1· , \ '.\ d \.,\-.. .... 1\ \ L." ~)' \\ _:\ \JVl!.:lfl U ::i lU L>U l l d cl 1·ec..; (c.1 n u u 1.

-,r
-; \ \":'.l.. r \ \ \\' r---c_,,~::, .:; l l _l l, o , _J : : 1 ) h 1~ .. , - f i ick , , n .•;: . ,_, .__,
'-t~1 1.,,_,,,•.1 \\· "\ -<-. , ~, .... , .. l , . \ ~•• -_.- ,l( "'I :. , ,, , . ., _, , , ,., , < 1 • • ~•

ts:
i
I· Defin itio n· o f Probablilty

Number of possible outcomes

Total number of equally likely outcomes

Therefore if E is an event in a sample space S the probability of E occurring is P(E) n(E) .

n(S)
=
•I
Example: A card is drawn from a pack of ordinary playing cards. What is the probability that the card
drawn is a red queen?
Answer· P(red queen) = _£_
:....::...:.==-· 52
= ...1..
26

I· Independent events
If A and Bare independent events then: P(A and B) = P(A) x P(B)
Example: When a coin is tossed (A) and a dice is rolled (B) then : P(a tail and an odd number)
P(tail) x P(odd number) = J x ¾= ¾
I• Mutually exclusive events
If A and B are mutually exclusive events it means that if A happens B cannot happen and if B happens A
cannot happen therefore P(A and B) = 0 .
Example: When one card is drawn from a pac k of cards then
P(red queen and an ace) = P(A) x P(B) = 0
BUT
P(red queen or an ace) = P(A) + P(B)

I· The addition rule

For any two events A and B: P(A or B) = P(A) + P(B) - P(A and B)
This can be written as: P(Au B) = P(A) + P(B) - P(An B)
In the event of mutually exclusive events P(An B) = 0.
Example:
Given: P(A) = 0 ,3 ; P(B) = 0,7 and P(Au B) = 0,9.
(i) Are the events A and B mutually exclusive? Motivate your answer.
(ii) Are the events A and B independent? Motivate your answer.
Answer: Use the addition rule as follows:
P(Au B) = P(A) + P(B) - P(An B) : . 0,9 = 0,3 + 0,7 - P(An B) .-. P(An B) = 0 , 1
Events are not mutually exclusive, because P(An B) *0
(ii) P(A) x P(B) = 0, 7 x 0,3 = 0,21 * P(An B) Therefore events are not independent.

I• The complementary rule

The sum of the probabilities of mutually exclusive and complementary events is 1. Therefore:
1
P(A or A 1) = P(A) + P(A1) = 1 OR P(A ) == 1 - P(A)
Example: A coin is tossed three times. Find the probability of getting a tail at least once.

Answer: First find the probability of not getting a tail at all.

1 1
_3..-- H 1 1 _]-- K
~ H---------- T 3--' K ----------
1
2

<;T-:·.·
2
~

... T
1
~
2
>
.... ....
M-· ···.. ·. ·.:.· M

P(H/H/H) = 1l P(K/K/K) == i
76 Mod 5
( ;,
l ,,.
-: .

71
Mod5

Therefore the probability of getting a tail at least once is 1 - ½== ½= 0,875.

The fundamental counting principle
• . . . c options then the total number of combined
If successive choices can be made from C1, c2, C3, · · · · n '
options is given by the product: c1 x c2 x C3 x . ... x Cn
Example: A meal can be made up as follows;
Choice 1: lamb, beef or chicken
Choice 2: rice, roasted vegetables or baked potato
Choice 3: coffee or juice .
How many different ways can a meal be assembled using these choices
?
Answer: 3 x 3 x 2 = 18 ways

I• Factorial notation: n!
Example: The number of ways the five vowels of the alphabet can be
arranged without any repetition is
5 x 4 x 3 x 2 x 1 = 120. This product can be written as 5!
and is called five factorial.

I• Permut ations
Permut ations are ordered arrangements where order is important. A
permutation represents the number
of ways that r items can be arranged (where order is important) from
a total of n differen t items (n > r)
and is given by n!
(n - r)!
Examp l~: There are 15 players in a rugby team. All hope to become
either captain of vice captain . How
many differen t appoint ments can be made?

Answe r: 15 x 14 = 210. This can also be written as ___Jfil_

(15 - 2)\
On the calcula tor you can do this calculat ion by using the key nP r

I· Identical items
The numbe r of ways n items can be arrange d ifs of them are the
same is n!
s!
Examp le: In how many ways can the letters in the word PETER be
arrange d?
Answe r: ~: = 60 differen t ways .

I· Seating in a row
If m males and f females are seated in a row:

(i) (m + f)l is the number of ways they can be seated in any order
(ii) 2 x fl x ml is the number of ways they can be seated if all the females
an all the males must be
seated next to each other
(iii) (f + 1) x ml x fl is the number of ways they can be seated of all the
males are to be seated next to
each other
(iv) (m + 1) x fl x m! is the number of ways they can be seated of all
the females are to be seated next
to each other
Example: In how many ways can 4 boys and 3 girls be seated in a
row if all the girls must be seated
next to each other?
Answer: (4 + 1) x 4! x 3! = 720 ways.
 78 Mod SA

[ WAYS OF DETERMINING PROBABILITY

ff ree diagrams . ..
drawn simultaneously from a pack of 52 playing cards. What 1s the probability
Example: Two cards are
of drawing at least one ace?

Answer:
Outcome
3
5i" Ace A/A
4 Ace
_ NoAce A/N
51
4
{ syAce N/A
- No ace/
52 ~
47~No Ace N/N
51
• Along the branches we multiply the probabilities because the events can be considered to be
independent.

• The probabilities at the end of each branch of the tree are mutually exclusive. The probabilities
are therefore added. The sum of these probabilities add up to 1.
Answer:
1 - P(N/N) = 1 - 48
52 X 51 = .li.
47
221 -- 0I 149

!Two way contingency tables

Example: The following two way contingency table is given. Determine whether trains that leave on time
are independent of the station from which they leave. Give a reason for your answer.
Diaz station Gap station Total
Trains left on time 85 35 120
Trains left late 65 25 90
Total 150 60 210
Answer:
P(trains left on time)= P(A) = 1;g
P(Diaz station)= P(B) = 1~g
P(trains left on time n Diaz station) = 1;g x 1~g ""0,41
P(AnB) = 2a150 ""0,4

P(A) x P(B)"" P(AnB) :. Events are independent.

!Venn diagrams
Example: If A and Bare independent items, find the values of x and y. All working must be shown.
[Source: IEB; 2010 Paper 3]

Answer
B AnB = (x + 0,1)(0,1 + 0,3) = (x + 0,1)(0,4) = 0,1
:. 0,4x + 0,04 = 0, 1
0,3
:. 0,4x = 0,1 - 0,04 = 0,06
y
:. x = 0,15 and y = 1 - (0,15 + 0,1 + 0,3) =0,45

The fundamental counting principle in probability problems

Exa~81e: Supp_ ose we take_ ~II the letters in the word JOHN and arrange them in any order without
repet1l1on. What Is the probab1hty that a word will start with a J and end with a N?
Answer: P(E) = n(E) = 2! = J_ "" 0,083
n(S) 41 12
IP m a se •~:w aw;; ,, • t , ~
 '_f .~, ' .' ~.

Cexe Rc1se s A l.;

ems based on
N B : Includ es rev ision of grade 11 work.
See Pythagoras 11 CAPS for extensive probl
could also be answ ered in other ways ). . c,
grade 1 1 Probability. (Some questions
IN
Tree diagr ams the
a pack of ordinary playing cards . Whal is Q
1. Two cards are drawn simultaneously from
probability of drawing: D
1. 1 two kings a
1.2 no kings 1.3 at least one king
1
1.4 exactly one king?
fine is 25%. If the weat her is fine , the probability
2. The probability that tomorrow's weat her will be
that John will play tennis is 90%. bility
If the weat her is not fine, the proba bility that
John will play tennis drops to 30% . Find the proba
that John will play tenni s tomo rrow.
are red .
shap e and size. Howe ver 9 are green and 3
3. There are 12 discs in a bag - all of the same gettin g discs of
repla ceme nt. Find the proba bility of
Two discs are chos en at rando m, but witho ut
both colou rs.
in dry
is 63% . A child has a 12% chan ce of faffing
4. The proba bility that it will rain on a given day
wet weat her.
weat her and is three times as likely to faff in
sent all outco mes of the abov e inform ation .
4.1 Draw a tree diagr am to repre
a child will fall in dry weat her?
4.2 Wha t is the proba bility that on any given day
on any given day? [Sou rce: NSC 20111
4.3 Wha t is the proba bility that a child will not fall

Venn diag rams that

nt will pass Chem istry is 0 ,6 . The proba bility
5. The proba bility that a rand omly chos en stude is 0,5. Dete rmine the
bility that she pass es both
she will pass Math emat ics is 0,7 and the proba
prob abilit y that she
5.1 only pass es Chem istry
5.2 fails both subje cts
5.3 pass es at /east one of the subje cts.
nce; 120 take
in grad e 12. Of these 96 take Phys ical Scie
6. At Font ys Colle ge there are 146 stude nts Math emat ics.
stude nts takin g Phys ical Scie nce also take
Math emat ics and 70 take Biolo gy. All the as well as
Scie nce and 50 of them take Math emat ics
40 of the stude nts take Biolo gy and Phys ical
chos en at rand om takes :
Biolo gy. Find the prob abilit y that a stude nt

6.1 Math emat ics but not Phys ical Scie nce
6.2 Math emat ics but not Biolo gy
6.3 Math ema tics or Biolo gy or both .
brea d ; 120
Of thes e 125 boug ht mea t; 100 boug ht
7. At a Supe r Spar , 260 peop le boug ht food. r and brea d
; 30 boug ht mea t a nd brea d ; 50 boug ht suga
boug ht suga r; 60 boug ht m e at a nd s ugar pers on chos en
and brea d. Wha t is the prob abilit y that a
and 25 boug ht all three i.e. mea t, suga r
at rand om boug ht
7.1 none of the spec ified item s?
7.2 brea d and suga r but no m eat?
7.3 brea d or mea t?
an unev en
Let A be the even t of draw ing a card with
8. A pack of ten card s are mark ed 11 to 20.
rd with a prim e num be r.
num ber a nd B the e v e nt of dra wing a c a
Dete rmin e:
8.1 P(A or B)
1
8.2 P(A and B)
8.3 P (An B)

79 Mod SA
 Two way contingency tables
9. A group of 5 0 0 0 people were tes te d for colour b li ndness and the resu lts are shown in the g iven t able·.
Male Female Total
Colour blind 350 58 408
Normal 2230 236 2 4 592
the Total 2580 2 4 20 5000

Does the evidence support the statement that colour blindness is independent of gender? Support your
answer with the necessary calculation(s).

10. The following two way contingency table is given. The information reflects the observed values of
,ility 500 persons.
Brown eves Eves not brown Total
1bility 240
Blonde hair 84 156
Dark hair 186 74 260
red. Totals 270 230 500
~s of
10.1 Redraw the table and give the expected values in each cell . NB: The total values must remain
the same.
jry
10.2 By comparing the observed and the expected values, decide whether the colour of a person's eyes
is independent of the colour of his/her hair. Motivate your answer.

11. The following two way contingency table reflects the observed information collected from 260
011] persons living either in urban or rural areas.
Urban Rural Totals
Had a smart phone 50 110
,at 60
he Did not have a smart phone 80 70 150
Totals 140 120 260

11.1 Redraw the table and give the expected values in each cell. NB: The total values must remain
the same .
11.2 By comparing the observed and the expected values, decide whether having a smart phone is
take
independent of whether a person lives in a rural or urban area. Motivate your answer.
tics.
II as The Fundamental Counting Principle
12. Assuming that any combination of letters forms a word, how many different words can be formed
by the following letters?
12.1 RAT 12.2 MOUSE 12.3 ELEPHANT

13. How many 5 digit numbers can be made by using the digits 1 to 8 if:
20 13.1 no digit may be repeated? 13.2 digits may be repeated?
ad
en 14. In how many ways can a captain and a vice-captain be chosen from a team of 12 players?

15. There are 7 seats in uncle Jack's 4x4 Land Rover, 2 in the front and 5 at the back. Uncle Jack
drives his four children to school every day. In how many ways can the children seat themselves in
the 6 available seats?

en 16. In how many ways can the results of 4 netball games be predicted if each can be either a win or a
loss or a draw?

17. A bank code can be made up starting with 3 digits and followed by 2 letters. The digits may be
chosen from O to 9 and the letters from the alphabet excluding the vowels.
How many different codes can be found if:
17.1 the digits as well as the letters may be repeated?
17.2 neither the letters nor the digits may be repeated?
17.3 only the letters may be repeated?

cl SA 80 Mod SA
~~ .•.,-- . ..... 411

81 Mod 5A

Ben has 6 T-shirts, 4 pairs ~f shorts and_2 ~airs of running shoes. How many different ways can he 31 . I r,r,,,,
18.
dress by using one of each item of clothing . 3'1:\ \r, hr, ,
31 .2 W h at
In music there are seven basic notes: doh, re, me, fah, so, la and ti. next I
19.
How many different seven-note songs can be composed tf:
32 . Four
19.1 notes may be repeated? end
19.2 notes may not be repeated and must start with me and end with doh?
\ EXERCIS
20. In how many ways can the letters of the following words be arranged?
MATHEMATICS 20.2 MISSISIPPI 20.3 UNNECESSARY 1. A l
20.1
be'
An hotel has 20 bedrooms available. On occasion 5 ~eople make ~ re~ervation. Each person is to
21.
have his or her own room. How many different allocations are possible . 2. If '
wi
Using the fundamental counting principle in probability problems . .
22. Suppose we take the letters in the word FRANCE . and arrange them tn any order. What ts the 3. If
probability that the word will start with a F and end with an E?
3.1 F
Suppose we take the letters in the word MATHEMATIC~ and arrange them in any order. What is 3.2 f
23.
the probability that the word will start with a M and end with a S?
4.
24. A main course and dessert are chosen from the following menu:
Main course: Fish; Steak; Lamb chops
Dessert: Ice cream; Cheese cake; Malva pudding 4.1
What is the probability that the choice will contain: 4.2
24.1 Steak and Malva pudding?
5.
24.2 Steak or Malva pudding?

25. Your dad would like to purchase a Toyota Fortuner. They are available in three different colours:
white, grey and black. The upholstery could be leather, or two different types of material. A 5.1
manual or automatic model is available in all of the above choices. 6.
25.1 How many different combinations are there to choose from? 6.1
25.2 What is the probability that your dad will choose a model that is automatic, not black and with 6.3
leather upholstery? 7.
26. Four different Mathematics books, three different German books and three different Spanish books
are randomly arranged on a shelf. What is the probability that all books of the same language land
8.
up next to each other?

27. A number plate is designed starting by using three letters of the alphabet excluding the vowels
and then followed by any three digits 1 to 9. Repetition is allowed. Calculate the probability that a
number plate, chosen at random 9.
27.1 starts with a D and ends with a 7.
27.2 has exactly one D.
27.3 has at least one 7.

28. The letters in the word IMMEDIATELY are randomly arranged. Assuming that all the words have 9.
meaning, 9
28.1 How many different words can be formed?
28.2 What is the probability that the word will start and end with the same letter?

29. What is the probability that a randomly arrangement of letters in the word \NNOCEN, will form a
word that starts and ends with the letter N?

30. In a certain province number plates are designed with 3 alphabetical letters next to each other
(excluding the letter 0), followed by any three digits from Oto 9. The digits may be repeated but
not the letters.
30.1 How many unique number plates can be designed?
30.2 What is the probability that the number plate will st art with an X?
 82 Mo dS A& B

ed in a row of 5 chairs.
31. Three boys and two girls are to be seat
31 .1 In how many ways can they be seated? the boys be seat ed
be seated next to each other and all
31 .2 Whal is the probability that all the girls
next to each other?
the girls will all
up in a row. What is the probability that
32. Four boys and three girls are to be lined
end up next to each other?

/ EXERCISE 5 8 (Mixed exercise)

person has a choice
that at exactly 6 different places a
1. A maze is constructed in such a way maz e?
y different routes are there through the
between turning left or right. How man
ability that the word
are randomly arranged , what is the prob
2. If the letters in the word NECESSARY
will start and end with the same letter?

3. If P(A) = j and P(B) = f, find:

events.
3.1 P(Au B) if A and 8 are mutually exclusive [Source: IEB pap er 3; 2009 ]
inde pend ent even ts.
3.2 P(Au B) if A and 8 are

4. The format of a car's number plate is: well as the digit s

rs (exc ludin g the vowe ls), follo wed by 4 numerical digits. The letters as
3 lette
may NOT be repeated .
possible?
4.1 How many different number plates are
plate starts with a B and ends with 9?
4.2 What is the probability that the number
r Q), follo wed by 3
rs (excluding the vowels and the lette
5. A number plate is assembled using 3 lette ber plate chos en at
ed. Calculate the probability that a num
numerical digits. Repetitions are allow
random has:
exactly one six
5.2 at least one six
5.1
n. What is the probability of
6. Two conventional six sided dice are throw
6.2 getting at least one 5?
6.1 not getting at least one 5?
6.3 getting a double 5?
s. Ass ume all the
arranged randomly to form diffe rent word
7. The letters of the word STAPLER are end with a cons onan t?
ability that the word will start and
words have meaning. What is the prob
take n out one at a
blue and the others are red. If they are
8. '.here are 8 discs in a bag. Some are if ther e are:
y different results could be recorded
time , and the colour recorded, how man
8.2 6 red 8.3 4 of each colo ur in the bag ?
8.1 3 blue

9. In rolling a die consider the following:

Sample space: {1 ; 2; 3; 4; 5; 6}
Event A: {2; 4; 6}
Event B: {1; 2; 3; 5}

e.
9.1 Draw a Venn diagram to illustrate the abov
1 (iii) P(A'nB)
9.2 Determine: (i) P(A') (ii) P(An B )

10. of these envelopes each have 5

A drawer contains 10 envelopes. Four blue and 2 red shee ts of
each cont ains 4 blue a d 3 d h
paper. The other six envelopes n r~ s eets of pape r. One enve lope
is
a sh
chosen at random. From this envelo e rand om . Wha t is the
eet of pape r ,s chosen at
probability that this sheet of paper is bl!?
11. play each othe r twi
In a schools' rugby competition 10 teams ce - once at hom e and once awa y.
Each team represents another school.
11.1 com petit ion?
How many games are played in total in the .
11.2
The Maroons and the Blues are two well
kno each
ion? Wha t ,s the prob abili ty that they play
wn team s.
other in the first match of the com petit
g _q L
 2 1. ln ;!.
c;;,,-
t ce ntr e in
tes ts tak en at a tes 21 .1 bo·
ma ris es the res ults of all dri vin g
· en cy t a bl e sum 21 .2 th•
Th e two wa y con ting
th e ti,rst we ek of Jan
ua ry 20 10 To tal
_,
12 .
wn d ·
urm g Fe ma le 75 22 .
c: ·
Ca pe To
Ma le 43 23 22 .1 Ir
r 32 15 98 22 .2 p
Pa ss 8 58
40 n
Fa il .
t week of Ja nu ary 2010
1 To tal
fro m tho se wh o too k their test du_nng the firs 23. 1
dom
A person is chosen at ran who failed .
the person was a fem~~e
V
pro ba bili ty tha t passed the test. [So urc e: IEB 20 10 ]
12.1 Fin d the
cho sen is a ma le. Fin d the probability that he 23.1 ~
12.2 The person sition is g. During a 23 .2 \
sh oo t a goal from a cert~in ~o
cer player , Joh n, can m ter ms of g for the
The probability that a soc fro m this po siti on . Wr ite down an expression 24 .
13. shots
practice he takes three
probability tha t he 13.3 shoot no goals
on e goal.
13.1 sho ots thr ee go als
ses (in tha t order) 13 .4 shoots at least
scores an d the n mis
13 .2 misses , s onto the field in a ran
do m 25 .
pla yer s. Bre nto n Hig h School's first team run
A netball team has 9
14. tain always runs first.
order except that the cap m run onto the field? to the
ferent ways can the tea the first three to run on
14.1 In ho w many dif t on a pa rtic ula r da y the centre is amongst
bability tha
14.2 What is the pro 26 .
field?
d on a shelf.
bo oks an d six ide ntic al red books are arrange
Three identical green anged?
15. ys can the books be arr to each oth er?
15.1 In ho w ma ny different wa bo oks of the sa me colour land up next
pro ba bili ty tha t all the 27 .
15.2 What is the be the ev en t tha t the
lar ity of two the atre productions. Let A iew ed
A survey was conducted
into the po pu that the person int erv
16. d for the on e pro duction and event B bo ok ed for the on e
person interviewed bo
oke
of the 12 0 pe op le interviewed, 65 28 .
It is given that er or both productions
.
booked for the oth er. d 90 booked for one or the oth
for the oth er an
production, 75
16.1 Us e the formula: n B).
n(A nB ) to calculate n(A
n(Au B) = n(A ) + n(B ) -
dings.
gra m to illustrate the fin the ca teg ory in
16 .2 Dr aw a Venn dia ch of the fol low ing ca teg ori es an d de scr ibe 29 .
mb er of an sw ers in ea
16.3 Calculate the nu 29.1
words : 1 16 .3. 3 n(Au B)' 29 .2
1 16.3.2 n(B )
16.3.1 n(A n 8) pro ba bil ity tha t
0 pe op le wh o we re int erv iew ed . Fin d the
the 12
16 .4 A person is ch
os en at ran do m from 30.
t bo ok for eit he r of the pro du ctio ns .
this person did no ze ro
firs t dig it mu st be a
ttin g a ten dig it cel l ph on e nu mb er if the
ity of ge
17. Determine the pro ba bil
ne of the dig its ma y be rep ea ted .
an d no
e rec ep tio nis t ha s to
ble . On e nig ht 5 pe op le co me to sta y. Th
roo ms av aila oc ati on s are av ail ab le?
18. A g_u es t ho us e has 8 Ho w ma ny dif fer en t all
the m to a roo m.
assign each of En gli sh bo ok s are
Xh os a bo ok s an d thr ee dif fer en t
s bo ok s, five dif fer en t me lan gu ag e lan d
19. Fo ur different Afr ika an the pro ba bil ity tha t all the bo ok s of the sa
a shelf. Wh at is
randomly arr an ge d on
up ne xt to ea ch oth er?
be
Th e clo the s ha ve to
dif fer en t pa irs of tro us ers in a cu pb oa rd.
20. irts an d 4
There are 7 di~ ere nt_sh
hung on a str aig ht rail. the rai l? 31
the s be arr an ge d on
20 .1 ;n ho w ma ny different wa ys ca n the clo to be hu ng ne xt to 31
clo the s be arr an ge d if all the sh irts are 31
20.2 wayfs ca n the the rai l?
e:! o:t h:: ~~ Jit ~e ren ~ ne xt to ea ch oth er on
,rs o tro us_e rs are to be hu ng irt wil l
What . e ~a
- g of the rail an d a sh
ha ng at the be gin nin
~~ tt:: t~~bo~~~~r:~l~t
20 .3 a pair of tro us ers wil l
hang ]
[So urc e: NS C 2011

83 M od SB
 \ \ '\ \ \ ow 1\1,11 , y \JHmes cffe µlayed In tolal 111 the competition?
·v\ :•. ·, \,e M ,""lr o o n s a nd the B lues a r e two w e ll known teams. Wh a t Is the p r oba bility that thoy p lay u rrr. l,
<°'\ h n ( \ n \\,c ) \ '\ \ -..·. \ n"\ a \ c h o f lho c o mpetition?

21 . /n a car park there are exactly 50 cars. There are 32 TOYOTA's and 18 VOLVO's. Assume that two
• Ill cars are stolen - one in the morning and the other in the afternoon. Determine the probability that
2 1. 1 both stolen cars are VOLVO's.
21.2 the first is a VOLVO and the second a TOYOTA.

22. Six people are to be seated in a row of six chairs .

22.1 In how many ways can they be seated in the row?
22.2 Peter and John are two of the six people. What is the probability that Peter and John will be seated
next to each other?
23. The letters in the word CURRICULUM are rearranged to form different words. Assuming all
words have meaning:
23.1 How many words will start and end with the letter R?
23.2 What is the probability that a word will start and end with the letter U?
r
24. There are 3 green pens and b blue pens in a bag. Thabo takes a pen from the bag, puts it back
and then takes another pen from the bag. The probability that the two pens are of the same colour
is ¾.
Calculate the possible number of blue pens in the bag i.e. the value of b.

25. A librarian wants to classify the books in his library. An inventory system has been developed to
allocate a specific code to each book. The code starts with a letter followed by a series of digits.
The digits 1 to 9 are allowed and may be repeated. How many digits must the code have to ensure
that all 90 000 books in the library have a unique code?

26. Assuming that it is equally likely to be born in any of the 12 months of the year, what is the r
probability that in a group of six at least 2 people have birthdays in the same month? I

[Source: CAPS, grade 12] '

f
27. A special 10 character code is made from 3 zeroes, 4 crosses and 3 dashes. Event A is that a
~
c·
C
code formed at random will start with 2 zeroes and end with 3 crosses. Find P(A). ,;,

~
28. The following Venn diagram refers to probabilities. y
If X and Y are independent events, find the values
of a and b. Show all your calculations. a
b ..,r

29. Three sets of twins, John and James, Joan and Ruth and Elias and Sidego, are seated on a bench.
29.1 How many different arrangements are possible?
29.2 Find the probability that Elias and Sidego land up sitting next to each other.

30. A school organized a camp for the 103 grade 12 learners. The learners were asked to indicate
their food preferences for the camp. They had to choose from chicken, vegetables and fish. The
following information was collected:
• 2 /earners do not eat chicken, fish or vegetables
• 5 /earners eat only vegetables
• 2 learners eat only chicken
• 21 learners do not eat fish
• 3 learners eat only fish
• 66 learners eat chicken and fish
• 75 learners eat vegetables and fish

Let x be the number of /earners who eat chicken, fish and vegetables.

30.1 Draw an appropriate Venn diagram to illustrate the given information.

30.2 Calculate x .
30.3 Calculate the probability that a /earner chosen at random:
(i) eats only chicken and fish and no vegetables
(ii) eats only two of the given food choices:
chicken , fish and vegetables . [Source: NSC 2010]

84 Mod 58
 Mod6
85

g th~ 5 vowels, _next

In Gauteng number plates are designed with 3 alphabetical letters, excludin GP Is constant m all
31 . other.
r and then any three digits from O to 9 next to each d d. ·t The
b t d ·
to one ano th e 191 s may e repea e m a
Gauteng number plates, for example TTT 012 GP. Letters an NB

number plate. ~
31.1 How many unique number plates are available? 1.
31.2 What is the probability that a car's number plate will start with a Y?
31 .3 What is the probability that a car's number plate will contain only one 7?
are not repeated?
31.4 How many unique number plates will be available if the letters and digits (Source : NSC 20101

on a straight washing
32. Five identical white shirts and three identical black shirts are to be hung
line. The shirts are hung randomly.
32.1 In how many ways can the eight shirts be hung?
32.2 What is the probability that there is a black shirt at each end?
32.3 What is the probability that the black shirts are all next to each other?
the probab ility that:
33. Three boys and four girls are to be seated randomly in a row . What is
33.1 the row will have a girl at each end?
33.2 the row has boys and girls sitting in alternate positions?
33.3 all the girls sit next to each other?

I TEST YOURSELF 5 = 24)

[381 \

(NB: All answer s containing factorials must be simplified, eg. 4!

no other cars. During the
1. A car park has 14 VOLKS WAGEN cars and 18 BMW's . There are
afternoon two cars are stolen - one early afternoon, the other later.

Determ ine the probability that:

1.1 both cars were BMW's
1.2 the first one stolen was a BMW and the second one a VOLKS WAGE
N. [61

d in any order without

2. Consid er the letters in the word INTERE STING. The letters are arrange
ng all the words have meanin g.
repetitio ns, but using all 11 letters to form different words. Assumi
(2)
2.1 How many differen t arrange ments are possible?
end with a T? (3)
2.2 What is the probab ility that the new word formed will start with a N and
(3)
2.3 What is the probab ility that the new word will start with the letter S?
[8]
Calcula te the probab ility
3. Accord ing to populat ion statistic s 55% of all South Africans are female.
d by a girl and then anothe r boy. [31
that a couple plannin g to have children will have 2 boys followe

differen t Biology books are

4. Four differen t History books, three differen t Science books and six
arrange d on a shelf.
(1)
4.1 In how many ways can the books be arrange d on the shelf?
next to each other? (3)
4.2 What is the probab ility that the books of the same subject land up
[41
of the year, what is the
5. If it is equally likely for people to be born in any of the 12 months
in the same month? [51
probab ility that at least two out of a group of five people are born

is the probab ility that:

6. Eight boys and seven girls are to be seated random ly in a row. What (3)
6.1 the row has a girl at each end? (3)
6.2 the row has girls and boys sitting in alterna te position s? (3)
6 .3 all the boys sit next to each other? (3)
6.4 two particu lar boys end up sitting next to each other? [12]
TOTAL : [38]