space for the possible grades of the
student
Question14
1)List the outcomes in the events
a){ x=0}
b){x>-1}
c){x > 1 }
2
d){x <-1}
e){x1}
f){-1  x <2}
The student estimates the probability of passing any course is
0.7, of failing is 0.1,and of withdrawing is 0.2.
Question14
2) Using the above figures, express in tabular form the
probability function f(x) induced by X. A assume independence
of grades between classes.
3)Graph the probability of X.
Answer14
X-{-2,-1,0,1,2}
S={FF.FP,FW,PP,PF,PW,WW,WF,WP}
1)
a){ FP, PF, WW}
b){X=0,1,2}={PW,WP,FP,PF,PP,WW}
c){X=1,2}={PW,WP,PP}
d){X=-2}={FF}
e){X=0,-1,-2}={FP,PF,FW,WF,FF}
f){X=-1,1}={FP,PF,FW,WF}
P(P)=.7,P(F)=.1,P(W)=.2
2)
X
-2
-1
0
F(X)
.01
.04
.18
1
2
.28
.49
Question15
Starting at a fixed time, we observe the sex of each newborn child
at a certain hospital until a boy(B) is born. Let p=P(B), assume
that successive births are independent, and define the random
variable X by X=number of births observed. Find the probability
mass function of X.
Answer15
X=1,2,3,….
F(1)=P(B)=P
F(2)=P(GB)=(1-P)P
F(3)=P(GGB)=(1-P) P
.
.
.
F(X)=
Question16
c
Suppose that f(x)= 3c x=1.2….
Is the probability function for a random variable X.
Determine c
Find P(2 X <5).
Find P(X  3).
Answer16


c
1
c

  c 
1 1
 c 1 
a)1   x 1 x  c  x 1    c   2  3  ....  


1
2
3
3
3
3
3
 


1   2  c  2
 3
2
 f ( x)  x
3
2 2 2 26
b) p(2  x  5)  2  3  4  4  0.32099
3 3 3
3
 c
2 2 2

c) p( x  3)  3 x   3  4  5  ....
3
3 3 3

2  1 1

 3 1   2 .....
3  3 3

1

9
x
Question17
The probability mass function of X is given as
F(X)=
Find
1.k
2.the cumulative function of X.
3. P(X>2)
4.P(-0.4<X<2)
5.P(X>1)
Answer17
2
2
4
1) 1 f ( x)  0   
k k k
2) F(X)==
4 1
3) 
8 2
2 1
4) 
8 2
6 3
5) 
8 4

8
k 8
k
Question18
Suppose X is random variable having density f given by
X
-3
-1
0
1
2
3
5
8
F(X)
0.1
0.2
0.15
0.2
0.1
0.15
0.05
0.05
Compute the following probabilities
a)X is negative
b) X takes a value between 1 and 8 inclusive
Answer18
a) f (3)  f (1)  0.3
b) p(1  x  8)  0.55
Question19
A fair die is tossed. Let X denote twice the number appearing
,and let Y denote 1 or 3 according as an odd or an even number
appears. Find the probability distribution of X and Y.
Answer19
x
2
4
6
8
10
12
F(x)
1/6
1/6
1/6
1/6
1/6
1/6
1
y
1
3
F(y)
1/2
1/2
Question20
Suppose a box has 12 balls labeled 1,2,…,12 Two independent
repetitions are made of the experiment of selecting a ball at
random from the box. Let X denote the larger of the two numbers
on the balls selected. Compute the density of X.
Question20
Suppose a box has 12 balls labeled 1,2,…,12 Two independent
repetitions are made of the experiment of selecting a ball at
random from the box. Let X denote the larger of the two numbers
on the balls selected. Compute the density of X.
STAT 111
Chapter Five
Question1
Among the 16 application for a job, ten have college degree. If
three of the application are randomly chosen for interviews, what
are the probability that
a)non has a college degree.
b) one has a college degree.
c)all three have college degree.
Answer1
10
6
a) c0 16c3  0.0357
c
b) c c
c
c) c c
c
3
10
6
1
2
16
 0.268
3
10
6
3
0
16
3
 0.214
Question2
Find the probability that 7 of 10 persons will recover from a
tropical disease, where the probability is 0.8 that any one of them
will recover from the disease.
Answer2
P(x  7)  c7 (0.8) 7 (0.2) 3  0.201
10
Question3
The average number of days school is closed due to snow during
the winter in a certain city is 4. What is the probability that the
schools in this city will close for 6 days during a winter.
Answer3
e 1 46
P(x  6) 
 0.1042
6!
Question4
A manufacturer of automobile tires reports that among a shipment
of 500 sent to a local distributor, 1000 are slightly blemished. If
one purchases 10 of these tires at random from the distribution,
what is the probability that exactly 3 will be blemished?
Answer4
P(x  3)  c3 (0.8) 3 (0.2) 7  0.201
10
Question5
The probability that a certain kind of component will survive a
given shock test is 3/4.
Find the probability that exactly 2 of the next 4 components
tested survive?
Answer5
1
4 3
P(x  2)  c2 ( ) 2 ( ) 2  0.2109
4 4
Question6
Suppose X has a geometric distribution with p=0.8. Compute the
probability of the following events.
a) X  3
b) 4  x  7
c)3  X  5 or 7  X  10
Answer6
a) p( x  3)  (1  p)3  0.008
b) p(4  X  7)  p( x  4)  p( x  7)  0.008
c) p(3  x  5( p(7  x  10)  0.04
Question7
In a manufacturing process in which glass items are being
produced, defects or bubbles occur, occasionally rendering the
piece undesirable for marketing. It is known that on the average
1 in every 1000 of these items produced has one or more bubbles.
What is the probability that a random sample of 8000 will yield
fewer than 7 items possessing bubbles?
Answer7
p( x  7)  p( X  6)  0.3134
Question8
Let X be uniformly distributed on 0,1,…,99. Calculate
a) P( X  25)
b) P(2.6  x  12.2)
c) P(8  X  10 or 2  X  32
d ) P(25  X  30)
Answer8
1
1
a ) p ( x  25)  p ( x  26)..... p ( x  99) 
 ... 
 0.75
100
100
10
b) p (3  x  12)  p ( x  3)  ..  p ( x  12) 
 0. 1
100
c) p (3  x  32)  p (4  x  32)  p ( x  4)  .. p ( x  32)  0.29
d ) p ( x  25)  p ( x  26)  ..  p ( x  30)  0.06
Question9
As part of air pollution survey , an inspector decides to examine
the exhaust of 6 of a company's 24 trucks. If 4 of the company's
trucks emit excessive amounts of pollutants, what is the
probability that none of them will be included in the inspector's
sample?
Answer9
4
20
6
p( x  0)  c0 c
 0.288
24
c
6
Question10
A fair die is rolled 4 times. Find
a) The probability of obtaining exactly one 6.
b)The probability of obtaining no 6.
c)The probability of obtaining at least one 6.
Answer10
3
1 5
a) p( x  1)  c1       0.386
8 6
4
0
4
4 1 
5
b) p( x  0)  c0       0.482
6 6
c) p( x  1orx  2orx  3orx  4)  1  p( x  0)  0.518
Question11
Of a population of consumers, 60% is reputed to prefer a
particular brand A of toothpaste. If a group of consumers is
interviewed, what is the probability that exactly five people have
to be interviewed to encounter the first consumer who prefers
brand A.
Answer11
p( x  5)  0.60.4  0.01536
4
Question11
Of a population of consumers, 60% is reputed to prefer a
particular brand A of toothpaste. If a group of consumers is
interviewed, what is the probability that exactly five people have
to be interviewed to encounter the first consumer who prefers
brand A.
Answer11
p( x  5)  0.60.4  0.01536
4
Question12
Team A has probability 2/5 of winning whenever it plays. If A
plays 4 games, find the probability that A ins
a)2 games.
b)at least 1 game
c)more than half of the games.
Answer12
2
2
4 2   3
a) p( x  2)  c2      0.3456
 5 5
4
3
b) p( x  1)  1  p( x  1)  1  p( x  0)  1     0.8704
5
c) p( x  2)  1  p( x  2)  1  0.8208 0.1792
Question13
The telephone company reports that among 5000 telephones
installed in a new subdivision 4000 have push-buttons. If 10
people are called at random, what is the probability that exactly 3
will be talking on dial telephones?
Answer13
3
7
10  1   4 
p( x  3)  c3      0.201
5  5
Question14
Suppose that 30% of the application for a certain industrial job
have advanced training in computer programming. Application
are interviewed sequentially and are selected at random from the
pool. Find the probability that the first application having
advanced in programming is found on the fifth interview.
Answer14
p( x  5)  (0.3) (0.7)4  0.07203
Question15
Suppose 2% of the items made by a factory are defective. Find
the probability that there are 3 defective items in a sample of 100
items.
Answer15
e 2 23
p( x  3) 
 0.180
3!
Question16
From a group of twenty PhD engineers, ten are selected for
employment . What is the probability that the ten selected include
all the five best engineers in the group of twenty.
Answer16
x~ hyp(n, N , K )  hyp(10,20,5)
 5 15
  
5  5 

f (5) 
 0.0163
 20
 
10 
Question17
If the probability is 0.40 that a child exposed to a certain
contagious disease will catch it, what is the probability that the
tenth child exposed to the disease will be the third to catch it.
Answer17
x~ Nb(3,0.4)
9
3
7
f (10)   0.4 0.6  0.0645
 2
Question18
Past experience has shown that the occurrence of defects in a
telephone line being produced by a certain machine generated a
Poisson process with 5 defects per kilometer occurring on the
average.
a) what is the probability that there will be 5 or less defects in 2
kilometers of cable?
b)what is the probability that there will be exactly 3 defects in ¼
kilometers of cable?
Answer18
10y e 10
a) f ( x) 
y!
p( y  5)  f (5)  0.0671
5
4
5
e  
 4   0.093
b) f (3)  p( z  3) 
3!
Question19
An inspector in a television manufacturing plant has observed
that defective tuners occur at a rate of 3 per 100 sets inspected.
What is the probability that in 30 sets inspected, 2 or few will
have defective tuners?
Answer19
p( x  2)  0.9371
Question20
The painted light bulbs produced. By a company are 50% red,
30% blue and 20% green. In a sample of 5 bulbs, find the
probability that 2 are red, 1 is green and 2 are blue.
Answer20
5 
0.52 0.31  0.135
p( A)  
 212
Question21
Find the probability of getting 5 heads and 7 tails in 12 of flips a
balanced coin?
Answer21
1
x ~ bin(12, )
2
5
7
12  1   1 
p( x  5)  c5      0.193
2 2
Question22
If the probability is 0.75 that an application for a driver's license
will pass the road test on any given try, what is the probability
that an application will finally pass the test on the fourth try.
Answer22
x ~ G(0.75)
p( x  4)  (0.75)(0.25)3  0.0117
Question23
The manufacturer of parts that are needed in an electronic device
guarantees that a box of its parts will contain at most two
defective parts. If the box holds 20 parts and experience has
shown that the manufacturer process produces 2 percent defective
items, what is the probability that a box of the parts will satisfy
the guarantee?
Answer23
x ~ b(20,0.02)
x ~ position(0.4)
f (2))  p( x  2)  0.9921
Question24
In an assembly process, the finished items are inspected by a
vision sensor, the image data is processed , and a determination is
made by computer as to whether or not a unt is satisfactory. If it
is assumed that 2% of the units will be rejected, then what is the
probability that the thirtieth unit observed will be second rejected
unit?
Answer24
x ~ Nb(2,0.02)
P( X  30)  c1 (0.02) 2 (0.98) 28  0.0066
29
Question25
In an interactive time-sharing environment it is found that, on
average, a job arrives for CPU service every 6 seconds. What is
the probability that there will be less than or equal to 4 arrivals in
a given minute? What is the probability that there will be
inclusively between 8 and 12 jobs arriving in a given minute?
Answer25
p( x  4)  0.0293
p(8  x  12)  f (12)  f (7)  0.5714
Question26
Lots of 40 components each are called acceptable if they contain
no more than 3 defective. The produce for sampling the lot is to
select 5 components at random and to reject the lot if a defective
is found. What is the probability that exactly 1 defective will be
found in the sample if there are 3 defective in the entire lot?
Answer26
x ~ hyp(5,40,3)
3
37
4
p( x  1)  c1 c
 0.30111
40
c
5
Question27
Find the probability of obtaining exactly three 2's if an ordinary
die is tossed 5 times.
Answer27
1
x ~ b(5, )
6
 5  1 
p( x  3)    
 3  6 
3
2
5
   0.03215
6
Question28
Find the probability that a person tossing three coins will get
either all heads or all tails for the second time on the fifth.
Answer28
2
x ~ Nb(2, )
8
p  p( HHH  TTT )  P( HHH )  P(TTT ) 
 4  2   6 
p( x  5)        0.1055
1  8   8 
2
3
2
8
Question29
A geological study indicates that an exploratory oil well drilled in
a particular region should strike oil with probability 0.2. Find the
probability that the third oil strike comes on the fifth well drilled.
Answer29
x ~ Nb(3,0.2)
p( x  5)  c2 (0.2)3 (0.8) 2  0.03072
4
STAT 111
Chapter six
Question1
Two refills for a ballpoint pen are selected at random from a box
that contains 3 blue refills, 2 red refills, and 3 green refills. If X is
the number of blue refills and Y is the number of red refills
selected, find
a) the joint probability function.
b) P{(X,Y)} where A is the region { (x,y):x+y1}.
Answer1
Y
0
3/28
9/28
3/28
15/28
1
6/28
6/28
0
12/28
2
1/28
0
0
1/28
F(X)
10/28
15/28
3/28
1
18
A  f (0,0)  f (1,0)  f (0,1) 
26
Question2
From a sack of fruit containing 3 oranges, 2 apples, and 3
bananas a random sample of 4 pieces of fruit is selected. If X is
the number of oranges and Y is the number of apples in the
sample, find
a)the joint probability distribution of X and Y;
b)P[(X,Y), where A is the region {(x,y) x+y ≤ 2}
Answer2
x/y
0
1
2
3
f(y)
0
0
3/70
3/70
3/70
15/70
1
2/70
18/70
18/70
2/70
40/70
2
3/70
9/70
3/70
0
15/70
F(X)
5/70
30/70
30/70
5/70
1
A  f (0,0)  f (0,1)  f (0.2)  f (1,0)  f (1,1)  f (2,0)  0.5
Question3
Suppose an experiment consists of three flips of a fair coin, with
each outcome being equally likely. Let X denote the number of
heads on the last flip. Y, the total number of heads for the three
tosses. Find the joint probability mass function.
Answer3
x/y
0
1
2
0
1/8
0
1/8
1
2/8
1/8
3/8
2
1/8
2/8
3/8
3
0
1/8
1/8
F(X)
4/8
4/8
1
S  {HHH , HTH , THH , HHT , TTH , THT , HTT , TTT }
X
1
1
1
0
1
0
0 0
Y
3
2
2
2
1
1
1
0
Question4
Two tablets are selected at random from a bottle containing 3
aspirin, 2 sedative, and 4 laxative tablets, If X and Y are ,
respective, the number of aspirin tablets and the number of
sedative tablets included among the two tablets drawn from the
bottle, find
a)the probabilities associated with all possible pairs of
values(x,y).
b)the marginal distribution of X and Y.
c)the conditional distribution of X given Y=1
Answer4
x/y
0
1
2
f(y)
0
6/36
12/36
3/36
21/36
1
8/36
6/36
0
14/36
2
1/36
0
0
1/36
F(X)
15/36
18/36
3/36
1
Y
0
1
2
F(Y)
21/36
14/36
1/36
X
0
1
2
F(X)
15/36
18/36
3/36
x
0
1
2
F(x)
8/14
6/14
0
X:num of as
,
y:num of se
1
1
1
Question5
Lets X and Y denote the number of black and white balls,
respectively, that will be obtained in drawing two balls from a
bag that contains two black and two white balls. Find the joint
probability .mass function of X and Y.
Answer5
x/y
0
1
2
f(y)
0
0
0
1/6
1/6
1
0
4/6
0
4/6
2
1/6
0
0
1/6
F(X)
1/6
4/6
1/6
1
X:black balls
Y:white balls
Question6
Suppose that X and Y have following joint probability function
y/x
1
2
3
1
0
1/6
1/12
2
1/5
1/9
0
3
2/15
1/4
1/18
Find
1.The marginal distribution of the random variable X.
2.The marginal distribution of the random variable Y.
3. P(Y=3\X=2)
Answer6
x/y
0
1
2
f(y)
1
0
30/180
15/180
45/180
2
36/180
20/180
0
56/180
3
24/180
45/180
10/180
79/180
F(x)
60/180
95/180
25/180
1
x
1
2
3
F(x)
60/180
95/180
25/180
y
1
2
3
F(y)
45/180
56/180
79/180
45
f ( x  2, y  3) 180 45
3.

 0.474
95 180
f ( x  2)
180
1
1
Question7
Consider an experiment that consists of 2 rolls of a balanced die.
If X is the number of 4's and Y is the number of 5's obtained in
the 2 rolls of the die, find
a)the joint probability distribution of X and Y;
b)P[(X,Y)  A] where A is the region given by {(x,y): 2x+y<3}
Answer7
x/y
0
1
2
f(y)
0
16/36
8/36
1/36
25/36
1
8/36
2/36
0
10/36
2
1/36
0
0
1/36
F(X)
25/36
10/36
1/36
1
16
f (0.0)  , f (0,1)  f {(1,5), (2,5), (3,5), (6,5)
36
8
(5,1), (5,2), (5,3), (6,5)} 
36
33
p{( x, y ) : 2 x  y  3}  f (0,0)  f (0,1)  f (0,2), f (1,0) 
36
Question8
A fair coin is tossed three times. Let X denote 0 or 1 according as
a head or a tail occurs on the first toss, and let Y denote the
number of heads which occur.
Determine
a)the distribution of X and the distribution of Y.
b) the joint probability mass function
Answer8
x/y
0
1
f(y)
0
0
1/8
1/8
1
1/8
2/8
3/8
2
2/8
1/8
3/8
3
1/8
0
1/8
F(X)
4/8
4/8
1
x
0
1
F(x)
1/2
1/2
1
y
0
1
2
3
F(y)
1/8
3/8
3/8
1/8
1
Question9
From a group of three Republicans, two Democrats, and one one
Independent, a committee of two people is to be randomly
selected, Let X denote the number of Republicans and Y the
number of Democrats on the committee. Find
A) the joint probability distribution of X and Y, and then find the
marginal distribution of X.
b)the conditional distribution of X given that Y=1.
Answer9
x/y
0
1
2
f(y)
0
0
3/15
3/15
6/15
1
2/15
6/15
0
8/15
2
1/15
0
0
1/15
F(X)
3/15
9/15
3/15
1
f ( x,1) f ( x,1)
f ( x / y  1) 

8
f ( y,1)
15
x
0
1
2
F(x)
2/8
6/8
0
1
Question10
Consider the joint probability distribution defined by the formula
1
f ( x, y ) 
( x  2 y)
27
x=0,1,2
y=0,1,2
Find the marginal distribution of X and Y, and f(x/y )
Answer10
x/y
0
1
2
f(y)
0
0
1/27
2/27
3/27
1
2/27
3/27
4/27
9/27
2
4/27
5/27
6/27
15/27
F(X)
6/27
9/27
12/27
1
If(y/x)=f(x,y)/f(x)
If x=0
y
0
1
2
Fy(x=0)
0
1/3
21/3
if x=1
if x=2
y
0
1
2
Fy(x
=1)
1/9
3/9
5/9
1
y
0
1
2
Fx/x
=2
1/6
2/6
3/6
1
Question11
The joint probability function of two random variables X and Y is
given as
F(x ,y )=c(2x+y) x=0,1,2 y=0,1,2,3
a)Find the value of the constant c
b)Find P(X=2,Y=1)
C)Find P(X 1, Y 2)
d)Find the marginal distribution of X and Y.
e)Find f() , and P(Y=1\X=2)
f)Determine whether the random variables X and Y are
independent .
Answer11
x/y
0
1
2
3
0
0
2c
4c
6c
1
c
3c
5c
9c
2
2c
4c
6c
12c
3
3c
5c
7c
15c
F(X)
6c
14c
22c
42c
Answer11
a ) x

y
f ( x, y )  1  c 
b) f (2,1)  5c 
1
42
5
42
c) f (1,2)  f (1,1)  f (1,0)  f (2,2)  f (2,1)  f (2,0)  24c 
f ( y,2)
fx(2)
f (2,1) 5
e)

fx(2) 22
f ) no, for example: f(0,0)=0 but fx(0)fy(0)#0
d ) f ( y,2) 
24
42
Question12
Let the joint probability mass function of X and Y is given in the
following table
Find
P (X  2, Y  2)
P (X  1)
P (X  Y  4)
P (X  2, Y  1)
F(-1,2)
F(1.5,2)
F(5,7)
X/Y
1
2
3
4
1
0.1
0
0.1
0
2
0.3
0
0.1
.20
3
0
0.2
0
0
Answer12
X/Y
1
2
3
4
F(x)
1
0.1
0
0.1
0
0.2
2
0.3
0
0.1
.20
0.6
3
0
0.2
0
0
0.2
F(Y)
0.4
0.2
0.2
0.2
1
1) 0 .5
2 ) 0 .6
3) 0 .1
4)0
5) 0
6 ) 0 .1
7 )1
Question13
Let X and Y be independent random variables with the following
distribution ;
X
1
2
Y
5
10
15
F(X)
0.6
0.4
F(Y)
0,2
0.5
0.3
Find the joint distribution of X and Y.
Answer13
x/y
0
1
f(y)
5
0.12
0.08
0.2
10
0.3
0.2
0.5
15
0.18
0.12
0.2
F(X)
0.6
0.4
1
f (1,5)  0.12
f (1,10)  0.30
f (1,15)  0.18
f (2,5( 0.08
f (2,10)  0.20
f (2,15)  0.12
Question14
Suppose that the joint probability mass function of X and Y, is
given by
F(0,0)=0.4
f(0,1)=0.2
f(1,0)=0.1
f(1,1)=0.3
Calculate the conditional probability mass function of X, given
that Y=1
Answer14
x/y
0
1
f(y)
0
0.4
0.1
0.5
1
0.2
0.3
0.5
F(X)
0.6
0.4
1
f y (1)  x f ( x,1)  0.2  0.3 0.5
 0.2


0
.
4
,
x

0

f ( x,1) x f ( x,1) 
 0.5

f x ( x / y) 



0
.
5
f y (1)
0.5

 0.6, x  1 


 0.5

Question15
Suppose that X and Y have following joint probability function
X/Y
1
2
3
4
1
0.1
0
0.1
0
2
0.3
0
0.1
0.2
3
0
0.2
0
0
Find
The marginal distribution of the random variable X;
The marginal distribution of the random variable y;
Determine whether the random variables X and Y are
independent.
Answer15
X
1
2
3
F(x)
0.2
0.6
0.2
1
y
1
2
3
4
F(y)
0.4
0.2
0.2
0.2
No,for example: f(1,2)=0, but fx(1)=0x0.2#1
1
STAT 111
Chapter seven
Question1
Let (x,y) have the following joint distribution function
x/y
1
2
3
sum
1
0
1/6
1/6
1/3
2
1/6
0
1/6
1/3
3
1/6
1/6
0
1/3
sum
1/3
1/3
1/3
1
Find
a)the probability mass function of X+Y.
b)the probability mass function of XY.
c)the probability mass function of X2.
d)the probability mass function of Y2.
Answer1
a)
b)
X+y
2
3
4
5
6
sum
F(x+y)
0
2/6
2/6
2/6
0
1
xy
1
2
3
4
6
9
sum
F(xy)
0
2/6
2/6
0
2/6
0
1
x2
1
4
9
sum
F(x2)
1/3
1/3
1/3
1
y2
1
4
9
sum
F(y2)
1/3
1/3
1/3
1
c)
d)
Question2
Lets X be a random variable with probability distribution
F(x)=
Find the probability distribution of the random variable Y=2X-1
Answer2
x
1
2
3
sum
y
1
3
5
sum
F(y)
1/3
1/3
1/3
1
Y=2x-1, x=
-1
= g (y)=>fy (y)=fx (g (y))
Question3
Let X1 and X2 be discrete random variable with joint probability
distribution
F(x)=
Find the probability distribution of the random variable Y=X1X2
Answer3
y
1
2
3
4
6
sum
Fy(y)
1/18
4/18
3/18
4/18
6/18
1
X1/x2
1
2
sum
1
1/18
2/18
3/18
2
2/18
4/18
6/18
3
3/18
6/18
9/18
sum
8/18
12/8
1
Question4
Let X be a random variable with probability distribution
F(x)=
Find the probability distribution of the random variable Y=X2
Answer4
x
1
2
3
F(x)
1/n
1/n
1/n
……
n
1/n
y
1
4
9
F(x)
1/n
1/n
1/n
……25
N2
1/n
Question5
Let X be a random variable with probability distribution
F(x)=
Find the probability distribution of the random variable Y=X2
Answer5
x
1
…..
2
n
y
1
4
9
n
Fy(Y)
2/2n
2/2n
2/2n
2/2n
STAT 111
Chapter eight
Question1
A lot twelve television sets includes two that are defective. If
three of the sets are chosen at random, how many defective sets
can they expect?
Answer1
x
0
1
2
sum
F(x)
6/11
9/11
1/22
1
Xf(x)
0
9/22
2/22
11/22
 2 10 
 

x  s  x 

f ( x) 
12
 
x 
x ~ hyp(3,12,2)
nk
2 1
E ( x) 
3 
N
12 2
Question2
If the random variable X is the two faces when rolling a pair of
balanced dice, find the expected value of X.
Answer2
X
2
3
4
5
6
7
8
9
10
F(X)
1/36
2/36
3/36
4/36
5/36
6/36
5/36
4/36
3/36
11
12
SUM
2/3 1/36
6
1
Question3
Let X have the following probability mass function
F(x)=
Find E(X),E(X2), and e(x+2)2
Answer3
X  uniform n  5
E(X)  3
E(X2 )  11
E(X  2) 2  27
Question4
The probability mass function of the random variable X is given
by
x
1
2
3
4
F(x)
4/10
1/10
3/10
2/10
Find the expected value of X.
sum
Answer3
X
1
2
3
4
SUM
F(X)
4/10
1/10
3/10
2/10
1
XF(X)
4/10
2/10
9/10
8/10
2.3>E(x)
Question5
The probability mass function of the random variable X is given
by
x
4
5
6
7
8
9
F(x)
1/12
1/12
1/4
1/4
1/6
1/6
Find the expected value of g(x)=2x-1.
sum
Answer5
x
4
5
6
7
8
9
sum
F(x)
1/12
1/12
1/4
1/4
1/6
1/6
1
E(x)=Xf(x)
4/12
5/12
6/4
7/4
8/6
9/6
82/1
2
E(X)=82/12=41/6
E(2X-1)=2E(X)-1=2(41/6)-1=12.7
Question6
Let the random variable X represent the number of automobiles that are
used for official business purposes on any give workday. The
probability distribution for company A is given by
x
1
2
3
F(x)
0.3
0.4
0.3
And for company B is given by
x
0
1
2
3
4
F(x)
0.2
0.1
0.3
0.3
0.1
Show that the variance of the probability distribution for company B is
greater than that of company A
Answer6
x
1
2
3
sum
F(x)
0.3
0.4
0.3
1
E(X)=X f(x)
0.3
0.8
0.9
2
E(X2)=X2 f(x)
0.3
1.6
2.7
4.6
Va=4.6-22 =0.6
X
0
1
2
3
4
F(X)
0.2
0.1
0.3
0.3
0.1
XF(X)
0
0.1
0.6
0.9
0.4
X2 f(x)
0
0.1
1.2
2.7
1.6 5.6->E(X 2)
Vb=5.6-22 =1.6
Vb(x)> Va(x)
1.6
0.6
SUM
2->E(X)
Question7
Find the moment generating function of the discrete random variable X
has the probability distribution
F(x)=2x
x=1,2,…
And use it to find µ1 µ2
Answer7
Question8
An urn contains nine chips, five red and four white. Three are drawn
out at random without replacement . Let X denote the number of red
chips in the sample. Find E(X).
Answer8
x ~ hyp(3,9,5)
 5  4 
 

x  3  x 

f ( x) 
9
 
3
 n  5
E ( x)  n   3   1.667
 N  9
Question9
Suppose that a sequence of independent tosses are made with a coin for
which the probability of obtaining a head on any given toss is 1/30.
Find the expected number of tosses that will be required in order to five
heads?
Answer9
X:number of heads
Y: number of tosses
1
X  bin(n, )
30
E ( x )  np  5
n
 5  n  150
30
Question10
Let X be binomial distribution random variable with E(X)=2 , and
V(X)=4/3 . Find the distribution on X.
Answer10
X  4 bin (n,p)
n? p?
E(x)  np  2......
equation(1)
3
V(x)  npq  ....... equation(2)
4
4
4
2
q 
3
6
3
2 1
p  1 q  1 
3
3
2q 
1
2n6
3
1
 x ~ bin(6,
3)
n
Question11
Let x1,x2 ,and x3 be independent random variables having finite
positive variances 21 , 22, 23, respectively. Find the correlation
between x1-x2 and x2+x3.
Answer11
cor(x1 - x2 , x2  x3 ) 
cov(x1  x2 , x2  x3 )
v( ( x1  x2 )v( x2  x3 )
Question12
A box has 3 red balls and 2 black balls. A random sample of size 2 is
drawn without replacement. Let U be the number of red balls selected
let V be the number of black balls selected. Compute (U,V).
Answer12
UR/VB
0
1
2
Fv(V)
0
0
0
3/10
3/10
1
0
6/10
0
6/10
2
1/10
0
0
1/10
Fu(U)
1/10
6/10
3/10
1
E(U)=6/5
E(U.U)=18/10
V(U)=.36
E(UV)=-.36
 (U,V)=-1
,
,
,
E(V)=4/5
E(V.V)=1
V(V)=.36

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