Post on 03-Jun-2018
8/12/2019 STPM Probability Notes & Exe
1/15
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)
8/12/2019 STPM Probability Notes & Exe
2/15
ProbabilityMathematics S & T
Question (
6ind the total number of four digits numbers formed byusing the digits 1, $, $, ' and
>( )>
n
r n r
$>+
$>+
8/12/2019 STPM Probability Notes & Exe
3/15
ProbabilityMathematics S & T
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
8/12/2019 STPM Probability Notes & Exe
4/15
ProbabilityMathematics S & T
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
8/12/2019 STPM Probability Notes & Exe
5/15
ProbabilityMathematics S & T
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$)/ 7'W8"
8/12/2019 STPM Probability Notes & Exe
6/15
ProbabilityMathematics S & T
Are the eents P and 2 inde#endent of eachotherB Are the eents P and 5 inde#endent ofeach otherB Rie your reasons for each ans!er/
b) T!o members of the committee are chosen
randomly/ Oet P$be the eent that both members
hae grey hair and 2$is the eent that both
smo3es/ 6ind
i) P(P$), 7
8/12/2019 STPM Probability Notes & Exe
7/15
ProbabilityMathematics S & T
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$9
18/ A delegation consists of < member is selectedfrom < males and . females/ Co! many differentdelegation can be formed if it must consists of atleast 1 femaleB 71$
8/12/2019 STPM Probability Notes & Exe
8/15
ProbabilityMathematics S & T
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
8/12/2019 STPM Probability Notes & Exe
9/15
ProbabilityMathematics S & T
ta3en from the second bag and #laced in the first/6ind the #robability that the first bag still containseactly . red discs/ 7$8W
8/12/2019 STPM Probability Notes & Exe
10/15
8/12/2019 STPM Probability Notes & Exe
11/15
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
8/12/2019 STPM Probability Notes & Exe
12/15
ProbabilityMathematics S & T
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
8/12/2019 STPM Probability Notes & Exe
13/15
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 *)/
(b) A bo is filled !ith < balls and the balls arelabeled !ith numbers 1, $, ', ., and
8/12/2019 STPM Probability Notes & Exe
14/15
ProbabilityMathematics S & T
8/12/2019 STPM Probability Notes & Exe
15/15
ProbabilityMathematics S & T
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