ProbabilityMathematics S & T
PPROBABILITY At the end of the lesson, students should should be able to: 1
Techniques of counting
$
%ents and #robabilities
'
Mutually eclusie eents
./
0nde 0nde#e #end nden entt and and cond condit itio iona nall eents
a) b) c) d) e) f) g) h) i)
use the counting rules for finiite sets, including the inclusion-eclusion rule, for t!o or three sets" use the formul formulae ae for #ermut #ermutati ations ons and combi combinat nation ions" s" understand the conce#t of sam#le s#aces, eents, and #robabilities" unders understan tand d the meanin meaning g of com# com#lem lement entary ary and ec eclus lusie ie eents eents"" calc calcul ulat ate e the the #rob #robab abil ility ity of an een eent" t" understand the meaning of mutually eclusie eents" use use the the form formul ula a P(A P(A *) + P(A P(A)) P(* P(*)) P(A P(A n *) *) unders understan tand d the the meani meaning ng of of inde# inde#end endent ent and and cond conditi itiona onall eent eents" s" use use the the form formul ula a PA PA n *) + P(A P(A)) P(* P(* A)
Question !
Techniques of counting
Permutations and combinations: 1/
Multi Multi#l #lic icat atio ion n 2ule 2ule : 0f an an o#er o#erat atio ion n can can ha## ha##en en in in r !ays, and the follo!ing o#eration can ha##en in s !ays, then the total number of different !ays for these t!o o#eration to ha##en is r s/
Question 1 0f there are ' different routes !hich lin3 to!n A to to!n * and there are 4 different routes !hich lin3 to!n * to to!n 5/ 6ind the total number of different !ays !here a #erson can trael from to!n A to 5/ 7189 Question 2 There are 8 contestants ta3ing #art in a com#etition/ 6ind the total number of !ays !here the first, second and and thi third rd #ri #ries es can can be be !on !on by the the con conte test stan ants ts// 7''4 7''499
6rom the !ord @DATSE, find a) the the numb number er of of diff differ eren entt !or !ords ds tha thatt can can be form formed ed by using all the al#habets / b) the the numb number er of of !ord !ords s form formed ed ifif the the al#h al#hab abet ets s SE SE must al!ays together/ c) the the num numbe berr of of !or !ords ds that that can can be be for forme med d ifif the the ' al#habets SE must al!ays se#arated/ d) the the numb number er of of !ord !ords s form formed ed !hi !hich ch beg begin in !it !ith h the the al#habet D and end !ith the al#habet E/ e) the the numb number er of of ' al# al#ha habe bets ts !or !ords ds !hi !hich ch can can be formed/ Question " Co! many . digits numbers can be formed by using =, 1, $, ',F, G if re#etition is not allo!ed/ 6ind, ho! many of these numbers (i) are greater than .===/ (ii) (ii) star startt !ith ith the the digi digitt ' (iii) (iii) can be dii diided ded com#le com#letely tely by
Permutations $/
'/
Question #
A #er #ermu muta tati tion on of of a set set of of diff differ eren entt ob;e ob;ect cts s is an arrangement of all or #art of the ob;ects in a s#ecific order !ithout re#etition/ The The tota totall arra arrang ngem emen entt of n dif differ feren entt ob; ob;ec ects ts by ta3ing r ob;ects ob;ects at a time is nPr , !here n! n Pr + + / ( n r ) !
6our girls and t!o boys are sitting in a straight line/ Co! many different arrangements can be formed, a) if no limi limita tati tion on in the the !ay !ays s of of sit sitti ting ng// b) if the the $ boys boys must must sit sit ne nett to to eac each h oth other er,, c) if the the $ boy boys mus mustt not not sit sit tog toget ethe her/ r/
−
%am#les: (a)
<
(b) ./
P' +
5! 5.4.3! + + $= 3! 3!
Permutation $ith identica% ob&ects
8
P8 +
H/
The The tot total al numb number er of #erm #ermut utat atio ion n of of all all the the n n different ob;ects is Pn + n >/ ?ues ?uesti tion ons s ino inol lin ing g #erm #ermut utat atio ions ns,, norma normally lly use use the !ord: @arrange or arrangements, @#ermutations and @formed/
The The num numbe berr of of arr arran ange geme ment nts s of of n ob;e ob;ect cts, s, of !hich # of one ty#e are ali3e, q of a second ty#e are ali3e, r of a third ty#e are ali3e, and son, is n> p > q > r > // ////
Question ' Question 0n ho! many many !ays, the ' al#habets A, * and 5 can can be arrangedB arrangedB 749 4/
0n #ermutations, the order of arrangements is arrangements is im#ortant/ Cence, A*5 and A5* is regarded as different arrangements/
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The letters of the !ord MATC%MAT05S MATC%MAT05S are !ritten, one on each of 11 se#erate cards/ The cards are laid on a line/ 5alculate the number of different arrangements of these letters/ 6ind the number of arrangements, !here the o!els are all #laced together/ 7(a). G8G 4==, (b)1$= G4=9
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ProbabilityMathematics S & T Question (
numbers of arrangements if the red and !hite beads are net to each other/ 71$9
6ind the total number of four digits numbers formed by using the digits 1, $, $, ' and <, if re#etition are not allo!ed/ Co! many of these are greater than '===B
,ombinations
11/
Permutation $ith re)etition a%%o$ed 8/
The total number of arrangements of n distinct ob;ects, by ta3ing r ob;ect at a time !ith re#etition r
allo!ed is n /
A* A5 *5 There are ' !ays/ Eote : the order of selection is not im#ortant/
Question * Seen digits numbers are formed by using the digits K=, 1, $, F/, GL / Determine the total number of H digits numbers formed/ Co! many of these numbers begin !ith the digit H/ Co! many are there if no re#etition are allo!ed/
G/
Cence selection of A* or *A is consider as one selection only/ *ut in #ermutations, A* and *A are consider as se#arate arrangements/ 1$/
Arrangements in a circ%e
A combination is a selection of one or more ob;ects from a grou# of ob;ects, !ithout ta3ng into consideration their order / %am#les: (i) 6ind the total number of !ays to select $ letters from the letters A, *, and 5/
The number of combinations of r ob;ects from n different ob;ects is n5r or
The number of arrangements of n different ob;ects in a circle !here cloc3!ise and anticloc3!ise arrangements are different , is (n-1)>/ hen students sit in a circle, the anticloc3!ise or cloc3!ise sitting are consider as different/
n
5r +
<
5' +
Question 1+ 6our boys and four girls are required to sit in a circle/ a) Determine the number of !ays !hich they can do so/ b) 6ind the total number of !ays !here they can be sit if they hae to sit boys and girls alternately/ c) 0f the most senior boy and girl must sit together/ 6ind the total number of !ays !here they can do so/ d) 6ind also the total number of !ays !here the most senior boy and girl must sit se#arately/ e) 0f $ boys, A and * and a girl 5 cannot sit together, find the total number of !ays !here they can sit if all the boys and girls must sit alternately/
1=/
The number of arrangements of n different ob;ects in a circle !here cloc3!ise and anticloc3!ise arrangements are the same, is
(n − 1)> / $
8
5$ +
G
5H +
1'/
n r ÷ , !here
n> r >(n − r )> <> $> '>$> 1/$ 8.7 1.2 9.8 1.2
+ $8 + '4
7 Eote: n5r + n5r 1 9
?uestions inoling combinations, normally use the !ord: @choose, @select , @formed, or @combination/
Question 1 A delegation of ' students are to be chosen from ten students/ 0n ho! many !ays can this be doneB Question 1! %ight #oints are mar3ed on a #lane such that no three #oints are on a straight line/ 6ind the total number of triangle that can be formed by dra!ing lines connecting any three #oints/
6or eam#le if beads !hich are threaded on a ring, then the cloc3!ise and anticloc3!ise arrangements are the same/
Question 1"
Question 11 Si bulbs are #lanted in a ring and t!o do not gro!/ 6ind the total arrangements !here these t!o bulbs are together/ 7.89
Ten students are diided into ' grou#s !ith grou# A consists of $ students, grou# * consists of ' students and remaining students ;oin grou# 5/ 6ind the total number of !ays to diide the students into ' grou#s/
Question 12
Question 1#
Nne !hite, one blue, one red and t!o yello! beads are threaded on a ring to ma3e a bracelet/ 6ind the total
A committee of si is chosen from H males and . females/
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ProbabilityMathematics S & T a) Determine ho! many !ays can the committee be chosen/ b) Co! many of these committee chosen, consist of eactly . males/ c) 0f the committee formed, must consists at least one female/ Determine the total number of committee that can be formed/ d) The most senior male or female, but not both, must be chosen/ Co! many committee can be chosen/ e) 0f $ males, A and * and a female, 5 must be selected, ho! many of these committee can be f ormed/
%am#le A coin is tossed ' times consecutiely/ An eent A is defined as the heat a##ears t!ice/ A + KCCT, CTC, TCCL 1H/
as P(A), and is defined as
n ( A)
/
n ( S )
!here =
≤ P(A) ≤ 1
The #robability of eent A not ha##ening is: P( A ) + 1 P(A) !here A or A is read as @not A
-e%ection of at %east one ob&ect from n ob&ects. 1./
The #robability of an eent A ha##ens is denoted
The number of selections of at least one ob;ect from n different ob;ects is $n 1 %ery ob;ect has t!o o#tions, that is to be selected or not selected at all, hence the total number of combinations for the n ob;ects is
$ x $ x $ x //////x $ 1 4 44 2 4 4 43 $n
*ut this !ill include the case !here all the ob;ects are re;ected/ Cence !e hae to minus this case out/ $n 1
Question 1* State ho! to read each of the follo!ing notations: P(A ∪ *) P(A ∩ *) P(A ) P(A ∩ * ) P(A ∩ * )
seful result relating t!o eents A and * Question 1' Co! many !ays can a boy inites at least one of his si friends to his #arty/
(a)
Probability A but not * P(A ∩ * ) + P(A) - P( A ∩ *)
Question 1(
A
0n ho! many !ays can a student select at least a boo3 from < different boo3s/
A
*
A
1
-am)%e s)ace0Ruang -am)e%, often denoted by S, is the set of all #ossible outcomes of a e#eriment/ %am#les: a) A coin is thro! ' times, list all the sam#el s#ace/ S + K (TTT, TTC, TCT, CTT, TCC, CTC, CCT, CCC) ,!here C stands for #icture(Cead) and T stands for number(Tail)/ n(S) + 4 b)
A dice is thro!n t!ice consecutiely/ Oists all the #ossible outcomes/ 7'49
c)
6our digit numbers are formed by using digits from K=, 1, $, ', ., <, 4, H, 8, GL/ 6ind the sie of sam#el s#ace if (i) no re#etition is allo!ed/ (ii) re#etition is allo!ed, and the number can begin !ith ero/ 71====9 14/
An eent(#eristi!a) is a subset of the sam#el s#ace/
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∩
P (A
Probabi%it/
-'-
A , 1
B ' )
B
1
∩
P (A
B )
(b) Probability neither A nor * P(A ∩ * ) + 1 P (A *) AA
B
*
A
A ,1
1
Question 2+ 0n a surey, 1
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ProbabilityMathematics S & T
Question 21 3utua%%/ 45c%usie 4ents
A bo contain 1= identical red balls, H identical blac3 balls, and . identical green balls a) A ball is dra!n randomly from the bo, find the #robability that i) the ball is blac3 in colour, ii) the ball is neither red nor green, iii) the ball is not green/ b) Three balls is dra!n randomly from the bo, find the #robability that, i) all the three balls are different in colour, i i) the first ball is red, the second ball is blac3 and the third ball is green/ iii) all the three balls are the same colour/ i) the third ball is green in colour c) 6ind the ans!ers for b(i) until b(i) if the ball is re#laced bac3 to the bo before the net ball is dra!n/
$=/
A
$1/
,ombine eents 18/
0f A and * are $ eents such that P(A) ≠ = and P(*) ≠ =, then P(A *) + P(A) P(*) P(A n *)
A
Rien P() + and P(U) + V / 0f and U are mutually eclusie, find i) P( U) ii) P(U n )
A , 1
$$/
1
0f A , * and 5 are ' eents such that P(A) ≠ =, P(*) ≠ =, and P(5) ≠ = then P(A ∪ * ∪ 5) + P(A) P(*) P(5) P(A ∩*) - P( A ∩ 5) P(* ∩ 5) P(A ∩ * ∩ 5)
Question 22 %ents A and * are such that P(A) + P(A∪*) + i) ii) iii)
3 5
Mutually eclusie eents are eents that cannot occur at the same time/ %am#les Oet say A stands for eent that you are standing and * stands for eent that you are sitting/ *oth A and * can ha##en, but not at the same time/
*
A
1G/
*
Question 2!
B
A
T!o eents A and * are said to be mutually eclusie if A ∩ * + ∅ , that is P (A ∩ * ) + =/ Cence, for eclusie eents : P(A ∪ *) + P(A) P(*)
17 25
, P(*) +
1 5
, and
/ 6ind
P(A ∩ *) P(A ∩ *) P( A or * but not both)
To sho! t!o eents A and * are mutually eclusie, !e need to sho! one of the follo!ing statements is true: a) P (A ∪ * ) + P(A) P(*), or b) P(A ∩ *) + = or A ∩ * + ∅/
Question 2" T!o dice, one red and one blue, are to be rolled once/ a) 6ind the #robabilities of the follo!ing eents: %ent A: the number sho!ing on the red dice !ill be a < or a 4 %ent *: the total of the numbers sho!ing on the t!o dice !ill be H, %ent 5: the total of the numbers sho!ing on the t!o dice !ill be 8/ b) State, !ith reason, !hich t!o of the eents A, * and 5 are mutually eclusie/
,onditiona% )robabi%it/ and inde)endent eents Question 2 A grou# of <= #eo#le !as as3ed !hich of three ne!s#a#ers, A, * or 5 they read/ The results sho!ed that $< read A, 14 read *, 1. read 5, < read both A and *, . read both * and 5, 4 read both 5 and A, and $ read all '/ 6ind the #robability that a #erson selected at random from these grou# reads i) at least 1 of the ne!s#a#ers, 7=/8.9 ii) only one of the ne!s#a#ers, 7=/4$9 iii) only A 7=/'$9
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$'/
6rom the tree diagram, P(A ∩ *) + P(A) P(* A) !here P(* A) is the conditional #robability and is read as P( *, gien A) or @#robability * ha##ens gien A has ha##ened/ 6rom
P(A ∩ *) + P(A) P(* A)
P(* A) +
P ( A ∩ B ) P ( A)
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ProbabilityMathematics S & T P(A ∩ *) + P(A) P(*A) or P(A ∩ *) + P(*) P(A*)
Similarly,
*
P(* A) P(A)
A P(* A)
P(A ∩ *) + P(A) P(* A) Question 2'
*
The eents A and * are inde#endent and are such that P(A) + , P(*) + =/$ and (A ∩ *) + =/1
P(A) A
* *
P(A *) +
$4/
P ( A ∩ B ) P ( B )
To test !hether $ eents A and * are inde#endent !e need to erify that one of the follo!ing statements is true: a) P( ∩ U) + P() P(U) b) P() + P( U) or P(U) + P(U)
Question 2# A bag P contains < blac3 balls and ' !hite balls/ A bag ? contains . blac3 balls and H !hite balls/ A ball is remoed randomly from bag P and #laced into bag ?, and a ball is then remoed randomly from bag ?/ The eents 1, *1, $, *$ are defined as 1 : The ball dra!n from bag P is !hite/ *1 : The ball dra!n from bag P is blac3/ $ : The ball remoed from bag ? is !hite/ *$: The ball remoed from bag ? is blac3 6ind P(1), P(*1), P(*$ *1), P($) and P(*1 $)/ 7'W8"
0f either of the eents A and * can occur !ithout being affected by the other, then the t!o eents are inde#endents/ %am#les of inde#endent eents: 1) A coin is tossed t!ice/ 0f eents C1 and T$ are defined as:
$
$
Inde)endent eents P(
∩ U) + P() P(U)
P() + P( U) or P(U) + P(U
Question 2( 0f A and * are $ eents such that P(A) + 8W1< , P(A and *) + 1W', P(A *) + .WH, calculate P(*), P(* A) and P(* A ) !here A is the eent X Eot A ha##ensX/ State, !ith reasons !hether eents A and * are inde#endent, mutually eclusie/ 7HW1$"
Inde)endent 4ents $./
1
3utua%%/ e5c%usie eents P (A ∪ * ) + P(A) P(*), P(A ∩ *) + = or A ∩ * + ∅/
Miscellaneous Problems
C1: getting a head in the first thro!
Question 2*06enn 7iagram
T$: getting a tail in the second thro!
A committee has $$ members of !hich H hae dar3 hair, are non-smo3ers, and do not !ear glasses" < hae grey hair, are non-smo3ers, and do not !ear glasses" . hae grey hair, smo3e and !ear glasses" ' hae dar3 hair, smo3e and do not !ear glasses" $ hae grey hair, !ear glasses and do not smo3e" 1 has dar3 hair, smo3es and !ear glasses/ a) A member of the committee is chosen randomly/ Oet P be the eent that the chosen member has grey hair, 5 is the eent that this member !ear glasses, and 2 is the eent that this member smo3es/ 6ind i) P(P), ii) P(P02), 71W$9 iii) P(P05), 74WH9 i) the #robability that this memberhas grey hair or !ear glasses (but not both), gien that it is 3no!n that his member smo3es/ 71W89
0f A and * are inde#endent eents, then P(A, gien * has occured) is #recisely the same as P(A), since A is not affected by */ P(A *) + P(A) 0t is also true that, P(* A) + P(*) *ut from aboe, P(A ∩ *) + P(A *) P(*), if A and * are inde#endent eents, then P(A ∩ *) + P(A) P(*) 5onclusion, if A and * are inde#endent eents, then (A ∩ *) + P(A) P(*) if A and * are de#endent eents, then
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ProbabilityMathematics S & T
b)
Are the eents P and 2 inde#endent of each otherB Are the eents P and 5 inde#endent of each otherB Rie your reasons for each ans!er/ T!o members of the committee are chosen randomly/ Oet P$ be the eent that both members
'/
0n ho! many !ays can 1< marbles can arrange in a straight line if 1= of them is red colour, $ are green, $ are yello! and 1 is blue colourB 7G==G=9
./
6ind the total number of arrangements for each of the follo!ing !ords:: a) MATC%MAT05S 7.G8G4==9 b) PA2AM%T%2 7.<'4=9 c) STAT0ST0J 71<1$=9
Sailors arrange colour flags in a line to re#resent instruction codes/ Co! many different instruction codes can be formed by using to $ identical blue flags and ' identical red flagsB 7''9
4/
Co! many . digits numbers !hich are greater than $=== can be formed by using digits 1, $, ', < if re#etition is allo!ed/ 71G$9
H/
6ind the number of !ays !here G students can sit in a circle/ 7.='$=9
8/
6ind the number of different !ays !here ' red balls, . !hite balls and 1 blue can be arranged in a straight line if i) all the ' red balls must together, ii) all the ' red balls must se#erated/ 7(i)'=" (ii)$<=9
G/
Co! many four and fie digits numbers can be formed by using the digits 1, $, ', ., < (re#etition is not allo!ed)B 7$.=9/ (i) Co! many of these numbers are (ii) greater than <===, 71..9 (iii) een numbersB 7G49
1=/
There are < seats including the driers seat in a Proton Saga car/ 6ind ho! many !ays < #ersons can be seated inside the car if only t!o of them can drieB 7.89
11/
6ind the total number of fie digits een numbers that can be formed by using digits 1, $, ', ., < if re#etition is allo!ed/ 71$<=9
1$/
6ind the total number of !ays !here the !ord MALAYSIA can be arranged/ Co! many of these arrangements !here the three al#habets A must be togetherB 74H$=9 7H$=9
1'/
0n ho! many !ays the letters in the !ord MINERAL can be arranged such that the three o!els must in een #lacing B 71..9
1./
6ind the number of 4-digits numbers that can be formed by using the digits ., ', ', ', 8, 8/ Co! many of these numbers are een numbersB 74=" '=9
1
0n ho! many !ays can a soccer team !hich consists of 11 #layers can be selected from 14 #layersB 7.'489
14/
%ight #layers in a game sha3e hand !ith each other before the game started/ Co! many times a #layer has to sha3e hands !ith other #layersB 7$89
hae grey hair and 2$ is the eent that both smo3es/ 6ind i) P(P$), 7
45ercise 1 0 Permutations and ,ombinations 1/
$/
0n ho! many !ays the letters in the !ord KERUSI can be arrangedB Co! many of these arrangements end !ith the letter K B 7H$="1$=9 A grou# of 11 boys are required to line u# in a straight line such that the tallest boy must be at the end of the line/ 6ind the number of !ays they can do so/ 7'4$88==9
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ProbabilityMathematics S & T
1H/
The committee of the Teachers and Parents Association of a school consists of 4 member and is selected from H #arents, . teachers and the schools #rinci#al/ 0n ho! many !ays this committee can be formed such that it consists of not more than ' #arentsB 7.4$9
18/
A delegation consists of < member is selected from < males and . females/ Co! many different delegation can be formed if it must consists of at least 1 femaleB 71$<9
1G/
A bag contain $ !hite balls and ' red balls/ 0n ho! many !ays can ' balls be selected from the bag if, (i) it must has at least one !hite ball, 7G9 (ii) it must has at least one red ballB 71=9
$=/
A committee of Parents and Teachers Association !hich consists of 4 member is formed by selecting its member from H #arents, . teachers and a school #rinci#al/ 0n ho! many !ays can this committee be formed if it must consists of (i) the school #rinci#al, 7.4$9 (ii) eactly . #arents, 7'<=9 (iii) not more than . #arentsB781$9
$1/
6ind ho! many ' al#habets !ord-code can be formed from $4 al#habets if (i) no re#etition is allo!ed, (ii) re#etition is allo!ed, (iii) eery !ord-code must consists of 1 o!el and re#etition is not allo!ed/ 7(i) 1<4==" (ii) 1H
include at most one of A and */ 0n ho! many !ays can the 1 committee no! be chosenB 7H==" <<=9 7E$==1WP1WH9 $4/
a) %ight #eo#le go to the theatre and sit in a #articular grou# of eight ad;acent resered seats in the front ro!/ Three of the eight belong to one family and sit together/ i) 0f the other fie #eo#le do not mind !here they sit, find the number of #ossible seating arrangements for all eight #eo#le/ 7.'$=9 ii) 0f the other fie #eo#le do not mind !here they sit , e#ect that t!o of them refuse to sit together, find the number of #ossible seating arrangements for all eight #eo#le/ 7$88=9
$H/
6our men, t!o !omen and a child sit at a round table/ 6ind the number of arranging the seen #eo#le if the child is seated a) bet!een the t!o !omen, b) the bet!een t!o men/ 7(a) .8, (b) $889 7H.WP1W19
45ercise 2: Probabi%it/ 1/
A boy has three bags P, ?, R, each of !hich contains $= balls/ P contains < blac3 balls, ? contains 1= blac3 balls and R contains 1< blac3 balls/ The rest of the balls are !hite/ 0f he dra!s a blac3 ball from P his net dra!s from ?, other!ise he dra!s from R. 0f he dra!s a blac3 ball from ? his net dra!s from R, other!ise he dra!s from P/ 0f he dra!s a blac3 ball from R his net dra!s from P, other!ise he dra!s from ?/ heneer a ball is dra!n it is al!ays re#laced before another ball is dra!n/ 0f he al!ays starts !ith bag P, find a) the #robability that the first four balls he dra!s are blac3, 7'W1$89 b) the #robability that, after fie dra!s, he has not dra!n from bag 2/ 71W4.9 7YH'WP$W19
$$/
0n ho! many !ays can ' different boo3s be diided among 1= students if, i) no limitation in the number of boo3s gien to the students, 71===9 ii) no student is allo!ed to get more than a boo3, 7H$=9 iii) no students can get more than $ boo3s/ 7GG=9
$'/
Co! many numbers can be formed by using . of the < digits 1, $, ', ., < i) if re#etition is not allo!ed, ii) if re#etition is allo!edB 0f re#etition is not allo!ed, ho! many of these . digits numbers iii) begin !ith $, i) end !ith $
$/
T!o #oints, A and * are mar3ed on a straight line/ Another line, !hich is #arallel to the first line is dra!n/ Eine different #oint are then mar3ed on the second line/ i) 5alculate the total number of triangles !hich can be dra!n by ;oining any three #oints as the ertices of the triangle/ 7819 ii) Co! many of these triangles are haing #oint A as its erteB 7.<9
0 hae a choice of t!o routes to get to !or3/ The #robability that 0 choose the first route on any day is =/4, and the #robabilities of my being delayed on the ;ourney are =/1 for the first route and =/$ for the second/ 5alculate the #robability that 0 get to !or3 !ithout being delayed, and hence sho! that the #robability of my being delayed #recisely once in three days is ;ust oer =/'1/ 7=/84" =/'1=49 7EH'WP$W$9
'/
Three men A, *, clan 5 agree to meet at a cinema/ A cannot remember if it is the 5athay or Ndeon cinema, and tosses a fair coin to decide/ * also tosses a coin to decide !hether to go the Ndeon or 2e/ 5 tosses a coin to decide if he should go to 5athay or not, and in the latter case, he !ould toss a coin again to choose bet!een Ndeon and 2e/ 6ind the #robability that i) A and * !ill meet, 71W.9 ii) * and 5 !ill meet, 71W.9 iii) A, * and 5 !ill all meet, 71W149
$./
$
0n ho! many !ays can a committee of ' men and ' !omen be chosen from a grou# of H men and 4 !omenB The oldest of the H men is A and the oldest of the 4 !omen is */ 0t is decided that the committee can
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*y JJC
ProbabilityMathematics S & T P(21 ∩ 2$), P(21 2$) and P(21 ∩ *$ ∩R3)" P(all the three balls are different colour)" P(the 1st and $nd ball is not red) 9 25 20 75 $ 7(i) " " (ii) (iii) 9 29 406 8H 29 812
i) A, * and 5 !ill go to different cinemas/ ) at least t!o of the men meet/ 7(i) V" () Z9 0f A and * are eents and P(*) + 1W4, P(A and *) + 1W1$, P(* 0 A) + 1W', calculate P(A), P(A 0 *) and P(A * ) !here * is the eent * does not occur/ State, !ith reasons, if A and * are i) inde#endent of each other, ii) mutually eclusie/ 7CS5XH'9 71W." " 1W< " not inde#endent " not mutually eclusie9
i) ii) iii)
./
%ight trees are #lanted randomly in a circle/ 0f t!o of the trees are attac3ed by irus and died, find the #robability that these t!o tree are ;ust net to each other/ 0f . of the trees are attac3ed by irus and died, find the #robability that i) at least t!o of these trees are net to each other, ii) all the . trees are net to each other/ 7$WH" '.W'<" .W'<9
4/
A bag contains . red and 4 blac3 balls/ Nne ball is dra!n at random" if it is blac3 it is re#laced in the bag, but if it is red it is not re#laced/ A second ball is then dra!n/ denotes the eent XThe first ball is redX and U denotes the eent XThe second ball is redX/ 6ind the #robabilities a) P(), 7$W<9 b) P(U gien ), 71W'9 c) P(U), 7$8WH<9 d) P(either or U but not both)/ 7'8WH<9
1=/
A regular octahedron, !hich has its eight triangular faces numbered from 1 to 8, is to be used as a dice/ The score for a thro! is the number on the face that comes u##ermost !hen the dice is thro!n on a horiontal table/ 0f t!o such dice are thro!n, find the #robability that a) the total score on the t!o dice is at least 1., 'W'$9 b) the difference bet!een the scores is $/ T!o Players, A and *, ta3e turn to thro! dice !ith A starting first/ The first #layer to obtain a difference of $ bet!een the scores on the dice !ins the game/ 6ind the #robability that c) A !in at his first thro!, d) * !ins at his first thro!, e) A !in at his third thro!/ 7(a) 'W'$" b9'W14" c) 'W14" d)'GW$<4" e) (' 1'.)W14<9 7EH
11/
Three dice, one red, one green and one blue are rolled simultaneously/ The eents R$, 2', S and T are defined as follo!s: R$ : the green dice sho!s the number $/ 2' : the red dice sho!s the number '/ S : the sum of the numbers on the red dice and the green dice is ./ T : the total of the numbers on the three dices is
7YH
0f A and * are eents and P (A) +
8 , P(A and *) 1<
1 . + , P(A *) + , ' H
7
calculate P(*), P(* 0 A) and P(* 0 A ), !here A is the eent A does not occur/ State !ith reasons, if A and * are i) inde#endent, ii) mutually eclusie/ 7HW1$"
A buc3et A contains < blac3 balls and ' !hite balls/ A buc3et * contains . blac3 balls and H !hite balls/ A ball is remoed randomly from buc3et A and #laced into buc3et *, and a ball is remoed randomly from buc3et */ The eents P1, C1, P$, C$ are defined as P1 : The ball #laced into buc3et * is !hite/ C1 : The ball #laced into buc3et * is blac3/ P$ : The ball remoed from buc3et * is !hite/ C$: The ball remoed from buc3et * is blac3/ 6ind P(P1), P(C1), P(C$ C1), P(P$) and P(C1 P$)/ 7'W8"
G/
A bag contains < red ball, 1= blue balls and 1< green balls/ Three balls is selected randomly from the bag, one by one !ithout re#lacing/ %ents 21, 2$, *$, R' are defined as belo!: 21 : the first ball dra!n is red" 2$ : the second ball dra!n is red" *$ : the second ball dra!n is blue" R' : the third ball dra!n is red/ 6ind
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1 36
,
1 6
;
1 3
;
11 36
;
1 36
;
1 2
9
7CS5 XH49
1$/
0n *altraia, there are three shi##ing com#anies A, * and 5/ These com#anies o!n shi#s of three ty#es" oil tan3ers, cargo shi#s, and #assenger liners/ The ty#es of shi# are described belo!/ Nil Passenger 5om#any 5argo shi# Tan3er shi# A 8= $= $= * .= .= $= 5 1= '= .= a) 6ind the #robability that a *altraia shi# chosen randomly is o!ned by 5om#any A/ 7=/.9 b) 6ind the #robability that a *altraia oil tan3er chosen randomly is o!ned by 5om#any A/ 7=/41<9 c) A shi# is #ic3ed randomly from each com#any/ 6ind the #robability that the three shi#s chosen are i) all oil tan3ers, 7=/=''9 ii) all of the same ty#e, 7=/=H<9 iii) all oil tan3ers, if it is 3no!n that they are of the same ty#e/ 7=/...9
1'/
T!o bags each contain ten discs !hich are indistinguishable a#art from their colour/ The first bag contains . red and 4 blac3 discs and the second, H red and ' blac3 discs/ A disc is chosen at random from the first bag and #laced in the second, Then, after thoroughly miing, a disc is
*y JJC
ProbabilityMathematics S & T ta3en from the second bag and #laced in the first/ 6ind the #robability that the first bag still contains eactly . red discs/ 7$8W<<9 7EH4WP1W19 1./
1
A man thro!s three dice and obseres the number on the to# face of each/ 6ind the #robabilities that a) all the three numbers are different, b) all the three numbers are the same, c) only t!o of the three numbers are the same, d) the sum of the three numbers is greater than 1
18/
A and B are eents, and AX denotes the com#lementary eent to A (i/e/ AX is the eent that occurs !heneer A does not occur)/ The follo!ing #robabilities are gien: P(A) + =/., P(B | A) + =/H, P(AX ∩ B) = =/'/ 6ind the #robabilities i) P(A ∩ *), 7=/$89 ii) P(*), 7=/<89 iii) P(A ∪ *), 7=/H9 i) P(A *)/ 7=/.89 State, !ith a reason, !hether or not A and B are inde#endent eents/ 7YHGWP 1W$9 7de#endent eents9
1G/
6our-figure numbers are to be formed from the digits ., <, 4, H, 8, G/ 6or each of the three cases belo!, find ho! many different four-figure numbers can be formed/ a) Any digit may a##ear u# to four times in the number/ 71$G49 b) Eo digit may a##ear more than once in the number/ 7'4=9 c) There is at least one re#eated digit, but no digit a##ears more than t!ice in the number/ 781=9 d) 6ind the #robability that a four-figure number chosen at random from the set of numbers in case (a) aboe contains at least one 4/ 74H1W1$G49 (EHGWP l W')
$=/
The eents A and * are such that P (A) + 1W$ P(*) + 1W. P(A or * but not both A and *) + 1W' / 5alculate P(A ∩ *), P(AX ∩ *), P(A 0 *) and P(* 0 AX ), !here AX is the eent A does not occurX/ State, giing your reasons, if A and * are i) inde#endent, ii) mutually eclusie/ 7
$1/
There are three families, A, * and 5" family A consists of 1 boy and ' girls, family * has ' boys and ' girls, family 5 has H boys and < girls/ A child is selected by selecting one family randomly, and then selecting a child randomly from the family/ a) 6ind the #robability that a girl is selected/ 7
$$/
A bo contains 1= ob;ects consisting of a red ball, $ !hite balls, ' red cubes and . !hite cubes/ Three ob;ects are dra!n randomly from the bo, one by one !ithout re#lacement/ The eents *$ and 21, are defined as follo!s: *$ : %actly t!o of the ob;ects dra!n are balls"
0f A and * are eents and '
P(*) +
, P(* A) +
<
$
P(not A or *) +
'
5alculate, P(A), P(A *), P(A State if A and * are: i) 0nde#endent, ii) Mutually eclusie/ 7
9 25
;
2 5
;
3 10
H
/
$<
B ), and
" (i) not inde#endent (ii) not
mutually eclusie9 14/
Are the eents P and 2 inde#endent of each otherB Are the eents P and 5 inde#endent of each otherB Rie your reasons for each ans!er/ T!o members of the committee are chosen randomly/ Oet P$ be the eent that both members hae grey hair and 2$ is the eent that both smo3es/ 6ind ) P(P$), 7
71GHH9
0n 5amelot, the rain does not fall on 6ridays, Saturdays, Sundays, and Mondays/ The #robability that rain falls on a Tuesday is 1W
%#lain the im#lications of the case α +β/ 7(i)
1 5
(α
+ 4β )(α − β ) β 9
1H/
A committee has $$ members of !hich H hae dar3 hair, are non-smo3ers, and do not !ear glasses" < hae grey hair, are non-smo3ers, and do not !ear glasses" . hae grey hair, smo3e and !ear glasses" ' hae dar3 hair, smo3e and do not !ear glasses" $ hae grey hair, !ear gl asses and do not smo3e" 1 has dar3 hair, smo3es and !ear glasses/ A member of the committee is chosen randomly/ Oet P be the eent that the chosen member has grey hair, 5 is the eent that this member !ear glasses, and 2 is the eent that this member smo3es/ 6ind i) P(P), ii) P(P02), 71W$9 iii) P(P05), 74WH9 i) the #robability that this member has grey hair or !ear glasses (but not both), gien that it is 3no!n that his member smo3es/ 71W89
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*y JJC
ProbabilityMathematics S & T 21 : %actly one of the ob;ects dra!n is red in colour/ Sho! that P(*$) + HW.= and calculate P(21 ), P(*$ ∩ 21), P(*$ 21), and P(*$ 21)/ 7 $'/
$./
$
$4/
1 11 H 11 9 " " " $ 1$= 1$ 4=
At the ninth hole on a golf course, there is a #ond/ A golfer hits a grade * ball into the #ond/ 0ncluding this ball, there are 4 grade 5 balls, 1= grade * balls, and . grade A balls in the #ond/ The golfer uses a net to XcatchX four balls/ The eents , U, and [ are defined as follo!s: : the catch consists of t!o grade A balls and t!o grade 5 balls/ U : the catch consists of t!o grade * balls and t!o other balls/ [ : the catch includes the golferXs ball/ *y assuming that the catch is a random selection of balls from the #ond, find (i) P()" (iii) P([)" (ii) P(U)" (i) P([ U) 6or each of the #airs and U, U and [, state !ith brief reasons, if the eents are mutually eclusie, inde#endent/ 1'< 4 7(i) (ii) (iii) 1W<" (i) 1W< '$' '$' (a) mutually eclusie/" not inde#endent" (b) not mutually eclusie/" inde#endent9 71GHG9 A com#etition bet!een t!o boats, A and *, consists of a series of inde#endent races and the com#etition !ill be !on by the first boat that !ins three races/ %ach race !ill be !on by either A or *, and their #robabilities of !inning are influenced by the !eather/ 0n bad !eather, the #robability that A !ins is =/G, !hile in fine !eather, the #robability that A !ins is =/./ 6or each race, the !eather is either fine or bad, and the #robability of bad !eather is =/$/ Sho! that the #robability of A !inning the first race is =/
)+
$ '
and P(A *) +
' !here H
1 $ *
, P(A
$H/
0n *robdignag, the !eather each day is either fine or rainy/ 0f the !eather is fine, then the #robability that the follo!ing day !ill also hae fine !eather is =/H/ 0f the !eather is rainy, then the #robability that the follo!ing day !ill also be rainy is =/8/ i) Su##ose that it is 3no!n that the !eather on 1 August !ill be fine/ Sho! that the #robability that the !eather on ' August !ill be fine is =/<<, and find the #robability that the !eather on < August !ill be fine/, ii) Rien that there is a #robability of =/. that the !eather on 1 Se#tember is fine, find the #robability that there !ill be fine !eather on $ Se#tember, and find also the #robability that there !ill be fine !eather on ' Se#tember/ 7(i) =/.'H< (ii) =/.9 71G8'A/1$9
$8/
At an underground station, trains can arrie from t!o directions, either from the !est or the east/ Assume that, at any instant, the net train !ill arrie from either one of these directions, and all arrials are inde#endent of one another/ The eents A, * and 5 are defined as follo!s: A : The net three trains all arrie from the east/ * : The net fie trains consist of eactly three from the east and t!o from the !est/ 5 : The net fie trains arrie alternately from the t!o directions/ 6ind P(A), P(*), P(5), P(A ∩ *), P(A ∪5), and P(* 5)/ State, giing your reasons in each case, !hether the eents a) A and * are mutually eclusie/ b) the eents * and 5 are inde#endent of one another/ 7
is the
- 1= -
1 < 1 ' ' 1 " " " " " 9 8 14 14 '$ 14 $
$G/
At a dinner, 1< bottled drin3s !ere o#ened/ There !ere H bottles of orange ;uice, ' of !hich are bro!n bottles, and the remaining !ere green bottles/ There !ere 8 bottles of lemonade, $ of !hich are bro!n !hile the rest !ere green/ These bottles !ere o#ened, one by one, in random order/ The eents A, * and 5 are defined as follo!s: A : The first $ bottles o#ened are bro!n/ * : The first $ bottles o#ened contain lemonade/ 5 : The last $ bottles o#ened are green and contain orange ;uice/ 6ind the alues of P(A), P(5), P(* A), P(5 A) and P(A ∩*X ), !here *X is the eent @not * / 7 '=/
eent X not * ha##eningX/ 6ind P(A ∩ *), P(A *), P(*) and P(* A)/
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State !ith reasons, !hether eents A and * are (i) inde#endent, (i) mutually eclusie 7=/', =/G, =/H, =/4" (i) inde#endent (ii) not mutually eclusie 9
$ $1
"
$ '<
"
1 1=
"
1 1'
"
' '<
9
71G8
A tetrahedron dice, !hich has its four faces numbered as 1, $, ' and ., is tossed ' times/ The score, S, for each thro!n is the sum of the numbers !hich are sho!n on the ' u##ermost faces of the dice !hen it is thro!n on a horiontal table/ %ents A, B and 5 are defined as belo!:
*y JJC
ProbabilityMathematics S & T A *
: The alue of S on the first thro!n is H/ : The sum of scores for $ thro!n is less than 1
1 4
'1/
;
3 8
;
1
;
2
3 4
;
15
;
16
11
9
12
There are $. stam#s inside an enelo#/ The details of the stam#s are sho!n belo!: 2ed '
*lue $
Uello! <
Rreen =
sed Stam#s nused $ 4 $ . Stam#s Three stam#s from the enelo# !ere dro# and lost/ %ents A, * and 5 are defined as: A :Eo green stam#s !as lost/ *: Stam#s that lost, consists of eactly one red and unused stam#s/ 5 :At least one used stam#s !as lost/ 6ind P(A), P(*), P(A ∩ *) and sho! that P(A *) + 14$W$<' i) 6ind P(5) and sho! that P(5 A) + 1HW1G/ ii) State, by #roiding reasons, !hether eents A and 5 are inde#endent or not/ 7(i)$8
''/
'./
A : The chosen digit is odd/ * : The chosen digit is bigger than ./ 6ind P(A), P(*) and P(A ∩ *)/ State, !ith reasons, if the eents A and * are de#endent eents/ b) A bo contains 1< !hite balls and 1= yello! balls/ *y assuming that selection is made randomly, one by one, !ithout re#lacement, find i) the #robability that the first ball chosen is !hite/ 7'W<9 ii) the #robability that the second ball #ic3ed is yello! gien that the first ball is !hite/ 7
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'
A surey on the interest in reading entertainment magaines and noels among the student in a secondary school !as conducted/ The results of the surey sho! that $GQ of the students li3e reading entertainment magaines, including 1HQ !ho do not en;oy reading noels/ The results also sho!ed that 'Q of the students do not li3e reading either entertainment magaines or noels/ A student is #ic3ed randomly from the students in the school/ The eents A, * and 5 are defined as follo!s: A : The selected student li3es to read entertainment magaines/ * : The selected student li3es to read noels/ 5 : The selected student li3es to read entertainment magaines or noels but not both/ 6ind P(A), P(*), P(5), P(* AX), P(* n 5)/ State, giing a reason, if the eents * and 5 are de#endent eents/ 7=/$G" =/8=, =/8<" =/G
'4/
%eryday, a fisherman has the choice of fishing at sea, in a rier, or a la3e/ The #robability that he fishes at sea, in the rier, and at a la3e are 1W$ , 1W. , and 1W. res#ectiely/ 0f he goes out to sea, his chances of catching some fish is 8=Q, !hile his chances at the rier and the la3e are .=Q and 4=Q res#ectiely/ i) 6ind the #robability that the fisherman catches fish on a randomly-selected day/ ii) 0f one day, the fisherman does not catch any fish, determine the #lace he most #robably !ent to fish/ Another fisherman !ho also goes fishing eeryday #ic3s one of the three #laces !ith a #robability of 1W' / 6ind the #robability that the t!o fisherman meet on a day #ic3ed randomly/ 7(i) 1'W$=" (ii) rier" 1W'9 71GG1A$/1$9
'H/
(a) The eents A and * occur !ith the #robabilities P(A) + $W< and P(*) +1W$ res#ectiely/ Determine !hether the eents A and * are mutually eclusie or not, and determine also if A and * are inde#endent eents, if i) P(A ∪ *) + GW1=, ii) P(A ∪ *) + HW1=/ (b) A qui contestant needs to ans!er three @true or false questions consecutiely/ 0f the contestant guess the ans!er randomly, find *y JJC
ProbabilityMathematics S & T iii)
the #robability that eactly t!o ans!ers are correct, i) the #robability that at least t!o are correct/ 7a9(i) mutually eclusie" de#endent" (ii) not mutually eclusie, inde#endent" (b)(i) 'W8" (ii) 9 71GG$A$/1$9 '8/
'G/
%ents A and * occur !ith the #robabilities P(A) + $W< and P(*) + V res#ectiely/ 5alculate P(A ∪ *) if i) the eents A and * are mutually eclusie, ii) eents A and * are inde#endent/ 7(a) 1'W$=" bHW$= 9 71GG'S$/$9
i)
ii)
.'/
%ents A and * are such that P(A) + $W<, P(*A) + .W<, and P(A *) + 'W
../
6rom the '== 6orm si students in a school, 1== read ne!s#a#er A and 8= read ne!s#a#er */ '= read both ne!s#a#er A and */ i) 5alculate the #robability that a form 4 student selected randomly from the school read either ne!s#a#er A or */ ii) 6ind the #robability that a 6orm 4 student in the school, !ho is selected randomly from the grou# of students !ho read ne!s#a#er *, also read ne!s#a#er A/ 7(i) =/< (ii) 'W89 71GG
.
A and * are $ eents !ith/ P(A) + Z and P(*) + $W' and P(A ∩ *) + / 5alculate the #robability that a) P(A ∩ *) 71W$9 b) P(A ∪ *)/ 71W1$9 71GG
.4/
%ents A, *, 5 and D are mutually eclusie !ith P(A) + P(*) + P(5) +P(D) + 1W4/ 0f % + A∪ * ∪ 5 and eent 6 + 5 ∪ D, find P(% ∩ 6) and P(% ∪ 6)/ 71W4" $W'9 71GG4S$/$9
.H/
%ents A and * are such that P(A) + , P(A| *) + 'W1=, and P(A ∪ *) + HW1=/ 6ind P(*), P(A ∩ *), and P(A ∩ *)/ 7 $WH" 'W'<" 1W<9 71GG4T$/$9
.8/
A family of ' members A,* and 5 stay in a house !hich has a tele#hone line/ heneer the tele#hone ring, the #robability that the call is for A, * and 5 are =/1, =/' and =/4 res#ectiely/ The #robability that A, * and 5 is at home !hen the #hone ring =/H, =/< and =/G res#ectiely/ Assume that all the #robabilities mentioned aboe are inde#endent, find the #robabilities that !hen the #hone ring, a) no one is at home, b) the #erson !anted by the call is at home, c) the call is for A and only A is not at home/ 7(a) =/=1<" (b) =/H4 (c) =/=1'<9 71GG4T$/11(i)9
.G/
There are fie bottles filled !ith grains/ Among the fie bottles, three are filled !ith grade A grains and the remainder filled !ith grade * grains/ 0f three bottles are selected randomly, find the #robability that i) both grade of grains are selected, ii) at least $ bottles filled !ith grade A grains are selected/ 7(i) GW1= (ii) HW1=9 71GGHT$/$9
<=/
(a) T!o digits and y are randomly selected from the set S + K=, 1, $, ', ., <, 4, H, 8, GL/ The eents A and * are defined as:
T!ele out of t!enty agricuture #roducts in an e#o are from MA2D0/ 0f t!o of these #roduct are selected randomly from this agriculture %#o, find the #robability that i) both of these t!o #roduct are from MA2D0, 7''WG<9 ii) only one #roduct is from MA2D0/ 7.8WG<9 71GG'T$/$9
6rom the #ast e#erience, a com#uter center in a uniersity estimated 1=Q of the com#uter system brea3do!n in the center is due to the #roblems in the hard!ares only,
the #robability that a ne! !or3ers of the com#any can com#lete his tas3s satisfactorily/ 7Y: =/H.9 the #robability that a ne! !or3er of the com#any has not undergone training gien that he is able to com#lete his tas3s satisfactorily/ 7=/=819 71GG.S/$9
.=/
.1/
.$/
A set is gien by S + K=, 1, $, ', . <, 4, H, 8, GL/ A three digit number is formed from the digits in S/ The eents A and * are defined as follo!s: A : %ent !here the number formed does not contain the digit =/ * : %ent !here the number formed begin !ith =/ 6ind P(A), P(*), P(A ∩ *), P(A ∩ * ), P(A ∩ *) and P(A ∩ * (ii) T!o digits and y are ta3en from S/ The eents 5 and D are defined as follo!s: 5 :%ent !here y + $/ D : %ent !here both and y are less than $/ 6ind P(5), P(D), P(5 ∩ D), P(5 ∪ D), and P(5 D)/ 7:i) =/H$G, =/1, =, =/H$G, =/1, 1H1W1=== (ii) =/=<, =/=., =/=1, =/=8, =/$<9 71GG.S$/1=9 6rom the #ast e#erience, the #robability that a ne! !or3er 0n a com#any can com#lete , his tas3s satisfactorily is =/8 if he has undergone the com#anyXs in-house training, and =/. if he has not undergone the training/ 0f 8
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*y JJC
ProbabilityMathematics S & T A
: the digits and y satisfy the relation + y $ * : both and y are greater than 4/ 6ind P(A), P(*), P7(A ∩ *) 9 and P(A *)/
<./
A Personal 0dentification Eumber (P0E) consists of . digits in order, each of !hich is one of the digits =, 0, $, ///, G/ Susie has difficulty remembering her P0E/ She tries to remember her P0E and !rites do!n !hat she thin3s it is/ The #robability that the first digit is correct is =/8 and the #robability that the second digit is correct is =/84/ The #robability that the first t!o digits are both correct is =/H$/ 6ind a) the #robability that the second digit is correct gien that the first digit is correct, 7=/G9 b) the #robability that the first digit is incorrect and the second digit is incorrect, 7=/=49 c) the #robability that the first digit is incorrect and the second digit is correct, 7=/1.9 d) the #robability that the second digit is incorrect gien that the first digit is incorrect/ 7=/'9 e) The #robability that all four digits are correct is =/H/ Nn 1$ se#arate occasions Susie !rites do!n inde#endently !hat she thin3s is her P0E/ 6ind the #robability that the number of occasions on !hich all four digits are correct is less than 1=/ 7=/H.H9(EG8WP$W4)
<
(a) %ents A and * are such that P(A) + =/', P(* A) + =/8, and P(* A) + =/., find P(A n *) and P(A *)/ 7=/=4,=/<89 (b) %ents 5 and D are $ inde#endent eents/ Sho! that, 5 and D are inde#endent eents/ Sho! also eents 5 and D are also inde#endent eents/ 71GG8T$/1=9
<4/
A comm#uter #rogram has errors and cause the #rogram fail to run smoothly/ To debug the errors, n #rogrammers are assigned and !or3 indiidually to detect the errors/ The #robability that each #rogrammer can detect the errors is =/8H
%ents A and * are such that P(A) + $W<, P(*) + V, and P(A n *) P(A n *) + 1W4/ 6ind P(A n *), and determine !hether A and * are inde#endent/ 7$GW1$=" de#endent971GGGS$/$9
<8/
A factory !hich #roduce electronic equi#ment, obtain its .=Q su##ly of electronic com#onents from su##lier A, $
(b) A bo is filled !ith < balls and the balls are labeled !ith numbers 1, $, ', ., and
A certain disease is #resent in 1 in 200 of the #o#ulation/ 0n a mass screening #rogramme a quic3 test for the disease is used, but the test is not totally reliable/ 6or someone !ho does hae the disease there is a #robability of 0.9 that the test !ill #roe #ositie, !hereas for someone !ho does not hae the disease there is a #robability of 0.02 that the test !ill #roe #ositie/ a) Nne #erson is selected at random and tested/ i) 6ind the #robability that the #erson has the disease and the test is #ositie/ 7GW$===9 ii) 6ind the #robability that the test is negatie/ 7$.'GW$<==9 i) Rien that the test is #ositie, find the #robability that the #erson has the disease/ 7.
<$/
<'/
A housing deelo#er has order sand, cement and bric3s from three different su##liers/ The #robabilities that the sand, cement and bric3s !hich he has ordered arrie at the #ro;ects site before or at the date agreed on are =/., =/H, and =/8/ 6ind the #robability that i) all the things ordered arrie in time at the site,/ ii) at least one thing arrie late to the site/ 7(i) =/$$. (ii) =/HH49 71GG8S$/$9 A husband and !ife sit for driing test/ 0f the #robability that, one of them, husband or !ife, #asses the driing test for each time they sit for it is =/8, i) 6ind the #robability that the husband or the !ife #asses the test after sitting for the test eactly t!o times" ii) 6ind the #robability that both husband and !ife #ass the test after sitting for the test more than t!o times/ 7(i) =/$G.. (ii) =/=H8.9 71GG8T$/$9
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*y JJC
ProbabilityMathematics S & T
0n a #robability e#eriment, three containers hae the follo!ing contents/ A ;ar contains $ !hite dice and ' blac3 dice/ A !hite bo contains < red balls and ' green balls/ A blac3 bo contains . red balls and ' green balls/ Nne dice is ta3en at random from the ;ar/ 0f the dice is !hite, t!o balls are ta3en from the !hite bo, at random and !ithout re#lacement/ 0f the dice is blac3, t!o balls are ta3en from the blac3 bo, at random and !ithout re#lacement/ %ents and M are defined as follo!s/ : A !hite dice is ta3en from the ;ar M : Nne red ball and one green ball are obtained/ Sho! that P(M ) + 1
4=/
A light bulbs factory used three ty#e of machines A, *, and 5 to #roduce the bulbs/ Machines A, *, and 5, each contribute $
41/
A and * are t!o eents !ith P(A) + =/<, P(*) + =/4, and P(A *) + =/8
4$/
According to a surey 4=Q of the house!ies hae !atch the adertisement of a ne! #roduct in the teleition/ The #robability that an house!ife that has seen the adertisement buy the #roduct is =/G, !hereas, an house!ife !ho has not seen the adertisement buys the #roduct is =/'/ i) 6ind the #robability that an house!ife !ill buys this ne! #roduct// ii) 6ind the #robability that an house!ife that bought this #roduct has seen the adertisement// iii) *y using a suitable a##roimation, find the #robability that out of <= house!ies, less than 1= house!ies has bought the #roduct !ithout !atching the adertisement// 7$===S$/1=9 7(i)=/44" (ii)=/81819
4'/
The #robability that Mimi ta3es u# Statistics tuition is =/'/ 0f she ta3es u# the tuition, the #robability that she #ass the Statistics #a#er is =/8/ 0f she does not ta3es u# the tuition, the #robability that she #ass the Statistics #a#er is =/4/ i) 6ind the #robability that Mimi #ass her Statistics #a#er/ ii) 6ind the #robability that Mimi ta3es u# tuition if she #assed her Statistics #a#er/
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7$==1S$/.9 4./
(a) %ents A and * are such that P(A) + 1W' , P(* 0A) + 1W. and P(A ∩ * ) + 1W4 / 6ind i) P(A *), 7
Three balls are selected randomly from one blue ball, three red balls and si !hite balls/ 6ind the #robability that all the three balls selected are of the same colour/ 7$==$T$/H9 7HW.=9
44/
A factory has '4 male !or3ers and 4. female !or3ers, !ith 1= male !or3ers earning less than 2M1===/== a month and 1H female !or3ers earning at least 2M1===/== a month/ At the end of the year, !or3ers earning less than 2M1===/== are gien a bonus of 2M1===/== !hereas the others receie a months salary/ i) 0f t!o !or3ers are randomly chosen, find the #robability that eactly one !or3er receies a bonus of one months salary/ ii) 0f a male !or3er and a female !or3er are randomly chosen, find the #robability that eactly one !or3er receie a bonus of one months salary/ 7i) 81HW14<=" (ii) $GW.89 7$=='T$/G9
4H/
The follo!ing table, based on a surey, sho!s the numbers of male and female ie!ers !ho #refer either documentary or drama #rogrammes on teleision/ Documentary Drama Male G4 .< 6emale .< 8< A teleision ie!er inoled in the surey is selected at random/ A is the eent that a female ie!er is selected, and * is the eent that a ie!er #refers documentary #rogrammes/ i) 6ind P(A n *) and P(A *) ii) Determine !hether A and * are inde#endent and !hether A and * are mutually eclusie/ 7$=='S$/49
48/
T!o transistors are chosen at random from a batch of transistors containing ninety good and ten defectie ones/ i) 6ind the #robability that at least one out of t!o transistors chosen is defectie/ ii) 0f at least one out of the t!o transistors chosen is defectie, find the #robability that both transistors are defectie/ 7i) $1W11=" (ii) 1W$19 7$==.T$/G9
*y JJC
ProbabilityMathematics S & T 4G/
According to a surey conducted in a com#any on ;ob satisfaction, salary and #ension benefits are t!o im#ortant issues/ 0t is found that H.Q of the em#loyees are of the o#inion that salary is im#ortant !hereas 4
H$/
The #robability that an em#loyee of a com#any is late for !or3 is =/1< in any !or3ing day and =/'< if it rains/ The #robability that it rains is =/$./ 5alculate , Ka) the #robability that it rains and the em#loyee is late, 7=/=8.9 (b) the #robability that it rains if the em#loyee is late, 71.W''9 (c) the #robability that the em#loyee is late on at least $ out of < consecutie !or3ing days/ 7=/$
H'/
T!o archers A and * ta3e turns to shoot, !ith archer A ta3ing the first shot/ The #robabilities of archers A and * hitting the bulls-eye in each shot are
1 6
and
1 5
tais are #ar3ed there, find the #robability that t!o red tais are #ar3ed net to each other/ 7' mar3s9 7Assume that a tai may be #ar3ed at any of the #ar3ing bays/9 7$==HT$/H971W.9 H4/
A study on 1== isitors to a boo3 fair sho!s that 4= isitors hae seen the adertisement about the fair/ Nut of .= isitors !ho ma3e #urchases, '= hae seen the adertisement/ 6ind the #robability that a isitor !ho has not seen the adertisement ma3es a #urchase/ 7. mar3s97$==HS$/$97=/$<9
HH/
T!o eents A and B are such that P(A) + + a)
b) c)
1 4
and P( A *) +
1
2
res#ectiely/ Sho! that the
H./
/
7. mar3s9
7$==44P$/H9
The #robability that it rains in a certain area is
1 5
/
The #robability that an accident occurs at a #articular corner of a road in that area is rains and
1 50
1 20
if it
if it does not rain/ 6ind the
#robability that it rains if an accident occurs at the corner/ 7< mar3s9 7$==4P$/897
There are eight #ar3ing bays in a ro! at a tai stand/ 0f one blue tai, t!o red tais and fie yello!
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8
, P(*)
/ 6 Sho! that the eents A and * are neither inde#endent nor mutually eclusie/ 7$ marks 6ind the #robability that at least one of the eents A and B occurs/ 7HW1$97' marks 6ind the #robability that either one of the eents A and B occurs/ 7. marks !"#$%&!%''%."'
#robability of archer A hitting the bulls-eye first is
1
3
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