STPM Probability Notes & Exe

8/12/2019 STPM Probability Notes & Exe

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ProbabilityMathematics S & T

PPROBABILITY

At the end of the lesson, students should be able to:

1 Techniques of counting a) use the counting rules for finiite sets, including the inclusion-eclusionrule, for t!o or three sets"

b) use the formulae for #ermutations and combinations"$ %ents and #robabilities c) understand the conce#t of sam#le s#aces, eents, and #robabilities"

d) understand the meaning of com#lementary and eclusie eents"

e) calculate the #robability of an eent"' Mutually eclusie eents f) understand the meaning of mutually eclusie eents"

g) use the formula P(A *) + P(A) P(*) P(A n *)./ 0nde#endent and conditional

eentsh) understand the meaning of inde#endent and conditional eents"i) use the formula PA n *) + P(A) P(* A)

Techniques of counting

Permutations and combinations:

1/ Multi#lication 2ule : 0f an o#eration can ha##en inr !ays, and the follo!ing o#eration can ha##en ins !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!n5/ 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, secondand third #ries can be !on by the contestants/ 7''49

Permutations

$/ A #ermutation of a set of different ob;ects is anarrangement of all or #art of the ob;ects in as#ecific order !ithout re#etition/

'/ The total arrangement of n different ob;ects byta3ing r ob;ects at a time is nPr , !here

nPr+

!)(

!

rn

n

/

%am#les:

(a)


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Question (

6ind the total number of four digits numbers formed byusing the digits 1, $, $, ' and

>( )>

n

r n r

$>+

$>+


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a) Determine ho! many !ays can the committee bechosen/b) Co! many of these committee chosen, consist ofeactly . males/c) 0f the committee formed, must consists at leastone female/ Determine the total number of committeethat 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 beselected, ho! many of these committee can be formed/

-e%ection of at %east one ob&ect from n ob&ects.

1./ The number of selections of at least one ob;ectfrom n different ob;ects is $n 1%ery ob;ect has t!o o#tions, that is to be selectedor not selected at all, hence the total number ofcombinations for the n ob;ects is

$

$ $ $ ////// $

n

x x x x1 4 44 2 4 4 43

*ut this !ill include the case !here all the ob;ectsare re;ected/ Cence !e hae to minus this caseout/

$n 1

Question 1'

Co! many !ays can a boy inites at least one of his sifriends to his #arty/

Question 1(

0n ho! many !ays can a student select at least a boo3from < different boo3s/

Probabi%it/

1


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Question 21

A bo contain 1= identical red balls, H identical blac3balls, and . identical green ballsa) 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, findthe #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 isre#laced bac3 to the bo before the net ball is dra!n/

,ombine eents

18/ 0f A and * are $ eents such that P(A) = and

P(*) =, then

P(A *) + P(A) P(*) P(A n *)

1G/ 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) +17

25, P(*) +

1

5, and

P(A*) +3

5/ 6ind

i) P(A *)ii) P(A *)iii) P( A or * but not both)

Question 2

A grou# of


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Similarly,

P(A *) +( )

( )

P A B

P B

Question 2#

A bag P contains < blac3 balls and ' !hite balls/ A bag? contains . blac3 balls and H !hite balls/ A ball isremoed randomly from bag P and #laced into bag ?,and a ball is then remoed randomly from bag ?/The eents 1, *1, $, *$are defined as1: 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(*$*1), P($) andP(*1$)/ 7'W8"


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Are the eents P and 2 inde#endent of eachotherB Are the eents P and 5 inde#endent ofeach otherB Rie your reasons for each ans!er/

b) 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

i) P(P$), 7


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1H/ The committee of the Teachers and ParentsAssociation of a school consists of 4 member andis selected from H #arents, . teachers and theschools #rinci#al/ 0n ho! many !ays thiscommittee can be formed such that it consists ofnot more than ' #arentsB 7.4$9

18/ A delegation consists of < member is selectedfrom < males and . females/ Co! many differentdelegation can be formed if it must consists of atleast 1 femaleB 71$


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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 *) andP(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'971W." " 1W< " not inde#endent " not mutually eclusie9


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ta3en from the second bag and #laced in the first/6ind the #robability that the first bag still containseactly . red discs/ 7$8W


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A : The alue of S on the first thro!n is H/* : The sum of scores for $ thro!n is less than

1


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iii) the #robability that eactly t!o ans!ers arecorrect,

i) the #robability that at least t!o are correct/7a9(i) mutually eclusie" de#endent" (ii) notmutually eclusie, inde#endent"(b)(i) 'W8" (ii) 9 71GG$A$/1$9

'8/ %ents A and * occur !ith the #robabilities P(A) +

$W< and P(*) + V res#ectiely/ 5alculate P(A *)ifi) the eents A and * are mutually eclusie,ii) eents A and * are inde#endent/7(a) 1'W$=" bHW$= 9 71GG'S$/$9

'G/ T!ele out of t!enty agricuture #roducts in ane#o are from MA2D0/ 0f t!o of these #roduct areselected randomly from this agriculture %#o, findthe #robability thati) both of these t!o #roduct are from MA2D0,

7''WG


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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 *)/

(b) A bo is filled !ith < balls and the balls arelabeled !ith numbers 1, $, ', ., and


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4G/ According to a surey conducted in a com#any on;ob satisfaction, salary and #ension benefits aret!o im#ortant issues/ 0t is found that H.Q of theem#loyees are of the o#inion that salary isim#ortant !hereas 4

STPM Probability Notes & Exe

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