Successfully reported this slideshow.
We use your LinkedIn profile and activity data to personalize ads and to show you more relevant ads. You can change your ad preferences anytime.
Stat 111
Collected by
Ayoob Aboobaker
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
possibl...
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)...
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),...
Question2
Suppose a family contains two children of different ages, and we
are interested in the sex of these children. Le...
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 =...
Question3
A total of 36 members of club play tennis, 28 play squash, and
18 play badminton. Furthermore, 22 of the members...
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)-
...
Question4
In a survey carried out in a school snack shop. The following
results were obtained. Of 100 boys questioned, 78 ...
Answer4
N(S)=100,
N(SW)=78,
N(I)=74,
N( C) =53,
N(SW ∩I)=57,
N(SW∩C)=4,
N(SW∩I ∩C)=31.
N(S)=N(SW ∪I ∪C)=N(SW+N(I)+N( C ) –...
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 ...
Answer1
6P4 = 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
bott...
3 X 4 = 12
Answer2
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 ...
Answer3
60
!1!2!3
!6

Proteins in living cells are composed of 20 different
kinds of amino acids. Most proteins consist of several
hundred amino...
Answer4
2
)20(
100
20.....202020


time
Find the number of subsets of a set X containing n
elements?
Question5
Either the subsets containing no element ,
1 elements 2 elements , n elements
Answer5






0
n






1
n
...
In how many can a person gathering data for a market
research organization interview 3 of the 20 families
living in a cert...
Answer6
1140
3
20 20
3






c
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. ...
3 X 5 = 15
Answer7
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 fa...
a) 2x2x2x2Xx….x2 = 20 = 1048,576
b)
c)
d)
Answer8
520,77
13,7
20
7
20 20
7












c
756,184
10
20
...
How many license plates may be formed beginning with
2 different letters of the Arabic alphabet following by 4
different d...
Without repetition :
28x27x10x9x8x7= 3,810,240.
With repetition :
28x28x10x10x10x10=7,840,000.
Answer9
A test has 10 true-false questions and 6 multiple-choice
questions with 5 possible choices for each. How many
possible set...
Answer10
000,000,1652
610

How many different 6-digits numbers may be formed
using the digits from the 4,4,5,6,6,6.
Question11
Answer11
60
!3!1!2
!6

A telephone company in a certain area has all telephone
numbers prefixed by either 465,475, or 482, followed
by 4 digits. ...
With repetition
Without repetition
Answer12
000,30103 4

520,245673 
A student may select one of 3 English classes, one of 2
mathematics classes ,and one of 2 history classes for his
program....
Answer13
12223
3
1
2
1
3
1
 ccc
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...
a)
b)
Answer14
361526
4
6
1
2
5
6
0
2
























26
5
6
0
2
3
6
2
2

...
How many ways can we arrange the letters in the word
cold?
Question15
4! = 24
Answer15
How many ways may 5 people be seated in a
5-passenger vehicle if one of two people must drive?
Question16
The drive seat must be filed in 2 ways, after that, the
remaining 4 can be arranged in 4!
Answer16
!4
1
2






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 ...
a)
b)
Answer17
120
10
7
c
110
5
2
5
5
5
3
5
4
5
4
5
3
cccccc
In how many ways can 2 oaks, 3 pines, and 2 maples be
arranged in a straight line if one does not distinguish
between tree...
Answer18
210
!2!3!2
!7

Ten persons have organized a club. How many different
committees consisting of 3 persons may be formed from
these 10 peopl...
a)
b)
Answer19
120
10
7
c
720
10
3
p
If eight persons are having dinner together, in how
many different ways can three order chicken, four order
steak, and one...
Answer 20
280
!4!3
!8

a)How many distinct permutations are there of the
letters in the word statistics?
b) How many of these begin and end with ...
1)
2)
Answer 21
400,50
!2!3!3
!10

360,3
!2!3
!8

In how many ways can 7 books be arranged on a shelf if
a) Any arrangement is possible,
b) 3 particular books must always s...
1)
2)
3)
Answer 22
040,5!7 
720!5!3 
240!52 
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) th...
a)
b)
Answer 23
600,7558765 
5044376 
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 ...
a)
b)
c)
Answer 24
120!5 
72!42!5 
12!3)!13( 
From a group of 8 women and 6 children a committee
consisting of 3 women and 3 children is to be formed.
How many differen...
a)
b)
c)
Answer 25
896
8
3
4
2
2
1
8
3
4
3
2
0
 cccccc
1000
2
6
1
2
3
6
0
2
3
6
















...
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...
3x2x1x2=12
Answer 26
CBA 
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)...
Answer 27
!1!1!4!6
!12
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 t...
a) 10! = 3628,800
b) 6! 4! 2 = 34,560
Answer 28
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 a...
Answer 29
240,30
5.....
10






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...
Answer 30
30
3
1
10
1
cc
How many functions defined on n points are possible if
each functional value is either 0 or 1 ?
Question31
Answer 31
n
timen
2
_
2...222


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.
Question...
Answer 32
24243 
How many sample points are in the sample space when
a pair of dice is thrown once?
Question33
6X6=36
S={(1,1),(1,2),…..(1,6)}
Answer 33
Calculate the number of permutations of the letters a, b,
c, d had taken two at a time?
Question34
Answer 34
12
4
2
p
four names are drawn from the 24 members of a club,
for the offices of president, vice - president , treasurer,
and secret...
Answer 35
024,255
24
4
p
In how many ways may 3 books be placed next to each
other on a shelf?
Question36
Answer 36
6!3 
Four different mathematics books, 6 different physics
books, and 2 different arrangements are possible if
a)The books in e...
no rest = 12!
a) 3! (4! 6! 2! ) = 207360
b) 4! 9! = 8,709,120
Answer 37
Four women and four children to be seated in a row of
chairs numbered 1 thought 8;
a) How many total arrangements are poss...
a) 8! = 40, 320
b) 2! (4! 4!) + 1,152
c)
d)
Answer 38
70
!4!4
!8

1680
!4
!8

From 4 chemists and 3 physicists find the number of
committees that be formed consisting of 2 chemists and
1 physicist?
Qu...
Answer 39
18
1
3
2
4












Form a group of 5 teachers and 7 students, how
different committees consisting of 2 teachers and 3
students can be formed?...
a)
b)
Answer 40
  300
300
350
5
2
2
1
5
3
2
0
5
2
5
2
5
2
2
1
5
2
5
3
2
0
7
3
5
2



ccccc
cccccc
cc
or
From 5 statisticians and 6 economists a committee
consisting of 3 statisticians and 2 economists is to be
formed. How many...
a)
b)
c)
Answer 41
100
45
150
5
2
5
3
6
2
3
1
2
2
6
2
5
3



cc
ccc
cc
A shipment of 10 television sets includes three that are
defective. In how many ways can a hotel purchase four
of these an...
Answer 42
70
7
1
3
3
7
2
3
2
 cccc
In how many ways can a set of four objects be
partitioned into three subsets containing, respectively,
2,1 and 1 of the ob...
Answer 43
12
1,1,2
4






In how many ways can seven scientists be assigned to
one triple and two double hotel rooms?
Question44
Answer 44
210
2,2,3
7






a)How many license plates are there if the first three
place are form the Arabic letters and the last three are
numbers?
b...
Answer 45
320,152,148910262728)
000,952,21101010282828)


b
a
If a travel agency offers special weekend trip to 12
different cities, by air, rail or bus. In how many different
ways can...
Answer 46
36312 
In an experiment consists of throwing a die and then
drawing a letter at random from the English alphabet,
how many points...
Answer 47
156266 
In a medical study are patients are classified in 8 way
according to whether they have blood type AB +, AB - ,
A+ , A - ,B...
Answer 48
2438 
If 4 teachers, 3 engineers, and 3 doctors are to be seated
in a row, how many seating arrangement are possible
when people...
Answer 49
184,5)!3!3!4(!3 
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 cour...
Answer 50
420
1,2,3,1
7






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...
Answer 51
300,33)
650,13)
760,38)
10
2
10
4
10
3
10
3
10
4
10
2
15
4
5
2
20
6



cccccc
cc
c
c
b
a
STAT 111
Chapter Two
The following data were given in a study of a group of
1000 subscribers to a certain magazine. In reference to
sex, marita...
Answer 1
11.057G)MrP(M
.25G)MrP(M
.147G)P(Mr
.042G)P(M
.525P(G)
.470P(MR)
312P(M)
CollegeG
MarriedMr
MaleM




...
The mathematics department consists of full professors,
15 associate professors, and 35 assistant professors; a
committee ...
Answer 2
029.
6
75
1
35
3
15
2
25
)(
00806.
6
75
6
35
)(
6
75
)(
























...
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).
Q...
Answer 3
5/6B)P(A
5/6)BP(A
1/6)BP(A
1/6B)P(A
c
cc
c




1/22/61/6
1/22/61/6
12/31/3
C
AA
B
C
B
If 3 books are picked at random from a shelf containing
5 mathematics, 3 books of statistics, and a chemistry,
what is the...
Answer 4
3571.
3
9
1
3
2
5
)
33.
3
9
2
8
1
1
)
































...
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 kn...
(The system will work)=
Answer 5
72.0)(
8.0)(
9.0)(



BAp
Bp
Ap
98.72.8.9.)()()()(  BAPBPAPBAp
p
Let A and B be events with P(A)=1/2 ,
P(A∪B)= 3/4,P(B )=5/8 . Find P(A∩B), P(A ∪ B )
,P(A ∩ B ),and P(B ∩A ).
Question6
c
...
Answer 6
2/8)AP(B
7/8)BP(A
1/4)BP(A
1/8B)P(A
c
cc
cc




A pair of fair dice is tossed. Find the probability that the
maximum of the two numbers is greater than 4?
Question7
Answer 7
5556.
30
20






p
Of 120 students, 60 are studying French, 50 are
studying Spanish, and 20 are studying French and
Spanish. If a student is ...
Answer8
120
30
)(
120
40
)(,
6
1
120
20
SP)P(F
120
50
P(SP),
120
60
P(F)
20SP)n(F50,n(SP)60,n(F)120,n(S)



...
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 t...
Answer9
2398.
5
15
2
9
3
6



















A jar contains 3 red, 2 green,4blue, and 2 white marbles.
Four marbles are selected at random without
replacement from thi...
Answer10
0727.
4
11
1
2
1
4
0
2
2
3































If 2 balls are randomly drawn from a bowl containing 6
white and 5 black balls, what is the probability that one
of the dr...
Answer11
5455.
2
11
1
5
1
6



















If the probability that a student A will fail a certain
statistics examination is 0.5, the probability that student
B will...
Answer12
.6B)P(A-P(B)P(A)B)P(A
.1B)P(A
.2P(B)
.5P(A)




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 t...
Answer13
.}T),T,(1,H),T,(1,T),H,(1,
H),H,{(1,H)}(6,H),(4,T),(2,H),{(2,S


If the probability are, respectively, 0.09, 0.15, 0.21, and
0.23, that a person purchasing a new automobile will
choose th...
Answer14
colures]thoseofoneleastator[
exclusive)(mutually
.68p(b))rp(p(w)p(g))brwP(g 
A die is loaded in such a way that the probability of any
particular face's showing is directly proportional to the
number...
Answer15
21
12
P(6)P(4)P(2)P(E)
21
1
w
121w16w5w4w3w2wwP(S) 1



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. ...
Answer16
12
4
1
12
1
4













It is known that a patient will respond to a treatment of
a particular disease with probability equal to 0.9. If
there pat...
P(at least one )=
Answer17
9.0)(1
)(1)(
321
331321


CCC
AAAP
AAAPAAAp
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 )
Q...
are indep also)
Answer18
CCCC
BABpApb
BpApBpApa
,(4.)()()
6.)()()()()


A mixture of candies contains 6 mint, 4 toffees, and 3
chocolates. If a person makes a random selection of one
candies, Fi...
or
Answer19
4615.
1
13
1
6
) 












a
13
6
A die is tossed 50 times. The following table gives the
six numbers and their frequency of occurrence
Find the relative fr...
Answer20
52.
50
989
)
48.
50
987
b)
14.
50
07
a)





c
Three women and three children sit in a row. Find the
probability that
a) the 3 children sit together.
b)the woman and chi...
Answer21
.1
6!
3!3!2!
b)
.2
6!
)4!(3!
a)


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).
Question22
Answer22
8
1
)BP(A
8
6
P(B)
8
3
P(A)
c



6/84/82/8
1/81/81/8
15/83/8
C
A
C
B
A
B
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 e...
.:not indep
Answer23
)()()()(
041.
36
61818
)()()(
083.
36
3
C)BP(A
(6,6)}(4,4),{(2,2),CBA
6)Cn(36,n(S)18,n(B)18,n(A)
,6)}...
Three names to be selected from a list of seven names
for use in a particular public opinion survey. Find the
probability ...
Answer24
4286.
3
7
2
6
1
1



















A hat contains twenty white slips of paper numbered from 1
through 20, ten red slips of paper numbered from 1 through 10,
...
Answer25
e).275
.1d)8/80
.1c)8/80
.25b)20/80
.37520)/80a)(10




Three studentsA,B and C are in a swimming race. A and B have
the same probability of winning and each is twice as likely t...
Answer26
.63/6b)p(c-p(b))p(cb)P(c
1/5)P(c
2/5P(b)
2/5P(a)





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 ha...
Answer27
4667.
2
10
2
7
0
3
)
06.
2
10
0
7
2
3
)































...
Consider families with two children. Let E be the event that a
randomly chosen family has at most one girl, and F ,the eve...
Answer 28
2/43/8
16/62/43/4P(E)P(E)
2/4F)P(E
BG}{GB,F
BB}BG,{GB,E
BG}GB,GG,{BB,S






Find the probability of getting three heads in three (independent)
tosses of a balanced coin.
Question29
Answer 29
8
1
P(HHH)
TTT}HTT,THT,TTH,THH,HTH,HHT,p(HHH,

Relays used in the construction of electric circuits function
properly with probability 0.9. A assuming that the circuits ...
For design(1):
=.9
For design(2):
=.96639
Answer 30
Find the probability of getting at least one head in 5 tossed of a
balanced coin?
Question31
P(at least one head)=1-p(HHH)=.969
Answer 31
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...
Answer 32
1471.
13
18
1
6
1
5
1
4
)
0123.
816
10
3
18
3
5
)


























...
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 co...
Answer 33





 













 












r
N
r
N
r
NM
Kr
N
K
M
1
1 or 1 ...
A single die is tossed. Find the probability of a 3 or 6 turning up?
Question34
Answer 34
.331/32/61/61/6p(6)p(3)6)orP(3 
If A and B are mutually exclusive events and P(A)=0.3,and
P(B)=0.5, find
Question35
)(.3
)(.2
)(.1
BAp
BAp
Ap
C
C


Answer 35
5.)()(.3
8.)()()(.2
7.0)(.1



BpBAp
BpApBAp
Ap
C
C
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. A...
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 )=...
STAT 111
Chapter Three
Find P(A|B)if
a) B is a student of A,
b)A and B are mutually exclusive.
Question1
Answer 1
 
0
)(
0
)()
1
)(
)()




BP
BApb
BP
BAP
BApa
A box contains three coins, one coin is fair, one coin is two-
headed, and one coin is weighted so that the probability of...
Answer 2
61111.0)()()()()()()(  IIIPIIIHPIIPIIHPIPIHPHP
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 ...
p(both white both same color) =
p (both white same color)
p(same color)
p (both white)
p( (both red)+p(both white)
Answer ...
The probability that three men hit a target are respectively , and
each shoots once at the target (independent) . a) Find ...
a) p(exactly one man)=
Answer 4
43.0
72
31
)()()(p(E) 321321321

 MMMPMMMPMMMP
CCCCCC
1935.0)(
)(
)(
)Ep(M 321
...
Only one in 1000 adults is affected with a rare disease for which
a diagnostic test has been developed. The test is such t...
Answer 5
0472.0
02097.0
001.099.0
)(
02097.0
999.002.0001.099.0
)()()()()(
)(
)()(
)(









vewithp
w...
The members of a consulting firm rent cars from three agencies;
60 percent from agency 1,30 percent from agency 2, and 10
...
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
Answer 6
Each of 2 cabinets identical in appearance has 2 drawers.
Cabinet A contains a silver coin in each drawer, and cabinet B
c...
Answer7
0.6667
)(
)()(
)|P(A
0.75
2
1
2
1
2
1
1)(
SP
APASp
S
SP


Suppose we have 10 coin such that if the ith coin is flipped, head
will appear with probability , i=1,…,10. When one of th...
Answer8
0.55|)P(H
0.09H)|P(5th


The probability that a regularly scheduled flight departs on time
is 0.83, the probability that it arrives on time is 0.82...
Answer9
0.95A)|b)P(D
0.94D)|a)P(A
0.78P(DA)
0.83P(A)
0.38P(D)





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 ...
Answer10
0.335)|P(Y 
Suppose that a fair coin is tossed until a head appears for the first
time. Determine the probability that exactly n tosse...
Answer11
n
XXHPXTXPTPHTTTP 






2
1
2
1
2
1
2
1
)(...)()(),,....,(
A die is tossed. If the number is odd, what is the probability that
it is prime?
Question12
Answer12
667.0
)(
)(
)( 
oddp
oddprp
oddprimeP
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 ...
Answer13
00877.0)( 321  DDDp
A number is picked at random from { 1,2,3,…,100}. Given that
the chosen number is divisible by 2 what is the probability i...
Answer14
0.03A5)A3P(A2
0.1A2)P(A5
0.16A2)P(A3
0.46A2)|A5?P(A3
5}bydivisiblewhichnumbers{theA5
3}bydivisiblewhichnumbers{th...
Three members of a private country club have been nominated
for the office of president. The probability that the first wi...
Answer15
b)0.2162
a)0.37
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...
Answer16
0.9524c)|P(m
0.02625P(c)
0.0025f)|P(c
0.05m)|P(c
1/2P(m)





In recent years much has been written about the possible link
between cigarette smoking and lung cancer. Suppose that in a...
Answer17
0.9364c)|P(s
0.4325)P(c


A laboratory blood test is 95 percent effective in detecting a
certain disease . when it is, in fact, percent. However, th...
Answer18
0.3231ve)|P(D
0.0147ve)P(


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
pe...
Answer19
c)0.5
b)0.4
a)0.375
twenty percent of the employees of a company are college
graduates. Of these, 75% are in supervisory position. Of those
wh...
Answer20
0.4839s)|P(c
0.8)c|P(s
0.2)c|P(s
0.8)P(c
0.75)c|P(s
0.2)P(c
1
22
21
2
11
1






STAT 111
Chapter Four
Two fair dice tossed, and let X denote the sum of the spots that
appears on the top face.
a)Obtain the probability distrib...
Answer1
12111098765432x
1/362/363/364/365/366/365/364/363/362/361/36F(x)
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 p...
Answer2
0F(3.5)
103.2)02(.)()4(F(5)
.904)98(.)()0(F(0)
9-5
5



DDDDDpyp
NNNNNpyp
A urn holds 5 white and 3 black marbles. If two marbles are
drawn at random without replacement and Z denote the number
of...
Answer3
sum210z
28/2810/2815/283/28
For each of the following functions determine if it is a
probability function or not and sketch the distribution
Question4...
a) Yes
b) Yes
c) No
Answer4
Two dice are rolled. Let X be the difference of the face numbers
showing, the higher minus the lower, and 0 for ties. Find...
Answer5
543210x
2/364/366/368/3610/366/36F(x)
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 ...
a)
b)
c)
d)
e)
Answer6
531-1x
3/81/81/83/8F(x)
2
1
4
1
8
5
8
2
)(,
8
1 

x
xfk
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
Find
Find the distribution...
F(x)=
Answer7
A box contains good defective items. If an item drawn is good.
We assign the number 1 to the drawing; otherwise, the numbe...
Answer8
p1)p(xF(1)
p-10)p(xF(0)


Consider the toss of balanced dice. Let X denote the random
variable representing the sum of the two faces,find
a)The prob...
Answer9
Suppose that a random variable X has a discrete distribution with
the following p.m.f
Find the value of the constant c.
Qu...
Answer10
1
2
1
1
1
2
.....
2
1
2
1
2
1
2
1
2
)(1 1 321x 1



















  





...
A fair coin is tossed until a head or five tails occurs. Find the
p.m.f and the c.d.f
Question11
Answer11
16
1
32
2
32
1
32
1
F(TTTTTT)F(TTTTH)F(5)
16
1
F(4)
8
1
F(TTH)F(3)
4
1
f(TH)F(2)
1/4F(1)





Independent trials, consisting of the flipping of a coin having
probability p of coming up heads, are continually performe...
Answer12
P)-(1F(N)
PP)-(11)-F(N
.
.
.
PP)-(1F(3)
P)P-(1F(2)
PF(1)





The distribution function of the random variable X is given by
1.Graph F(X)
2.Determine the p.m.f
3.Compute P(X<3),P(X 2)a...
Answer13
4321x
2/83/82/81/8F(x)
8
7
1.4)P(X
,
8
6
3)XP(1
,
8
7
2)P(X
,
8
3
3)P(X




a certain school gives only three letter grades in its course:
p(pass),F(fail),and W (withdrew) . In computing grade point...
1)List the outcomes in the events
a){ x=0}
b){x>-1}
c){x > }
d){x <-1}
e){x 1}
f){-1 x <2}
The student estimates the proba...
2) Using the above figures, express in tabular form the
probability function f(x) induced by X. A assume independence
of g...
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){...
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...
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)=
Answer15
Suppose that f(x)= x=1.2….
Is the probability function for a random variable X.
Determine c
Find P(2 X <5).
Find P(X 3).
Q...
Answer16
9
1
.....
3
1
3
1
1
3
2
....
3
2
3
2
3
2
3
)3()
32099.0
3
26
3
2
3
2
3
2
)52()
3
2
)(
22
3
1
1
1
3
....
3
1
3
1
3...
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(...
1)
2) F(X)==
Answer17
8
8422
0)(1  k
kkkk
xf
2
1
8
4
)3 
2
1
8
2
)4 
4
3
8
6
)5 
Suppose X is random variable having density f given by
Compute the following probabilities
a)X is negative
b) X takes a va...
Answer18
55.0)81()
3.0)1()3()


xpb
ffa
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 numb...
Answer19
12108642x
11/61/61/61/61/61/6F(x)
31y
1/21/2F(y)
Suppose a box has 12 balls labeled 1,2,…,12 Two independent
repetitions are made of the experiment of selecting a ball at
...
Suppose a box has 12 balls labeled 1,2,…,12 Two independent
repetitions are made of the experiment of selecting a ball at
...
STAT 111
Chapter Five
Among the 16 application for a job, ten have college degree. If
three of the application are randomly chosen for interview...
Answer1
214.0)
268.0)
0357.0a)
16
3
6
0
10
3
16
3
6
2
10
1
16
3
6
3
10
0c



c
cc
c
cc
c
c
c
b
Find the probability that 7 of 10 persons will recover from a
tropical disease, where the probability is 0.8 that any one ...
Answer2
0.201(0.2)(0.8)7)P(x 3710
7
 c
The average number of days school is closed due to snow during
the winter in a certain city is 4. What is the probability ...
Answer3
1042.0
!6
4
6)P(x
61


e
A manufacturer of automobile tires reports that among a shipment
of 500 sent to a local distributor, 1000 are slightly ble...
Answer4
0.201(0.2)(0.8)3)P(x 7310
3
 c
The probability that a certain kind of component will survive a
given shock test is 3/4.
Find the probability that exactly...
Answer5
0.2109)
4
1
()
4
3
(2)P(x 224
2
 c
Suppose X has a geometric distribution with p=0.8. Compute the
probability of the following events.
or
Question6
53)
74)
3...
Answer6
04.0)107((53()
008.0)7()4()74()
008.0)1()3() 3



xpxpc
xpxpXpb
pxpa
In a manufacturing process in which glass items are being
produced, defects or bubbles occur, occasionally rendering the
p...
Answer7
3134.0)6()7(  Xpxp
Let X be uniformly distributed on 0,1,…,99. Calculate
or
Question8
)3025()
108()
)2.126.2()
)25()




XPd
XPc
xPb
X...
Answer8
06.0)30(..)26()25()
29.0)32(..)4()324()323()
1.0
100
10
)12(..)3()123()
75.0
100
1
...
100
1
)99().....26()25()
...
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 c...
Answer9
288.0)0( 24
6
20
6
4
0

c
ccxp
A fair die is rolled 4 times. Find
a) The probability of obtaining exactly one 6.
b)The probability of obtaining no 6.
c)T...
Answer10
518.0)0(1)4321()
482.0
6
5
6
1
)0()
386.0
6
5
8
1
)1()
40
4
0
3
4
1

















...
Of a population of consumers, 60% is reputed to prefer a
particular brand A of toothpaste. If a group of consumers is
inte...
Answer11
  01536.04.06.0)5(
4
xp
Of a population of consumers, 60% is reputed to prefer a
particular brand A of toothpaste. If a group of consumers is
inte...
Answer11
  01536.04.06.0)5(
4
xp
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)...
Answer12
1792.08208.01)2(1)2()
8704.0
5
3
1)0(1)1(1)1()
3456.0
5
3
5
2
)2()
4
22
4
2










...
The telephone company reports that among 5000 telephones
installed in a new subdivision 4000 have push-buttons. If 10
peop...
Answer13
201.0
5
4
5
1
)3(
73
10
3












 cxp
Suppose that 30% of the application for a certain industrial job
have advanced training in computer programming. Applicati...
Answer14
07203.0)7.0()3.0()5( 4
xp
Suppose 2% of the items made by a factory are defective. Find
the probability that there are 3 defective items in a sample...
Answer15
180.0
!3
2
)3(
32


e
xp
From a group of twenty PhD engineers, ten are selected for
employment . What is the probability that the ten selected incl...
Answer16
0163.0
10
20
5
15
5
5
)5(
)5,20,10(),,(~





















f
hypKNnhypx
If the probability is 0.40 that a child exposed to a certain
contagious disease will catch it, what is the probability tha...
Answer17
    0645.06.04.0
2
9
)10(
)4.0,3(~
73






f
Nbx
Past experience has shown that the occurrence of defects in a
telephone line being produced by a certain machine generated...
Answer18
093.0
!3
4
5
)3()3()
0671.0)5()5(
!
10
)()
4
5
10












e
zpfb
fyp
y
e
xfa
y
An inspector in a television manufacturing plant has observed
that defective tuners occur at a rate of 3 per 100 sets insp...
Answer19
9371.0)2( xp
The painted light bulbs produced. By a company are 50% red,
30% blue and 20% green. In a sample of 5 bulbs, find the
proba...
Answer20
    135.03.05.0
212
5
)(
12






Ap
Find the probability of getting 5 heads and 7 tails in 12 of flips a
balanced coin?
Question21
Answer21
193.0
2
1
2
1
)5(
)
2
1
,12(~
75
12
5












 cxp
binx
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 th...
Answer22
0117.0)25.0)(75.0()4(
)75.0(~
3
xp
Gx
The manufacturer of parts that are needed in an electronic device
guarantees that a box of its parts will contain at most ...
Answer23
9921.0)2())2(
)4.0(~
)02.0,20(~
 xpf
positionx
bx
In an assembly process, the finished items are inspected by a
vision sensor, the image data is processed , and a determina...
Answer24
0066.0)98.0()02.0()30(
)02.0,2(~
28229
1
 cXP
Nbx
In an interactive time-sharing environment it is found that, on
average, a job arrives for CPU service every 6 seconds. Wh...
Answer25
5714.0)7()12()128(
0293.0)4(


ffxp
xp
Lots of 40 components each are called acceptable if they contain
no more than 3 defective. The produce for sampling the lo...
Answer26
30111.0)1(
)3,40,5(~
40
5
37
4
3
1

c
ccxp
hypx
Find the probability of obtaining exactly three 2's if an ordinary
die is tossed 5 times.
Question27
Answer27
03215.0
6
5
6
1
3
5
)3(
)
6
1
,5(~
23


















xp
bx
Find the probability that a person tossing three coins will get
either all heads or all tails for the second time on the f...
Answer28
1055.0
8
6
8
2
1
4
)5(
8
2
)()()(
)
8
2
,2(~
32




















xp
TTTPHHHPTTTHHH...
A geological study indicates that an exploratory oil well drilled in
a particular region should strike oil with probabilit...
Answer29
03072.0)8.0()2.0()5(
)2.0,3(~
234
2
 cxp
Nbx
STAT 111
Chapter six
Two refills for a ballpoint pen are selected at random from a box
that contains 3 blue refills, 2 red refills, and 3 green...
Answer1
Y
15/283/289/283/280
12/2806/286/281
1/28001/282
13/2815/2810/28F(X)
26
18
)1,0()0,1()0,0(  fffA
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 ...
Answer2
f(y)3210x/y
15/703/703/703/7000
40/702/7018/7018/702/701
15/7003/709/703/702
15/7030/7030/705/70F(X)
5.0)0,2()1,1(...
Suppose an experiment consists of three flips of a fair coin, with
each outcome being equally likely. Let X denote the num...
Answer3
210x/y
1/801/80
3/81/82/81
3/82/81/82
1/81/803
14/84/8F(X)
},,,,,,,{ TTTHTTTHTTTHHHTTHHHTHHHHS 
00010111X
0111222...
Two tablets are selected at random from a bottle containing 3
aspirin, 2 sedative, and 4 laxative tablets, If X and Y are ...
X:num of as , y:num of se
Answer4
f(y)210x/y
21/363/3612/366/360
14/3606/368/361
1/36001/362
13/3618/3615/36F(X)
210Y
11/3...
Lets X and Y denote the number of black and white balls,
respectively, that will be obtained in drawing two balls from a
b...
X:black balls
Y:white balls
Answer5
f(y)210x/y
1/61/6000
4/604/601
1/6001/62
11/64/61/6F(X)
Suppose that X and Y have following joint probability function
Find
1.The marginal distribution of the random variable X.
...
Answer6
f(y)210x/y
45/18015/18030/18001
56/180020/18036/1802
79/18010/18045/18024/1803
125/18095/18060/180F(x)
474.0
180
4...
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 ob...
Answer7
f(y)210x/y
25/361/368/3616/360
10/3602/368/361
1/36001/362
11/3610/3625/36F(X)
36
33
)0,1(),2,0()1,0()0,0(}32:),{(...
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 d...
Answer8
f(y)10x/y
1/81/800
3/82/81/81
3/81/82/82
1/801/83
14/84/8F(X)
10x
11/21/2F(x)
3210y
11/83/83/81/8F(y)
From a group of three Republicans, two Democrats, and one one
Independent, a committee of two people is to be randomly
sel...
Answer9
f(y)210x/y
6/153/153/1500
8/1506/152/151
1/15001/152
13/159/153/15F(X)
15
8
)1,(
)1,(
)1,(
)1/(
xf
yf
xf
yxf 
2...
Consider the joint probability distribution defined by the formula
x=0,1,2 y=0,1,2
Find the marginal distribution of X and...
If(y/x)=f(x,y)/f(x)
If x=0
if x=1 if x=2
Answer10
f(y)210x/y
3/272/271/2700
9/274/273/272/271
15/276/275/274/272
112/279/2...
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 v...
Answer11
3210x/y
6c4c2c00
9c5c3cc1
12c6c4c2c2
15c7c5c3c3
42c22c14c6cF(X)
no, for example: f(0,0)=0 but fx(0)fy(0)#0
Answer11
)
22
5
)2(
)1,2(
)
)2(
)2,(
)2,()
42
24
24)0,2()1,2()2,2()0,1()1,1()2,...
Let the joint probability mass function of X and Y is given in the
following table
Find
Question12
F(5,7)
F(1.5,2)
F(-1,2)...
Answer12
1)7
1.0)6
0)5
0)4
1.0)3
6.0)2
5.0)1
F(x)4321X/Y
0.200.100.11
0.6.200.100.32
0.2000.203
10.20.20.20.4F(Y)
Let X and Y be independent random variables with the following
distribution ;
Find the joint distribution of X and Y.
Ques...
Answer13
f(y)10x/y
0.20.080.125
0.50.20.310
0.20.120.1815
10.40.6F(X)
12.0)15,2(
20.0)10,2(
08.0(5,2(
18.0)15,1(
30.0)10,1...
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
Calcu...
Answer14
f(y)10x/y
0.50.10.40
0.50.30.21
10.40.6F(X)


















1,6.0
5.0
5.0
0,4.0
5.0
...
Suppose that X and Y have following joint probability function
Find
The marginal distribution of the random variable X;
Th...
No,for example: f(1,2)=0, but fx(1)=0x0.2#1
Answer15
321X
10.20.60.2F(x)
4321y
10.20.20.20.4F(y)
STAT 111
Chapter seven
Let (x,y) have the following joint distribution function
Find
a)the probability mass function of X+Y.
b)the probability ma...
a)
b)
c)
d)
sum65432X+y
102/62/62/60F(x+y)
sum964321xy
102/602/62/60F(xy)
sum941x2
11/31/31/3F(x2)
sum941y2
11/31/31/3F(y2...
Lets X be a random variable with probability distribution
Find the probability distribution of the random variable Y=2X-1
...
Y=2x-1, x= = g (y)=>f (y)=f (g (y))
sum321x
sum531y
11/31/31/3F(y)
-1
y x
Answer2
Let X1 and X2 be discrete random variable with joint probability
distribution
F(x)=
Find the probability distribution of t...
sum64321y
16/184/183/184/181/18Fy(y)
sum21X1/x2
3/182/181/181
6/184/182/182
9/186/183/183
112/88/18sum
Answer3
Let X be a random variable with probability distribution
F(x)=
Find the probability distribution of the random variable Y=...
n……321x
1/n1/n1/n1/nF(x)
N 2……25941y
1/n1/n1/n1/nF(x)
Answer4
Let X be a random variable with probability distribution
F(x)=
Find the probability distribution of the random variable Y=...
n…..21x
n941y
2/2n2/2n2/2n2/2nFy(Y)
Answer5
STAT 111
Chapter eight
A lot twelve television sets includes two that are defective. If
three of the sets are chosen at random, how many defectiv...
sum210x
11/229/116/11F(x)
11/222/229/220Xf(x)
2
1
12
2
3)(
)2,12,3(~
12
102
)(

















...
If the random variable X is the two faces when rolling a pair of
balanced dice, find the expected value of X.
Question2
SUM12111098765432X
11/362/3
6
3/364/365/366/365/364/363/362/361/36F(X)
Answer2
Let X have the following probability mass function
F(x)=
Find E(X),E(X2), and e(x+2)2
Question3
Answer3
272)E(X
11)E(X
3E(X)
5nuniformX
2
2




The probability mass function of the random variable X is given
by
Find the expected value of X.
sum4321x
2/103/101/104/10...
SUM4321X
12/103/101/104/10F(X)
2.3-
>E(x)
8/109/102/104/10XF(X)
Answer3
The probability mass function of the random variable X is given
by
Find the expected value of g(x)=2x-1.
sum987654x
1/61/6...
sum987654x
11/61/61/41/41/121/12F(x)
82/1
2
9/68/67/46/45/124/12E(x)=Xf(x)
E(X)=82/12=41/6
E(2X-1)=2E(X)-1=2(41/6)-1=12.7
...
Let the random variable X represent the number of automobiles that are
used for official business purposes on any give wor...
sum321x
10.30.40.3F(x)
20.90.80.3E(X)=X f(x)
4.62.71.60.3E(X2)=X2 f(x)
SUM43210X
0.10.30.30.10.2F(X)
2->E(X)0.40.90.60.10X...
Find the moment generating function of the discrete random variable X
has the probability distribution
F(x)=2x x=1,2,…
And...
Answer7
An urn contains nine chips, five red and four white. Three are drawn
out at random without replacement . Let X denote the ...
Answer8
667.1
9
5
3)(
3
9
3
45
)(
)5,9,3(~
































N
n
nx...
Suppose that a sequence of independent tosses are made with a coin for
which the probability of obtaining a head on any gi...
X:number of heads
Y: number of tosses
Answer9
1505
30
5)(
)
30
1
bin(n,X



n
n
npxE
Let X be binomial distribution random variable with E(X)=2 , and
V(X)=4/3 . Find the distribution on X.
Question10
equation(1)
equation(2)
Answer10
.......
4
3
npqV(x)
2......npE(x)
p?n?
p)(n,bin4X



)3
1
,6(~
62
3
1
3
1
3
2
11
3
...
Let x1,x2 ,and x3 be independent random variables having finite
positive variances 2
1 ,
2
2,
2
3, respectively. Find the ...
Answer11
)()((
),cov(
),-cor(
3221
3221
3221
xxvxxv
xxxx
xxxx



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 re...
E(U)=6/5 , E(V)=4/5
E(U.U)=18/10 , E(V.V)=1
V(U)=.36 , V(V)=.36
E(UV)=-.36
(U,V)=-1
Fv(V)210UR/VB
3/103/10000
6/1006/1001
...
Upcoming SlideShare
Loading in …5
×

U11007410 statistics 111

533 views

Published on

Statistics

Published in: Education
  • Be the first to comment

  • Be the first to like this

U11007410 statistics 111

  1. 1. Stat 111 Collected by Ayoob Aboobaker
  2. 2. STAT 111 Chapter Zero
  3. 3. 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.
  4. 4. 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).
  5. 5. 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 Cont..Answer1
  6. 6. 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
  7. 7. 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}
  8. 8. 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.
  9. 9. 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
  10. 10. 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?
  11. 11. Answer4 N(S)=100, N(SW)=78, N(I)=74, N( C) =53, N(SW ∩I)=57, 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. y 31 z x 26 6 15
  12. 12. STAT 111 Chapter One
  13. 13. 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?
  14. 14. Answer1 6P4 = 6 x 5 x 4 x 3= 360
  15. 15. 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?
  16. 16. 3 X 4 = 12 Answer2
  17. 17. 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? Question3
  18. 18. Answer3 60 !1!2!3 !6 
  19. 19. 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? Question4
  20. 20. Answer4 2 )20( 100 20.....202020   time
  21. 21. Find the number of subsets of a set X containing n elements? Question5
  22. 22. Either the subsets containing no element , 1 elements 2 elements , n elements Answer5       0 n       1 n       2 n       n n 2 )11(........ 10                   n nnn
  23. 23. 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? Question6
  24. 24. Answer6 1140 3 20 20 3       c
  25. 25. 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? Question7
  26. 26. 3 X 5 = 15 Answer7
  27. 27. 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? Question8
  28. 28. a) 2x2x2x2Xx….x2 = 20 = 1048,576 b) c) d) Answer8 520,77 13,7 20 7 20 20 7             c 756,184 10 20 10,10 20 20 10             c 1351 20 20 20 19 20 18 20 17  cccc
  29. 29. 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? Question9
  30. 30. Without repetition : 28x27x10x9x8x7= 3,810,240. With repetition : 28x28x10x10x10x10=7,840,000. Answer9
  31. 31. 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? Question10
  32. 32. Answer10 000,000,1652 610 
  33. 33. How many different 6-digits numbers may be formed using the digits from the 4,4,5,6,6,6. Question11
  34. 34. Answer11 60 !3!1!2 !6 
  35. 35. 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? Question12
  36. 36. With repetition Without repetition Answer12 000,30103 4  520,245673 
  37. 37. 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? Question13
  38. 38. Answer13 12223 3 1 2 1 3 1  ccc
  39. 39. 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? Question14
  40. 40. a) b) Answer14 361526 4 6 1 2 5 6 0 2                         26 5 6 0 2 3 6 2 2                        
  41. 41. How many ways can we arrange the letters in the word cold? Question15
  42. 42. 4! = 24 Answer15
  43. 43. How many ways may 5 people be seated in a 5-passenger vehicle if one of two people must drive? Question16
  44. 44. The drive seat must be filed in 2 ways, after that, the remaining 4 can be arranged in 4! Answer16 !4 1 2      
  45. 45. 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? Question17
  46. 46. a) b) Answer17 120 10 7 c 110 5 2 5 5 5 3 5 4 5 4 5 3 cccccc
  47. 47. 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? Question18
  48. 48. Answer18 210 !2!3!2 !7 
  49. 49. 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? Question19
  50. 50. a) b) Answer19 120 10 7 c 720 10 3 p
  51. 51. If eight persons are having dinner together, in how many different ways can three order chicken, four order steak, and one order lobster? Question20
  52. 52. Answer 20 280 !4!3 !8 
  53. 53. a)How many distinct permutations are there of the letters in the word statistics? b) How many of these begin and end with letters? Question21
  54. 54. 1) 2) Answer 21 400,50 !2!3!3 !10  360,3 !2!3 !8 
  55. 55. 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? Question22
  56. 56. 1) 2) 3) Answer 22 040,5!7  720!5!3  240!52 
  57. 57. 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? Question23
  58. 58. a) b) Answer 23 600,7558765  5044376 
  59. 59. 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? Question24
  60. 60. a) b) c) Answer 24 120!5  72!42!5  12!3)!13( 
  61. 61. 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? Question25
  62. 62. a) b) c) Answer 25 896 8 3 4 2 2 1 8 3 4 3 2 0  cccccc 1000 2 6 1 2 3 6 0 2 3 6                                     910 3 5 2 7 1 2 5 3 7 1 3 5 3 7 0 2                                          
  63. 63. 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? Question26
  64. 64. 3x2x1x2=12 Answer 26 CBA 
  65. 65. 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? Question27
  66. 66. Answer 27 !1!1!4!6 !12
  67. 67. 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? Question28
  68. 68. a) 10! = 3628,800 b) 6! 4! 2 = 34,560 Answer 28
  69. 69. 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? Question29
  70. 70. Answer 29 240,30 5..... 10      
  71. 71. 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? Question30
  72. 72. Answer 30 30 3 1 10 1 cc
  73. 73. How many functions defined on n points are possible if each functional value is either 0 or 1 ? Question31
  74. 74. Answer 31 n timen 2 _ 2...222  
  75. 75. 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. Question32
  76. 76. Answer 32 24243 
  77. 77. How many sample points are in the sample space when a pair of dice is thrown once? Question33
  78. 78. 6X6=36 S={(1,1),(1,2),…..(1,6)} Answer 33
  79. 79. Calculate the number of permutations of the letters a, b, c, d had taken two at a time? Question34
  80. 80. Answer 34 12 4 2 p
  81. 81. 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? Question35
  82. 82. Answer 35 024,255 24 4 p
  83. 83. In how many ways may 3 books be placed next to each other on a shelf? Question36
  84. 84. Answer 36 6!3 
  85. 85. 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? Question37
  86. 86. no rest = 12! a) 3! (4! 6! 2! ) = 207360 b) 4! 9! = 8,709,120 Answer 37
  87. 87. 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 Question38
  88. 88. a) 8! = 40, 320 b) 2! (4! 4!) + 1,152 c) d) Answer 38 70 !4!4 !8  1680 !4 !8 
  89. 89. From 4 chemists and 3 physicists find the number of committees that be formed consisting of 2 chemists and 1 physicist? Question39
  90. 90. Answer 39 18 1 3 2 4            
  91. 91. 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? Question40
  92. 92. a) b) Answer 40   300 300 350 5 2 2 1 5 3 2 0 5 2 5 2 5 2 2 1 5 2 5 3 2 0 7 3 5 2    ccccc cccccc cc or
  93. 93. 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. Question41
  94. 94. a) b) c) Answer 41 100 45 150 5 2 5 3 6 2 3 1 2 2 6 2 5 3    cc ccc cc
  95. 95. 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? Question42
  96. 96. Answer 42 70 7 1 3 3 7 2 3 2  cccc
  97. 97. In how many ways can a set of four objects be partitioned into three subsets containing, respectively, 2,1 and 1 of the objects? Question43
  98. 98. Answer 43 12 1,1,2 4      
  99. 99. In how many ways can seven scientists be assigned to one triple and two double hotel rooms? Question44
  100. 100. Answer 44 210 2,2,3 7      
  101. 101. 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? Question45
  102. 102. Answer 45 320,152,148910262728) 000,952,21101010282828)   b a
  103. 103. 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? Question46
  104. 104. Answer 46 36312 
  105. 105. In an experiment consists of throwing a die and then drawing a letter at random from the English alphabet, how many points are possible? Question47
  106. 106. Answer 47 156266 
  107. 107. 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? Question48
  108. 108. Answer 48 2438 
  109. 109. 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? Question49
  110. 110. Answer 49 184,5)!3!3!4(!3 
  111. 111. 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? Question50
  112. 112. Answer 50 420 1,2,3,1 7      
  113. 113. 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? Question51
  114. 114. Answer 51 300,33) 650,13) 760,38) 10 2 10 4 10 3 10 3 10 4 10 2 15 4 5 2 20 6    cccccc cc c c b a
  115. 115. STAT 111 Chapter Two
  116. 116. 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. Question1
  117. 117. Answer 1 11.057G)MrP(M .25G)MrP(M .147G)P(Mr .042G)P(M .525P(G) .470P(MR) 312P(M) CollegeG MarriedMr MaleM          
  118. 118. 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. Question2
  119. 119. Answer 2 029. 6 75 1 35 3 15 2 25 )( 00806. 6 75 6 35 )( 6 75 )(                                                Ap Ap sn
  120. 120. 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). Question3
  121. 121. Answer 3 5/6B)P(A 5/6)BP(A 1/6)BP(A 1/6B)P(A c cc c     1/22/61/6 1/22/61/6 12/31/3 C AA B C B
  122. 122. 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 Question4
  123. 123. Answer 4 3571. 3 9 1 3 2 5 ) 33. 3 9 2 8 1 1 )                                       b a
  124. 124. 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? Question5
  125. 125. (The system will work)= Answer 5 72.0)( 8.0)( 9.0)(    BAp Bp Ap 98.72.8.9.)()()()(  BAPBPAPBAp p
  126. 126. Let A and B be events with P(A)=1/2 , P(A∪B)= 3/4,P(B )=5/8 . Find P(A∩B), P(A ∪ B ) ,P(A ∩ B ),and P(B ∩A ). Question6 c c c c c c AA B B c c
  127. 127. Answer 6 2/8)AP(B 7/8)BP(A 1/4)BP(A 1/8B)P(A c cc cc    
  128. 128. A pair of fair dice is tossed. Find the probability that the maximum of the two numbers is greater than 4? Question7
  129. 129. Answer 7 5556. 30 20       p
  130. 130. 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. Question8
  131. 131. Answer8 120 30 )( 120 40 )(, 6 1 120 20 SP)P(F 120 50 P(SP), 120 60 P(F) 20SP)n(F50,n(SP)60,n(F)120,n(S)     SPFP SPFP C C 5833.)()() ..)(1)() 75.)()    SPFPSPFPc SPFPSPFPb SPFpa CC CC
  132. 132. 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? Question9
  133. 133. Answer9 2398. 5 15 2 9 3 6                   
  134. 134. 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? Question10
  135. 135. Answer10 0727. 4 11 1 2 1 4 0 2 2 3                               
  136. 136. 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? Question11
  137. 137. Answer11 5455. 2 11 1 5 1 6                   
  138. 138. 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? Question12
  139. 139. Answer12 .6B)P(A-P(B)P(A)B)P(A .1B)P(A .2P(B) .5P(A)    
  140. 140. 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? Question13
  141. 141. Answer13 .}T),T,(1,H),T,(1,T),H,(1, H),H,{(1,H)}(6,H),(4,T),(2,H),{(2,S  
  142. 142. 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? Question14
  143. 143. Answer14 colures]thoseofoneleastator[ exclusive)(mutually .68p(b))rp(p(w)p(g))brwP(g 
  144. 144. 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? Question15
  145. 145. Answer15 21 12 P(6)P(4)P(2)P(E) 21 1 w 121w16w5w4w3w2wwP(S) 1   
  146. 146. 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? Question16
  147. 147. Answer16 12 4 1 12 1 4             
  148. 148. 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? Question17
  149. 149. P(at least one )= Answer17 9.0)(1 )(1)( 321 331321   CCC AAAP AAAPAAAp
  150. 150. 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 ) Question18
  151. 151. are indep also) Answer18 CCCC BABpApb BpApBpApa ,(4.)()() 6.)()()()()  
  152. 152. 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. Question19
  153. 153. or Answer19 4615. 1 13 1 6 )              a 13 6
  154. 154. A die is tossed 50 times. The following table gives the six numbers and their frequency of occurrence Find the relative frequency of the event a) a 4 appears. b) and odd numbers appears. c)a prime number appears. Question20 654321Number 1097897Frequency
  155. 155. Answer20 52. 50 989 ) 48. 50 987 b) 14. 50 07 a)      c
  156. 156. 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. Question21
  157. 157. Answer21 .1 6! 3!3!2! b) .2 6! )4!(3! a)  
  158. 158. 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). Question22
  159. 159. Answer22 8 1 )BP(A 8 6 P(B) 8 3 P(A) c    6/84/82/8 1/81/81/8 15/83/8 C A C B A B
  160. 160. 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? Question23
  161. 161. .:not indep Answer23 )()()()( 041. 36 61818 )()()( 083. 36 3 C)BP(A (6,6)}(4,4),{(2,2),CBA 6)Cn(36,n(S)18,n(B)18,n(A) ,6)}(2,2),..(6{(1,1),C 6,6)}(1,6)....(4),(1,4)..(6,(6,2),{(1,2),...B .(6,6)}(6,1),....(4,6),(4,1),....6),{(1,2).(2,A CPBPAPCBAP CPBPAP           
  162. 162. 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? Question24
  163. 163. Answer24 4286. 3 7 2 6 1 1                   
  164. 164. 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. Question25
  165. 165. Answer25 e).275 .1d)8/80 .1c)8/80 .25b)20/80 .37520)/80a)(10    
  166. 166. Three studentsA,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. Question26
  167. 167. Answer26 .63/6b)p(c-p(b))p(cb)P(c 1/5)P(c 2/5P(b) 2/5P(a)     
  168. 168. 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. Question27
  169. 169. Answer27 4667. 2 10 2 7 0 3 ) 06. 2 10 0 7 2 3 )                                       b a 533. 2 10 0 7 2 3 1 7 1 3 )                                      c
  170. 170. 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. Question28
  171. 171. Answer 28 2/43/8 16/62/43/4P(E)P(E) 2/4F)P(E BG}{GB,F BB}BG,{GB,E BG}GB,GG,{BB,S      
  172. 172. Find the probability of getting three heads in three (independent) tosses of a balanced coin. Question29
  173. 173. Answer 29 8 1 P(HHH) TTT}HTT,THT,TTH,THH,HTH,HHT,p(HHH, 
  174. 174. 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? Question30
  175. 175. For design(1): =.9 For design(2): =.96639 Answer 30
  176. 176. Find the probability of getting at least one head in 5 tossed of a balanced coin? Question31
  177. 177. P(at least one head)=1-p(HHH)=.969 Answer 31
  178. 178. 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? Question32
  179. 179. Answer 32 1471. 13 18 1 6 1 5 1 4 ) 0123. 816 10 3 18 3 5 )                                       b a 041. 3 18 0 5 3 4 1 5 2 4 )                                c
  180. 180. 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? Question33
  181. 181. Answer 33                                   r N r N r NM Kr N K M 1 1 or 1 KMif
  182. 182. A single die is tossed. Find the probability of a 3 or 6 turning up? Question34
  183. 183. Answer 34 .331/32/61/61/6p(6)p(3)6)orP(3 
  184. 184. If A and B are mutually exclusive events and P(A)=0.3,and P(B)=0.5, find Question35 )(.3 )(.2 )(.1 BAp BAp Ap C C  
  185. 185. Answer 35 5.)()(.3 8.)()()(.2 7.0)(.1    BpBAp BpApBAp Ap C C
  186. 186. 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. Question36 1 2 3 4 1 3 2 4 A A B B
  187. 187. 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 Answer 36
  188. 188. STAT 111 Chapter Three
  189. 189. Find P(A|B)if a) B is a student of A, b)A and B are mutually exclusive. Question1
  190. 190. Answer 1   0 )( 0 )() 1 )( )()     BP BApb BP BAP BApa
  191. 191. A box contains three coins, one coin is fair, one coin is two- headed, 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. Question2
  192. 192. Answer 2 61111.0)()()()()()()(  IIIPIIIHPIIPIIHPIPIHPHP
  193. 193. 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? Question3
  194. 194. p(both white both same color) = p (both white same color) p(same color) p (both white) p( (both red)+p(both white) Answer 3 875.0 )()()()( 10 6 10 7 221221    wwpwpRRpRp = =
  195. 195. 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? Question4
  196. 196. a) p(exactly one man)= Answer 4 43.0 72 31 )()()(p(E) 321321321   MMMPMMMPMMMP CCCCCC 1935.0)( )( )( )Ep(M 321 1 1    CC MMMP EP EMP
  197. 197. 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? Question5
  198. 198. Answer 5 0472.0 02097.0 001.099.0 )( 02097.0 999.002.0001.099.0 )()()()()( )( )()( )(          vewithp withoutpwithoutvepwithpwithvepvep vep withpwithvep veWithP
  199. 199. 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 tune- up. Question6
  200. 200. 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 Answer 6
  201. 201. 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? Question7
  202. 202. Answer7 0.6667 )( )()( )|P(A 0.75 2 1 2 1 2 1 1)( SP APASp S SP  
  203. 203. 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? Question8
  204. 204. Answer8 0.55|)P(H 0.09H)|P(5th  
  205. 205. 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. Question9
  206. 206. Answer9 0.95A)|b)P(D 0.94D)|a)P(A 0.78P(DA) 0.83P(A) 0.38P(D)     
  207. 207. 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? Question10
  208. 208. Answer10 0.335)|P(Y 
  209. 209. 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. Question11
  210. 210. Answer11 n XXHPXTXPTPHTTTP        2 1 2 1 2 1 2 1 )(...)()(),,....,(
  211. 211. A die is tossed. If the number is odd, what is the probability that it is prime? Question12
  212. 212. Answer12 667.0 )( )( )(  oddp oddprp oddprimeP
  213. 213. 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? Question13
  214. 214. Answer13 00877.0)( 321  DDDp
  215. 215. 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? Question14
  216. 216. Answer14 0.03A5)A3P(A2 0.1A2)P(A5 0.16A2)P(A3 0.46A2)|A5?P(A3 5}bydivisiblewhichnumbers{theA5 3}bydivisiblewhichnumbers{theA3 2}bydivisiblewhichnumbers{theA1Let       
  217. 217. 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? Question15
  218. 218. Answer15 b)0.2162 a)0.37
  219. 219. 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? Question16
  220. 220. Answer16 0.9524c)|P(m 0.02625P(c) 0.0025f)|P(c 0.05m)|P(c 1/2P(m)     
  221. 221. 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? Question17
  222. 222. Answer17 0.9364c)|P(s 0.4325)P(c  
  223. 223. 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? Question18
  224. 224. Answer18 0.3231ve)|P(D 0.0147ve)P(  
  225. 225. 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. Question19
  226. 226. Answer19 c)0.5 b)0.4 a)0.375
  227. 227. 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? Question20
  228. 228. Answer20 0.4839s)|P(c 0.8)c|P(s 0.2)c|P(s 0.8)P(c 0.75)c|P(s 0.2)P(c 1 22 21 2 11 1      
  229. 229. STAT 111 Chapter Four
  230. 230. 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. Question1
  231. 231. Answer1 12111098765432x 1/362/363/364/365/366/365/364/363/362/361/36F(x)
  232. 232. 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)? Question2
  233. 233. Answer2 0F(3.5) 103.2)02(.)()4(F(5) .904)98(.)()0(F(0) 9-5 5    DDDDDpyp NNNNNpyp
  234. 234. 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. Question3
  235. 235. Answer3 sum210z 28/2810/2815/283/28
  236. 236. For each of the following functions determine if it is a probability function or not and sketch the distribution Question4 ( )= ( )= ( )=
  237. 237. a) Yes b) Yes c) No Answer4
  238. 238. 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. Question5
  239. 239. Answer5 543210x 2/364/366/368/3610/366/36F(x)
  240. 240. 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) Question6
  241. 241. a) b) c) d) e) Answer6 531-1x 3/81/81/83/8F(x) 2 1 4 1 8 5 8 2 )(, 8 1   x xfk
  242. 242. 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 Find Find the distribution F(x)for the random variable X Graph this distribution function. Question7 6ji1 jij)X(I, 
  243. 243. F(x)= Answer7
  244. 244. 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? Question8
  245. 245. Answer8 p1)p(xF(1) p-10)p(xF(0)  
  246. 246. 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). Question9
  247. 247. Answer9
  248. 248. Suppose that a random variable X has a discrete distribution with the following p.m.f Find the value of the constant c. Question10 (x)=
  249. 249. Answer10 1 2 1 1 1 2 ..... 2 1 2 1 2 1 2 1 2 )(1 1 321x 1                             cc c cc c xf x xx x
  250. 250. A fair coin is tossed until a head or five tails occurs. Find the p.m.f and the c.d.f Question11
  251. 251. Answer11 16 1 32 2 32 1 32 1 F(TTTTTT)F(TTTTH)F(5) 16 1 F(4) 8 1 F(TTH)F(3) 4 1 f(TH)F(2) 1/4F(1)     
  252. 252. 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. Question12
  253. 253. Answer12 P)-(1F(N) PP)-(11)-F(N . . . PP)-(1F(3) P)P-(1F(2) PF(1)     
  254. 254. The distribution function of the random variable X is given by 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) Question13 F(x)=
  255. 255. Answer13 4321x 2/83/82/81/8F(x) 8 7 1.4)P(X , 8 6 3)XP(1 , 8 7 2)P(X , 8 3 3)P(X    
  256. 256. 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
  257. 257. 1)List the outcomes in the events a){ x=0} b){x>-1} c){x > } 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 1  
  258. 258. 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. Question14
  259. 259. 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) Answer14 210-1-2X .49.28.18.04.01F(X)
  260. 260. 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. Question15
  261. 261. 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)= Answer15
  262. 262. Suppose that f(x)= x=1.2…. Is the probability function for a random variable X. Determine c Find P(2 X <5). Find P(X 3). Question16 c c 3  
  263. 263. Answer16 9 1 ..... 3 1 3 1 1 3 2 .... 3 2 3 2 3 2 3 )3() 32099.0 3 26 3 2 3 2 3 2 )52() 3 2 )( 22 3 1 1 1 3 .... 3 1 3 1 3 1 32 1) 23 3 543 4432 3211                                                   x x x x x x c xpc xpb xf c cc c c c c a
  264. 264. 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) Question17
  265. 265. 1) 2) F(X)== Answer17 8 8422 0)(1  k kkkk xf 2 1 8 4 )3  2 1 8 2 )4  4 3 8 6 )5 
  266. 266. Suppose X is random variable having density f given by Compute the following probabilities a)X is negative b) X takes a value between 1 and 8 inclusive Question18 853210-1-3X 0.050.050.150.10.20.150.20.1F(X)
  267. 267. Answer18 55.0)81() 3.0)1()3()   xpb ffa
  268. 268. 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. Question19
  269. 269. Answer19 12108642x 11/61/61/61/61/61/6F(x) 31y 1/21/2F(y)
  270. 270. 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
  271. 271. 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
  272. 272. STAT 111 Chapter Five
  273. 273. 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. Question1
  274. 274. Answer1 214.0) 268.0) 0357.0a) 16 3 6 0 10 3 16 3 6 2 10 1 16 3 6 3 10 0c    c cc c cc c c c b
  275. 275. 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. Question2
  276. 276. Answer2 0.201(0.2)(0.8)7)P(x 3710 7  c
  277. 277. 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. Question3
  278. 278. Answer3 1042.0 !6 4 6)P(x 61   e
  279. 279. 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? Question4
  280. 280. Answer4 0.201(0.2)(0.8)3)P(x 7310 3  c
  281. 281. 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? Question5
  282. 282. Answer5 0.2109) 4 1 () 4 3 (2)P(x 224 2  c
  283. 283. Suppose X has a geometric distribution with p=0.8. Compute the probability of the following events. or Question6 53) 74) 3)    Xc xb Xa 107  X
  284. 284. Answer6 04.0)107((53() 008.0)7()4()74() 008.0)1()3() 3    xpxpc xpxpXpb pxpa
  285. 285. 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? Question7
  286. 286. Answer7 3134.0)6()7(  Xpxp
  287. 287. Let X be uniformly distributed on 0,1,…,99. Calculate or Question8 )3025() 108() )2.126.2() )25()     XPd XPc xPb XPa 322  X
  288. 288. Answer8 06.0)30(..)26()25() 29.0)32(..)4()324()323() 1.0 100 10 )12(..)3()123() 75.0 100 1 ... 100 1 )99().....26()25()     xpxpxpd xpxpxpxpc xpxpxpb xpxpxpa
  289. 289. 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? Question9
  290. 290. Answer9 288.0)0( 24 6 20 6 4 0  c ccxp
  291. 291. 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. Question10
  292. 292. Answer10 518.0)0(1)4321() 482.0 6 5 6 1 )0() 386.0 6 5 8 1 )1() 40 4 0 3 4 1                            xporxorxorxxpc xpb xpa c c
  293. 293. 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. Question11
  294. 294. Answer11   01536.04.06.0)5( 4 xp
  295. 295. 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. Question11
  296. 296. Answer11   01536.04.06.0)5( 4 xp
  297. 297. 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. Question12
  298. 298. Answer12 1792.08208.01)2(1)2() 8704.0 5 3 1)0(1)1(1)1() 3456.0 5 3 5 2 )2() 4 22 4 2                      xpxpc xpxpxpb xpa c
  299. 299. 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? Question13
  300. 300. Answer13 201.0 5 4 5 1 )3( 73 10 3              cxp
  301. 301. 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. Question14
  302. 302. Answer14 07203.0)7.0()3.0()5( 4 xp
  303. 303. 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. Question15
  304. 304. Answer15 180.0 !3 2 )3( 32   e xp
  305. 305. 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. Question16
  306. 306. Answer16 0163.0 10 20 5 15 5 5 )5( )5,20,10(),,(~                      f hypKNnhypx
  307. 307. 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. Question17
  308. 308. Answer17     0645.06.04.0 2 9 )10( )4.0,3(~ 73       f Nbx
  309. 309. 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? Question18
  310. 310. Answer18 093.0 !3 4 5 )3()3() 0671.0)5()5( ! 10 )() 4 5 10             e zpfb fyp y e xfa y
  311. 311. 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? Question19
  312. 312. Answer19 9371.0)2( xp
  313. 313. 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. Question20
  314. 314. Answer20     135.03.05.0 212 5 )( 12       Ap
  315. 315. Find the probability of getting 5 heads and 7 tails in 12 of flips a balanced coin? Question21
  316. 316. Answer21 193.0 2 1 2 1 )5( ) 2 1 ,12(~ 75 12 5              cxp binx
  317. 317. 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. Question22
  318. 318. Answer22 0117.0)25.0)(75.0()4( )75.0(~ 3 xp Gx
  319. 319. 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? Question23
  320. 320. Answer23 9921.0)2())2( )4.0(~ )02.0,20(~  xpf positionx bx
  321. 321. 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? Question24
  322. 322. Answer24 0066.0)98.0()02.0()30( )02.0,2(~ 28229 1  cXP Nbx
  323. 323. 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? Question25
  324. 324. Answer25 5714.0)7()12()128( 0293.0)4(   ffxp xp
  325. 325. 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? Question26
  326. 326. Answer26 30111.0)1( )3,40,5(~ 40 5 37 4 3 1  c ccxp hypx
  327. 327. Find the probability of obtaining exactly three 2's if an ordinary die is tossed 5 times. Question27
  328. 328. Answer27 03215.0 6 5 6 1 3 5 )3( ) 6 1 ,5(~ 23                   xp bx
  329. 329. Find the probability that a person tossing three coins will get either all heads or all tails for the second time on the fifth. Question28
  330. 330. Answer28 1055.0 8 6 8 2 1 4 )5( 8 2 )()()( ) 8 2 ,2(~ 32                     xp TTTPHHHPTTTHHHpp Nbx
  331. 331. 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. Question29
  332. 332. Answer29 03072.0)8.0()2.0()5( )2.0,3(~ 234 2  cxp Nbx
  333. 333. STAT 111 Chapter six
  334. 334. 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}. Question1
  335. 335. Answer1 Y 15/283/289/283/280 12/2806/286/281 1/28001/282 13/2815/2810/28F(X) 26 18 )1,0()0,1()0,0(  fffA
  336. 336. 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} Question2
  337. 337. Answer2 f(y)3210x/y 15/703/703/703/7000 40/702/7018/7018/702/701 15/7003/709/703/702 15/7030/7030/705/70F(X) 5.0)0,2()1,1()0,1()2.0()1,0()0,0(  ffffffA
  338. 338. 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. Question3
  339. 339. Answer3 210x/y 1/801/80 3/81/82/81 3/82/81/82 1/81/803 14/84/8F(X) },,,,,,,{ TTTHTTTHTTTHHHTTHHHTHHHHS  00010111X 01112223Y
  340. 340. 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 Question4
  341. 341. X:num of as , y:num of se Answer4 f(y)210x/y 21/363/3612/366/360 14/3606/368/361 1/36001/362 13/3618/3615/36F(X) 210Y 11/3614/3621/36F(Y) 210X 13/3618/3615/36F(X) 210x 106/148/14F(x)
  342. 342. 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. Question5
  343. 343. X:black balls Y:white balls Answer5 f(y)210x/y 1/61/6000 4/604/601 1/6001/62 11/64/61/6F(X)
  344. 344. Suppose that X and Y have following joint probability function Find 1.The marginal distribution of the random variable X. 2.The marginal distribution of the random variable Y. 3. P(Y=3X=2) Question6 321y/x 1/121/601 01/91/52 1/181/42/153
  345. 345. Answer6 f(y)210x/y 45/18015/18030/18001 56/180020/18036/1802 79/18010/18045/18024/1803 125/18095/18060/180F(x) 474.0 180 45 180 95 180 45 )2( )3,2( .3    xf yxf 321x 125/18095/18060/180F(x) 321y 179/18056/18045/180F(y)
  346. 346. 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} Question7 
  347. 347. Answer7 f(y)210x/y 25/361/368/3616/360 10/3602/368/361 1/36001/362 11/3610/3625/36F(X) 36 33 )0,1(),2,0()1,0()0,0(}32:),{( 36 8 )}5,6(),3,5(),2,5(),1,5( )5,6(),5,3(),5,2(),5,1{()1,0(, 36 16 )0.0(    ffffyxyxp fff
  348. 348. 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 Question8
  349. 349. Answer8 f(y)10x/y 1/81/800 3/82/81/81 3/81/82/82 1/801/83 14/84/8F(X) 10x 11/21/2F(x) 3210y 11/83/83/81/8F(y)
  350. 350. 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. Question9
  351. 351. Answer9 f(y)210x/y 6/153/153/1500 8/1506/152/151 1/15001/152 13/159/153/15F(X) 15 8 )1,( )1,( )1,( )1/( xf yf xf yxf  210x 106/82/8F(x)
  352. 352. Consider the joint probability distribution defined by the formula x=0,1,2 y=0,1,2 Find the marginal distribution of X and Y, and f(x/y ) Question10 )2( 27 1 ),( yxyxf 
  353. 353. If(y/x)=f(x,y)/f(x) If x=0 if x=1 if x=2 Answer10 f(y)210x/y 3/272/271/2700 9/274/273/272/271 15/276/275/274/272 112/279/276/27F(X) 210y 21/31/30Fy(x=0) 210y 15/93/91/9Fy(x =1) 210y 13/62/61/6Fx/x =2
  354. 354. 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=1X=2) f)Determine whether the random variables X and Y are independent . Question11
  355. 355. Answer11 3210x/y 6c4c2c00 9c5c3cc1 12c6c4c2c2 15c7c5c3c3 42c22c14c6cF(X)
  356. 356. no, for example: f(0,0)=0 but fx(0)fy(0)#0 Answer11 ) 22 5 )2( )1,2( ) )2( )2,( )2,() 42 24 24)0,2()1,2()2,2()0,1()1,1()2,1() 42 5 5)1,2() 42 1 1),() f fx f e fx yf yfd cffffffc cfb cyxfa yx     
  357. 357. Let the joint probability mass function of X and Y is given in the following table Find Question12 F(5,7) F(1.5,2) F(-1,2) 1)Y2,P(X 4)YP(X 1)P(X 2)Y2,P(X     4321X/Y 00.100.11 .200.100.32 000.203
  358. 358. Answer12 1)7 1.0)6 0)5 0)4 1.0)3 6.0)2 5.0)1 F(x)4321X/Y 0.200.100.11 0.6.200.100.32 0.2000.203 10.20.20.20.4F(Y)
  359. 359. Let X and Y be independent random variables with the following distribution ; Find the joint distribution of X and Y. Question13 21X 0.40.6F(X) 15105Y 0.30.50,2F(Y)
  360. 360. Answer13 f(y)10x/y 0.20.080.125 0.50.20.310 0.20.120.1815 10.40.6F(X) 12.0)15,2( 20.0)10,2( 08.0(5,2( 18.0)15,1( 30.0)10,1( 12.0)5,1(       f f f f f f
  361. 361. 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 Question14
  362. 362. Answer14 f(y)10x/y 0.50.10.40 0.50.30.21 10.40.6F(X)                   1,6.0 5.0 5.0 0,4.0 5.0 2.0 5.0 )1,( )1( )1,( )/( 5.03.02.0)1,()1( x xxf f xf yxf xff x y x xy
  363. 363. Suppose that X and Y have following joint probability function 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. Question15 4321X/Y 00.100.11 0.20.100.32 000.203
  364. 364. No,for example: f(1,2)=0, but fx(1)=0x0.2#1 Answer15 321X 10.20.60.2F(x) 4321y 10.20.20.20.4F(y)
  365. 365. STAT 111 Chapter seven
  366. 366. Let (x,y) have the following joint distribution function 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. Question1 sum321x/y 1/31/61/601 1/31/601/62 1/301/61/63 11/31/31/3sum
  367. 367. a) b) c) d) sum65432X+y 102/62/62/60F(x+y) sum964321xy 102/602/62/60F(xy) sum941x2 11/31/31/3F(x2) sum941y2 11/31/31/3F(y2) Answer1
  368. 368. Lets X be a random variable with probability distribution Find the probability distribution of the random variable Y=2X-1 Question2 F(x)=
  369. 369. Y=2x-1, x= = g (y)=>f (y)=f (g (y)) sum321x sum531y 11/31/31/3F(y) -1 y x Answer2
  370. 370. Let X1 and X2 be discrete random variable with joint probability distribution F(x)= Find the probability distribution of the random variable Y=X1X2 Question3
  371. 371. sum64321y 16/184/183/184/181/18Fy(y) sum21X1/x2 3/182/181/181 6/184/182/182 9/186/183/183 112/88/18sum Answer3
  372. 372. Let X be a random variable with probability distribution F(x)= Find the probability distribution of the random variable Y=X2 Question4
  373. 373. n……321x 1/n1/n1/n1/nF(x) N 2……25941y 1/n1/n1/n1/nF(x) Answer4
  374. 374. Let X be a random variable with probability distribution F(x)= Find the probability distribution of the random variable Y=X2 Question5
  375. 375. n…..21x n941y 2/2n2/2n2/2n2/2nFy(Y) Answer5
  376. 376. STAT 111 Chapter eight
  377. 377. 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? Question1
  378. 378. sum210x 11/229/116/11F(x) 11/222/229/220Xf(x) 2 1 12 2 3)( )2,12,3(~ 12 102 )(                     N nk xE hypx x xsx xf Answer1
  379. 379. If the random variable X is the two faces when rolling a pair of balanced dice, find the expected value of X. Question2
  380. 380. SUM12111098765432X 11/362/3 6 3/364/365/366/365/364/363/362/361/36F(X) Answer2
  381. 381. Let X have the following probability mass function F(x)= Find E(X),E(X2), and e(x+2)2 Question3
  382. 382. Answer3 272)E(X 11)E(X 3E(X) 5nuniformX 2 2    
  383. 383. The probability mass function of the random variable X is given by Find the expected value of X. sum4321x 2/103/101/104/10F(x) Question4
  384. 384. SUM4321X 12/103/101/104/10F(X) 2.3- >E(x) 8/109/102/104/10XF(X) Answer3
  385. 385. The probability mass function of the random variable X is given by Find the expected value of g(x)=2x-1. sum987654x 1/61/61/41/41/121/12F(x) Question5
  386. 386. sum987654x 11/61/61/41/41/121/12F(x) 82/1 2 9/68/67/46/45/124/12E(x)=Xf(x) E(X)=82/12=41/6 E(2X-1)=2E(X)-1=2(41/6)-1=12.7 Answer5
  387. 387. 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 And for company B is given by Show that the variance of the probability distribution for company B is greater than that of company A 321x 0.30.40.3F(x) 43210x 0.10.30.30.10.2F(x) Question6
  388. 388. sum321x 10.30.40.3F(x) 20.90.80.3E(X)=X f(x) 4.62.71.60.3E(X2)=X2 f(x) SUM43210X 0.10.30.30.10.2F(X) 2->E(X)0.40.90.60.10XF(X) 5.6->E(X 2)1.62.71.20.10X2 f(x) Va=4.6-22 =0.6 Vb=5.6-22 =1.6 Vb(x)> Va(x) 1.6 0.6 Answer6
  389. 389. 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 Question7
  390. 390. Answer7
  391. 391. 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). Question8
  392. 392. Answer8 667.1 9 5 3)( 3 9 3 45 )( )5,9,3(~                                 N n nxE xx xf hypx
  393. 393. 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? Question9
  394. 394. X:number of heads Y: number of tosses Answer9 1505 30 5)( ) 30 1 bin(n,X    n n npxE
  395. 395. Let X be binomial distribution random variable with E(X)=2 , and V(X)=4/3 . Find the distribution on X. Question10
  396. 396. equation(1) equation(2) Answer10 ....... 4 3 npqV(x) 2......npE(x) p?n? p)(n,bin4X    )3 1 ,6(~ 62 3 1 3 1 3 2 11 3 2 6 4 3 4 2 binx nn qp qq    
  397. 397. Let x1,x2 ,and x3 be independent random variables having finite positive variances 2 1 , 2 2, 2 3, respectively. Find the correlation between x1-x2 and x2+x3. Question11   
  398. 398. Answer11 )()(( ),cov( ),-cor( 3221 3221 3221 xxvxxv xxxx xxxx   
  399. 399. 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). Question12 
  400. 400. E(U)=6/5 , E(V)=4/5 E(U.U)=18/10 , E(V.V)=1 V(U)=.36 , V(V)=.36 E(UV)=-.36 (U,V)=-1 Fv(V)210UR/VB 3/103/10000 6/1006/1001 1/10001/102 13/106/101/10Fu(U) Answer12 

×