50
Chapter2
Axiomsof Probability
which can be generalizedto give
"(!^')
:
,7
\-
/2-
P t\ r' 'tI, t'
- If
r6;a1 + III
P(AiA1A1,)
i.j.k
t<J
+ . . . +( - 1 ) ' + r P ( A t . - . A n \
If S is finite and ezrchone point set is assumedto have equal probability,then
p@):ry
tsl
where lEl denotesthe number of outcomesin the event E.
P(A) canbe interpretecleither as a long-run relative frequencyor as a measureof
one's desree of beliei.
PROBLEMS
1. A box contains 3 marblos: 1 red. I green, and 1
blue. Consider an experiment that consistsof taking I marble from the box and then replacing it
in the box and drawing a secondmarble from the
box. I)escribe the sarnplespace.Repeat when the
secondmarble is drawn without replacing thrl lirst
marble.
2. In an experiment. die is rolled continually until a
6 appears, at which point the experiment stops.
What is the sample spaceof this experiment? Let
E,, denote the event that n rolls are necessaryto
complete the experiment. What points of the sam-
plespace
arecontained
in E,'i Whatis lU f " 1 't
\r /
'Iwo
/o
3.
\'
dice are thrown. Let E be the event that the
sum of the dice is odd, let ,F be the event that
at least one of the dice lands on 1, and let G be
lhe event that the surn is 5. Describe the evonts
EF, E U F, FG, EF', ANdEFC,
4. A, B, and C take turns llipping a coin. The lirst one
to get a head wins. The sample spaceof this experiment can be delined by
"" - J 0l ,0o0t .0o. .o. l , o o o l . . . . ,
I
(a) Interpret the sample space.
(b) Define the following eventsin terms of S:
(i) zl wins :,,4.
(ii) B rvins:11.
(iii) t,,l i) B)".
Assume that,,1flips first, then B, then C,
then A. and so on.
5. A system is comprised of -5components.each of
which is either rvorking or failed, Consider an
experiment that consistsoI observing the statusof
eiich component,and let the outcome of the experiment be given by the vector (r1.,r2,x3,x.1,-r5).
where -v; is equal to I if cr:rnponent I is working
and is equal to 0 if component i is failed.
(a) Floiv many outcomes are in tl.resample space
of this experiment?
(b) Supposethat the system will work if components 1 ancl 2 arrl both rvorking, or if components 3 and 4 are both working. or if cornponents1.3. and 5 are all working.Let l{ be the
event that the systemwill rvork. Specifyall the
outcclmesin 1,V.
(c) Let zl be the event that components4 and .5
are troth failed. How many outcomr:sare containedin the event A?
(d) Write out all the outcomesin the eventAW.
6. A hospital administratorcodesirrcomingpatients
sutfering gunshot wounds according to whether
they have insurzrnce(coding 1 if they clo and 0
iI they clo not) and according to their condition,
which is ratecl as goocl (g), fair (f), or serious (s).
Consider an experinrentthat consistsnf the cocling
ol such a patient.
(a) Give the sample spaceof this experirnent.
(b) Let,4 be the event.that the patientis in scrious
condition.Specifythe outcomesin A.
(c) Let B be the event that the patient is uninsured.Specifythe outcomesin B.
(d) Give all the outcomesin the eventBc U A.
7. Consider an experirnent that consistsof detcrrnining the type o1 job-either blue-collar or whitecc;llar-and the political affiliation-Republican.
Democratic.or Independent-of the 15 mombcrs
of an adult soccerteam. Ilolv many outcomesare
(a) in the samplespace?
(b) in the event that at least one of the team members is a blue-collar worker?
',
*9o
3
3
3
/
:
3
s
L
J
"
Z
f
:
1*75'
Problems 51
(c) in the event that none of the team members
considershimself or herself an Independent?
8. Suppose that /4 and B are mtttually exclusive
events for which P(A) : .3 and P(B) : .5. What is
the probability that
(a) either A or B occurs'/
(b) r1 occurs but B does not'?
(c) both A t'nd B occur?
9. A retail establishrnentacceptseither the Arncrican
Expressor the VISA credit card.A total ol24 percent of its customers carry an American Express
card,6l percent carry a VISA card, and 11 percent carry both cards. What percentageof its customers carry a credit card that the establishment
will accept?
10. Sixty percent of the students at a certain school
wear neither a ring nor a necklace. Twenty percsnt wear a ring and 30 percent wear a necklace.
If one of the students is chosenrandomly. wlrat is
the probability that this student is wearing
(a) a ring or a necklace?
(b) a ring and a necklace'?
11. A total of 28 percent of American males smoke
cigarettes.7 percent smoke cigars,and 5 percent
smokc bo{.hcigarsanc{ci€larettes.
(a) What percentageof males smokes neither
cigarsnor cigarettes?
(b) What percentage smokes cigars but not
cigarettes?
An elementary school is offering 3 language
classes:one in Spanish,one in French.and one in
German. The classesare open to any of the 10t)
studentsin the school.There are 28 studentsin the
Spanishclass,26irr the F'renchclass,and I6 in the
German class.There ttre 12 studentsthat arc in
both Sparrishand French,4 that are in both Spanish and Gerrnan, and 6 that are in both French and
German. lrr addition, there are 2 studentstaking
all 3 classes.
(a) If a student is chosenranclomly.what is the
probability that ht: or she is not in any oi the
languageclasses?
(tr) if a student is chosen randotrly. what is the
probabilitythat he or sheis takingexactlyone
languageclass}
(c) lf 2 studentsare chosenrandomly,what is the
probabilitythat at least 1 is taking a language
class?
13. A certain town with a population c1f100,000has
3 nervspapers:I, Il, and lll. The proportions of
townspeoplewho read these papers are as follows:
[: 10 percent I and lI: 8 percent
I l : 3 0 p e r c e r r tI a n d I l l : 2 p e r c e n t
III: 5 percent II andIII:4 perccnt
I and II and
III: 1 percent
(Thc list tells us. for instance, that 8000 people
r e a dn e r v s p a p e rI sa n d I l . )
(a) Find tht: nunber of people lvho read only one
Ilewspaper.
(b) How many people read at least lwo
nervslrapcrs?
(c) lf I and III are morning papers and ll is an
evcningpaper,how many peopleread at least
one morning paper:plus an eveningpaper?
(d) How many people clo not rcacl any
newspapers'?
(e) Horv many people read onlv one morning
paper and one evening paper'l
14, The following clatawert: given in a study of a group
of I000 subscribersto a certain magazine:[n referenceto job. marital status,and education.there
were 312 professionals,470 married persons,525
college graduatcs, 42 professional collcge graduates, 147 married college graduates, {J6 married
prolessionals,and 25 tn:rrricd prot.essionalcollcgc
graduates.Show that the nunrbersreported in the
stuclymust be incorrect.
the
Hint:LeL M.W, and G den<ltc.respec{.ively.
set of professionals,manied persorls,and college
graduates.Assume tha( <sneo{ the I(l(Il persons
is chosen at raudom. and use Proposition4.4 to
show that if the given numbcrs are corrr:ct, then
P ( M u W u G ) > t .
poker hands are
"? )
ecluallylikely, what is the pr<lbabilitvoI beingdealt
(a) a flush'?(A hand is said to be a flush if all 5
carclsare o1 the samesuit.)
(b) one pair'? ('I'ltis occurs when the cards have
dcnominationsa. ri. b. c, d. r.vheren. b. c. and
r/ trreall distinct.)
(c) trvo pairs? (This occurs rvhen thc cards have
denominationsa, a, b. b, c, rvherea, b, and c
are all distinct.)
(d) three of a kincl? (This occurswhen the carcls
have denominations a, (t, e, b, <:.where a, b,
and c are all distincl.)
(e) four of a kincl'l ('I'his occurs when the cards
have denominationsa. a. a, a, b.)
Poker dice is playcd bv sirnultaneouslyrolling -s
clice. Show that
(a) P{no two alike} : .0926:
(tt) P{one pair} : .zffljg'
(c) P{two pair} : .2315;
(d) P{three alike} : .1J43;
(e) f' {full house} : .0386:
(f) P{four alike} : .0193'
(g) P{fivealike} : .0008.
17. If 8 rooks (castles) arc randomly placed on it
chessboar:d.
compute the prnbabilit;rthat nclneof
the rooks can capturc any of the others.That is.
15. If it is assumeclthat all (
52
C h a p t e r2
Axioms of Probability
compute the probability that no row or file contains more than one rook.
18. Two cards are randomly selcctedfront an ordinary
playing deck. What is the probability that they
form a blackjack? That is, what is the probability
that one of the cards is an ace and the other one is
either a ten. a jack, a queen,or a king?
19. Two syrnmetric dice have both had two of their
sides painted red, two painted black. one painted
yellow. and the other painted white. When this
pair of dice is rolled, what is the probability that
both dice land with the same color face up?
2{}. Supposethat you are playing blackjack against a
dealer. Irr a freshly shuffled deck, what is the probability that neither you nor the clealer is clealt a
blackjack?
21. A small community organization consists o1 20
families,of which 4 have one child,8 havetwo children, -5have three children, 2 have four children,
and t has five children.
(a) If one of these families is chosen at random,
what is the probability it has i children, I :
r.2.3,4.51
(b) If one oI the children is randomly chosen,
what is the probability that child corneslt'om
a f a m i l yh a v i n gj c h i l d r e n i, : 7 , 2 . 3 , 4 . 5 ' l
22. Consider the following technique lbr shullling a
deck of n cards: For any initial orc'leringo1 the
cards,go through the deck one card at a time and
at each carcl, Ilip a lair coin. If the coin comes
utrrheads, ttren leave the card where it is; if the
coin comes up tails, then movc that card to the
end of the deck. After the coin has been flipped n
times, say that one round has been completed.For
instance,if n : 4 and the initial orderingis 1. 2, 3,
4. then if the successivellips result in the outcome
h. t, t, h. then the ordering at the end of the round
is 1,4,2,3. Assumingthat all possibleoutcomesof
the sequenceof n coin fiips are equally likely, what
is the probability that the ordering after one round
is the same as the initial orclering?
23. A pair of fair dice is rolled. What is the probatrility that the seconddie lands on a higher value than
doesthe first?
24. If two dice are rolled, what is the probability that
the sum of the upturned facesequals l'?Find it lbr
i :2.3, ...,11,12
A pair of dice is rolled until a sum of either 5 or 7
appears.Find the probability that a 5 occurs first.
Hint: Let E,, denote the event that a 5 occurs on
the nth roli and no 5 or 7 occurson thclirst n - 1
rolls. CcrmputeP(Er) and argue that I
P(8,') is
thll desiredprobability.
26. The game of craps is played as follows: A player
rolls two dice. lf the sum of the dice is eithcr a 2,
3, or'12, the player loses;if the sutn is either a 7
or an 11, the player wins. lf the outcome is anything else,the player continuesto roll the diceuntil
she rolls either the initial outcorne or a 7. If the 7
comes first. the player loses.whereas if the initial
outcome reoccurs before the 7 appears.the player
wins. Cornpute the prolrability of a player winning
at craps.
flint: I-et f; denote the event that the initial outcome is I and the player wins. The clesiredprobability is I
ptgil. To compute P(Ei), dsline the
i-a
events E;,,1to be the event that the initial sum is
I and theplayer wins on the nth roll. Argue thal
P(Ei):
,,).
,LrP(Ei
27, An urn contains 3 red and 7 black balls. PlayersI
and B withdraw balls ffom the urn consecutively
until a red ball is selected.Find the probabilitythat
,4 selectsthe red ball. (,,1draws the first ball. then
B, anclso on. There is no replacementol the balls
drarvn.)
28. An urn contains 5 r:ed,6 blue, and 8 greerrballs.
If a set of 3 balls is randomly selected,what is the
probabilitythat each of the balls will be (a) of the
same color'l (b) of diflerent colors'?Repeat uuder
that whenever a ball is selectecl.its
the assumptior:r
color is noted and it is then replaced in the urn
before the next selection. This is known as satn'
p lin g with repIacentetfi.
29. An urn contains ruwhite and m blitck balls, where
n anclm are positive numbers.
(a) If two balls are randonly withdrawn, what is
the probability that they are the samecolor?
(tr) If a ball is randornly withdrawn and then
replacedbefore the secondone is drawn, what
is the probability that the withdrawn balls are
the same color?
(c) Show that the probability in part (b) is always
larger than the one in part (a).
30. The chessclubs of trvo schoolsconsist o[, respectively. 8 and 9 players. Four members from each
club are rtrndomly chosen to participate in zrcontest between the two schools. T'he chosen players lrorn one team arc then randomly paired with
thosefrom the other team, and eachpairing plays
a gamc oI chess.Supposethat Rebeccaand her sister Elise are on the chcssclubsitt dilferent schools.
What is the probability that
(a) Rebeccaarrd Elise will be pairecl?
(b) Rebeccaand Elise u'ill be chosento represent
their schoolsbut will not play eachother?
(c) either Rebecca or Rlise will be chosen to
representher school?
Problems 53
31. A 3-person basketball team consistsof a guard. a
forward. and a center.
(a) If a person is chosen at rand<lmfrom each of
three different such teams,what is the probability of selectinga complete team?
(b) What is the probability that all 3 players
selectedplay the same position?
32. A group of irrdividuals containing b boys and g
girls is lined up in random orden that is, each cll
the (b + g) ! per:rnutationsis assumedto be equally
likely. What is the probability that the person in
t h e l t h p o s i t i o n ,1 = i < b + g , i s a g i r l ?
33. A forest contains 20 elk, of which 5 are captured,
tagged,and then released.A certain time later, 4
of the 20 elk are captured.What is the probability
that 2 clf these 4 have been taggcd? What assumptions are you making?
34. The second Earl of Yarborough is reported to
have bet at odds of 1000to 1 that a bridge hand
of 13 cardswould contain at leastone card that is
ten or higher. (By ten or higher we mean thaf a
card is either a ten, a jack, a queen.a king, or an
ace.)Nowadays,we call a lrand that has no carcls
higher than 9 a Yttrborctul4lr.What is the probability that a randomly selected bridge hand is a
Yarborough?
35. Seven balls are randomly withdrawn from an urn
that contains12 rcd. 16 blue, and lu green balls.
Find the probability that
(a) 3 red, 2 blue, and 2 green balls are rvithdrawn;
(b) at least 2 red balls are withdrawr;
(c) all withdrawn balls are the same color;
(d) either exactly 3 red balls or exactly 3 blue balls
are withdrawn.
36. Two cards are chosenat random front a deck of -52
playing cards.What is the probability that they
(a) are both aces?
(b) have the same value?
37. An instructor gives her classa set of 10 problems
with the inlbrmation that the final exam will consist of a random selectionof 5 of them. If a student
hasligured out how to do 7 of the problems,what is
the probability that he or she will answercorrectly
(a) all,5 problems?
(b) at least 4 of the problems'/
38. There are n socks.3 of which are red, in a drawer.
What is the value of n if, lvhen 2 of the socks
are chosenrandomly, the probability that they are
both red is |'?
39. Thcre are 5 hotels in a certain town. If 3 people
check into hotels in a day, what is the probatrility
that they each check into a diff'erent hotel? What
assumptionsare you making?
40. A town contains 4 people who repair televisions.
lf 4 sets break dorvn, what is the probability that
exactly I of the repairers aro called? Solve the
problem for I : 1,2,3,4. What assumptionsare
you making?
41. If a die is rolled 4 times,what is the probabilitythat
6 comesup at leastonce?
'l"wo
42.
dice are thrown n times in succession.Compute the probability that double 6 appoarsat lcast
once. How large need n be to make this probability
a t l e a s t; ?
43. (a) If N people, including A and B, are randomly
arrangedin a line . what is the probabiiity that
A and Il are Ilext to each other?
(b) What would the probability be if the people
'l
rvcre randoml-varranged in a circle
44. Irive people. designated as A. I], C, D, E. are
arrangedin lincar order. Assumingthat eachpossible order is ecluall;,likely, rvhat is the probability that
(a) there is exactly one person betweenA and B'!
(b) there are exactly two people between A
and ,B?
(c) there are three people betweenA and B'!
45. A wornan has n keys, of which one will open
her door.
(a) If slre tries the ke5rsat random, discarding
those that do not work, rvhatis thc probability
that she will open the door on her ftth tr:y?
(b) What if she does not discard previously tried
kcys?
46. How many people have to be in a room in order
that the probability that at least two of thern celebrate their birthdav in the same mnnth is at least
j? Assumethat all possiblcmonthly outcomcsare
equally likely.
47. lf there are 12 strangers in a room, what is the
probability that no two c;f them celebratetheir
birthday in the samemonth?
48. Given 20 people, what is the probability that.
among the 12 months in thr: year. therc are 4
months containing exactly 2 birthctaysancl 4 containing exactly 3 birthdays'/
49. A gror-rpof 6 men ancl 6 women is ranclomly
divided into 2 groups of size 6 each. What is the
probability that both groups will have the same
number of rnen?
50. In a hand of briclge, fincl the probability that you
have 5 spadesand your partner has the remaitring 8.
51. Supposethat n balls are randomly distributed irrto
N compartments.Find the probability that n balls
will fall into the first compartment. Assume that all
N'l arrangementsare equally likely.
54
Chapter 2
Axioms of Probability
52. A closet contains 10 pairs of shoes. If 8 shoes
are randomly selected,what is the probability that
there will be
(a) no complete pair?
(b) exactly 1 complete pair?
53. If 4 married couplesare arrangedin a row, find the
probability that no husband sits next to his wife.
54. Cornpute the probability that a bridge hand is void
in at least one suit. Note that the answer is not
(Why not?)
I:ILnt: U se Proposition 4.4.
Compute the probability that a hand of 13 cards
contains
(a) the ace and king of at least one suit;
(b) all 4 of at least 1 of the 13 denominations.
Two players play the following game: Player ,,{
chooses one of the three spinners pictured in
Figure 2.6, and then plaver B choosesone of the
remaining two spinners. Both players then spin
their spinner, and the one that lands on the higher
number is declared the winner. Assuming that
each spinner is equally likely to land in any of its
3 regions,would you rather be player z1 or player
B? Explain your answer!
2.6: Spinners
FIGURE
EXERCISES
THEORETICAL
Prove the following relations:
. (V n,)n :[ r;r *no
l . E F C E C E U F .
2. ltE C .1. then F'' C E'.
3. F:FEU
F E ca n d E U F : E U
E'F.
(l'') UF':frru''"r
Answersto SelectedProblems
CHAPTER
1
1. 67,600,000;
19,656,000 2. 1296 4. 24:4 5. 144;18 6. 2401 7. 720;72.:
I44:
72 8. 120;1260:34,650 9.27,720 10. 40,320;10,080;
1.152;2880l.384l1..720l.
72: 144 12,24,300,000:1.7
,1,00,720L3. 190 14.2,598,960 16.42:94
17.604,800 18.600 19.896;1000;91020.36:26 21.35 22.18 23.48
25. 521/(Bl)4 27.27,720 28. 65,536:2520 29. 12,600:945 30. 564,480
31. 165;35 32. 1287;L4,lI2 33. 220;572
CHAPTER
2
9.74 10. .4;.1 11.70:2 12..5:.32;149119813.20,000;
12,000;
11,000;
68,000;
10,000 14. 1.057 15..0020;.4226:.0475:.0211:-.A{J024
x 10-o
17.9.1.0947
18. .048 19. 5/18 20. .9052 22. (n + D lzn 23.5tl2 25. .4 26. .492929
27. .0888;.2477;.1243;.2099 30. 1/18;1.16;
L12 3L. 219:Il9 33.701323
36. .0045;.0588 37. .0833;.5 38.4 39. .48 40.1164;21t64;36164:6164
41..5177 44..3;.2;.1 46.5 48. 1.0604x 10-3 49..4329 50.2.6084x 10-6
52..091,45:.426853.12135 54. .0511 55. .2198;.0343
CHAPTER
3
l. u3
2 . 1 . 1 6 ; 7 1 5 ; 1 1 4 : 1 1 3 : 13".1. 23 :3l 9 5 , 6 1 9 1 6 . 1 1 2 7 . 2 1 3
8.1,12 9.711,1"I0. .22 ll.ll17:1133 12..504;.3629 14.351768;2101768
15. .4848 16..9835 17..0792;.264 18. .331;.383:.286:48.6219.44.29:
41.18 20..4:1126 21..496;311.4;9162
22.519;116;5154
23.419;Il2 24.lt3:
112 26. 20121;4014128. 31128;2911,536
29. .0893 30.711.2;31533..76,
49176 34.2713L 35..62,101L936.112 37.113:1.15:1.
38.'1213739.461185
40.3113:5113:5152:15152
41. 431459 42.34.48 43.4le 45.1111. 48.213
52..1.1:16/89;
50.17.5;381L65;17133
51..65;56165;8165:1165:14135;72135;9135
12127:
315;9125 55. 9 57.(c) 213 60.213;113;314 6L 116:3120 65.9lI3;
ll2 69.9;9:18:L10;4; 4: B:1.20alloverL28 70. ll9;1lLB 7L 38164;13164;1,3164
73.1,116;1,132;5116;114;31132
74.9119 75.314,7112lt. p211t- 2p + 2p2)
79..5550 81..9530 83..5;.6;.B 84.911.9;6119;4119;7115;531L65;7133
89.97l l42: t5 126:3311,02
4
CHAPTER
l. p(4) : 6/91:p(2) : 8/9L:p(1) : 32/91;p(0): 1/91;p\-1) : 16/91,;
p(*2) :28/91 4. (a)1,12;5118:'5136:5184:51252;11252:0;0;0;0
5. n - 2i;
457
A
l. A classhas 10 boys and 6 girls. 3 studentsare selectedat randomfrom the class.What is the
probability that 2 of them are girls and 1 is a boy?
2.Dice arethrown until a six occurs.Computethe probabilityof the events
4 : {"Six" obtainedin first two throws}
B : {the numberof throws is odd}
3.. 4 fair coinsaretossed"Find the probabilitythat they are all headsif at leastone of coin is head.
4. In telephonenumber5 last numbersare cleaned.Find the probability of eventthat different
figuresarecleanedbesides1,3,5.
5. There are4 balls with numbers1,2,3,4in the urn. Chooseat randomone ball after anotherfrom
the urn. Find the probability that at least 1 of balls orderednumbercan be the samewith its own.
6. Coin, with the probability of "head" which is equalto l13 thrown until a "tail" occurs.Find the
probability of eventsthat:
A: {the numberof throwsis odd};
3 : {"tail" is obtainedin the first threethrows}
7.Dice thrown 10 times.Find probability f eventsthat:
A:{at leastoneof them"5"}; B:{ exactlytreeof them"5"}
8. Given is an urn containing 9w and2b balls.Find the probability of eventthat third takenball
is white if color of fist 2 takenballs without replacementunknown.
g.Thereare 30 ticket, 6 of them are "good". Two studentsone after another aretaking ticket. Find
the probabilityof eventsthat
4:{both havegot "good" ticket};B:{secondhavegot "good" ticket}.
10. 90 of 100 itemsare non-defective.Find the probabilityof eventthat at least2 of 10 taking
itemsare defective.
11. 80 of 100itemsare non-defective.Find the probabilityof eventthat 3 of 5 taking itemsare
defective.
"1"}, if it is known that the
12. 3 dice arethrown. Computethe probabilityof eventA:{occur two
sumoffaces equal4.
13. Find the probabilityof event,that the 5 digit numberof automobiletakenat randomhasonly
two identicallyfigures.
Find
14. Given is an um containing 4w and 3b balls. 1 ball took out of urn without replacement.
the probabilityof eventthat taking 2 balls with replacementarewhite.
15. Dice arethrown until a six occurs.Computethe probability of the following events:A : {A
six is obtainedin the first two throws),
B: {the numberof throwsis even}.
3
16. Threecoinsthrown 10 times.Find probabilityof eventthat at leastin one trial appearances
"heads".
...20) . Find the probabilityof eventthat 3 of
17 Chooseat random 4 numberfrom (1 2 ....
them even,1 odd and exactly 1 divided by 10.
30. 3 of 15 books in Russian.Calculateprobability of event thatat leastone among3 choosing
booksin Russian.
18. There are N studentsin a class.Each student'sbirthday is one of 365 days .What is the
probabilityP that therearc atleast2 studentswhosebirthdaycoincide?
19.ThereareM tickets numbered 1,2,...,M,of which n, numbered 1,2,...nwin prizes(M>:2n).
You buy n tickets. Find the probability of winning at leastone prize
20. Considera family with two children. Ask the probability that both children are boys, if: a) that
the older child is a boy; b) ) that at leastone of the childrenis a boy.
2l.Letwe havethe sequences
{a,,e2,...,a,},whereo,.{0,r,2,3\,
i =ln.Find the probabilitythat :1) the endsare0;
2) exactlym of them are"2".
22.Letwe havethe sequences{o,,ort...ten},
wherea, e{0,1,2\,t=tn. Find the probabilitythat 1)
first threeelementsare"l";
2) exactlym+2 of them are "0" , suchthat2 of them in the end.
23. Chooseat random 10 numbersfrom (1 2 ...30) . Find the probability of eventthatT of them
evenand exactly2 divided by 10.
24 .5 of 20 bookson "Probability Theory" .Calculateprobability of eventthat two of 4 choosing
bookson "ProbabilityTheory" .
25. Dice thrown twice. Computethe probabilityof the following events:A: {A "3" obtainedat
lastonetime);
B : {A sumof facesis not lessthan9}.