MATHEMATICS
MORE EXERCISES
Probability - 2 way.

Question  1
TV programme, R, S or T, they preferred.
The table summarises the information :

 Programme Boys (D) Girls (E) Total R 34 36 a S 44 b 102 T c 38 70 Totaal 110 d 250

1.1  Calculate the missing values in
the table.   [ A 1.1 ]

1.2  Are the events "Girl" and
"Prefers programme S" mutually
exclusive? Give a reason.  [ A 1.2 ]

1.3  Prove that the events "Girls" and
"Programme S" are independent. [ A 1.3 ]

1.4  A pupil is selected at random.
Calculate the probability, correct to
3 decimal digits, that the pupil

1.4.1  is a boy.   [ A 1.4.1 ]

1.4.2  is a boy that prefers programme R.
[ A 1.4.2 ]

1.4.3  prefers programme R.   [ A 1.4.3 ]

1.4.4  is a girl that prefers programme T.
[ A 1.4.4 ]

Question  2
The 100 employees of a firm have the following
or a Diploma or a degree.
The table summarises the information :

 Highest Qualification Male (M) Female (F) Total Only Senior Certificate 9 b 25 Diploma 18 42 60 Degree a 12 c Totaal d 70 100

2.1  Calculate the missing values in
the table.   [ A 2.1 ]

2.2  Are the events "Male" and
"Only Senior Certificate" mutually
exclusive? Give a reason. [ A 2.2 ]

2.3  Are the events "Female" and
"Diploma" independent?
Give a reason.   [ A 2.3 ]

2.4  A person is selcted at random.
Calculate the probability, correct to 2
decimal digits, that the person

2.4.1  does not have a degree.   [ A 2.4.1 ]
2.4.2  only has a Senior Certificate.   [ A 2.4.2 ]
2.4.3  is a female and only has a
Senior Certificate. [ A 2.4.3 ]
2.4.4  is a male having a degree.    [ A 2.4.4 ]

Question  3
A restaurant conducted a survey about the
preferences of their customers for
mutton (SM), chicken (HC), beef (B) and
pork (VP).
The table shows the results :

 Meat dish Male (M) Female (L) Total Mutton (SM) a 12 20 Chicken (HC) 6 22 b Beef (B) 12 c 26 Pork (VP) 4 2 6 Total d 50 80

3.1  Calculate the missing values in
the table.   [ A 3.1 ]
3.2  Are the events "Male" and
"Prefers mutton" mutually
exclusive? Give a reason. [ A 3.2 ]

3.3  Are the events "Female" and
"Prefers chicken" independent?
Give a reason. [ A 3.3 ]

3.4  A person is selected at random.
Calculate the probability, correct to 2
decimal digits, that the person

3.4.1  is a male.    [ A 3.4.1 ]
3.4.2  prefers pork. [ A 3.4.2 ]
3.4.3  is a female and prefers mutton.         [ A 3.4.3 ]
3.4.4  does not prefer beef.     [ A 3.4.4 ]

Question  4
In a survey conducted by 220 Grade 12 learners
in a school, the following data
were collected :

 Like ice-cream (L) Do not like ice-cream (D) Total Boys (B) 65 30 Girls (G) 70 55 Total

4.1  Determine the percentage of boys
that like ice-cream.      (2)     [ A 4.1 ]

4.2  Calculate the probability that a
randomly selected boy likes
ice-cream.      (2)   [ A 4.2 ]

4.3  Are the events of being a boy and
liking ice-cream independent or not?
Show your working.        (3)      [ A 4.3 ]

[ Mathematics Grade 11 Paper 1 September 2020

Question  5
The table below shows the data on the
monthly income of employed people
in two resedential areas.
Representative samples were used in
the collection of the data.

 Monthly Income (in Rand) Area 1 Area 2 Total x < 3 200 500 460 960 3 200 ≤ x ≤ 25 600 1 182 340 1 522 x ≥ 25 600 150 14 164 Total 1 832 814 2 646

5.1  What is the probability that a person
chosen randomly from the entire sample
will be
5.1.1  from Area 1?          (2) [ A 5.1.1 ]
5.1.2  from Area 2 and earn less than
R3 200 per month?            (1) [ A 5.1.2 ]
5.1.3  a person from Area 2 who earns
more than or equal to R3 200?      (2)
[ A 5.1.3 ]

5.2  Prove that earning an income of less
than R3 200 per month is not
independent of the area in which
a person resides.      (5)                           [ A 5.2 ]

5.3  What is more likely : a person from
Area 1 earning less than R3 200 or
a person from Area 2 earning less
than R3 200? Show calculations to

[ DBE Mathematics Grade 11 Paper 1 November 2015 ]

Question  6
On a flight passengers could choose
between a vegetarian snack and a chicken
snack. The snacks selected by the passengers
were recorded. The results are shown in the
table below.

 Snack Male (M) Female (F) Total Vegetarian (V) 12 20 32 Chicken (H) 55 63 118 Total 67 83 150

6.1  Was the choice of snack on this flight
independent of gender?
necessary calculations.        (5)               [ A 6.1 ]

[ DBE Mathematics Grade 11 Paper 1 November 2018 ]

$$\hspace*{6 mm}\mathrm{1.1\kern3mm\ a = 34 + 36 = 70\kern2mm\ }$$ $$\hspace*{15 mm}\mathrm{b = 102 − 44 = 58\kern2mm\ }$$ $$\hspace*{15 mm}\mathrm{c = 70 − 38 = 32\kern2mm\ }$$ $$\hspace*{15 mm}\mathrm{d = 250 − 110 = 140\kern2mm\ }$$
[ Q 1.1 ]

1.2      Event A = "Pupil is a girl"
Event B = "Programme S is preferred".
A and B are mutually exclusive if
P(A or B) = P(A) + P(B)
i.e. P(A ∩ B) = 0
P(A ∩ B) = 58/250 ≠ 0
A and B are not mutually exclusive.
[ Q 1.2 ]

1.3      Event A = "Girls"
Event B = "Programme S is preferred".
A and B are independent if
P(A ∩ B) = P(A) × P(B)
$$\hspace*{12 mm}\mathrm{P(A) = \frac{140}{250} = 0,56\kern2mm\ }$$ $$\hspace*{12 mm}\mathrm{P(B) = \frac{102}{250} = 0,41\kern2mm\ }$$ $$\hspace*{12 mm}\mathrm{P(A ∩ B) = \frac{58}{250} = 0,23\kern2mm\ }$$ $$\hspace*{12 mm}\mathrm{P(A)\ \times P(B) = 0,56 \times 0,41\kern2mm\ }$$ $$\hspace*{32 mm}\mathrm{= 0,23 = P(A ∩ B)\kern2mm\ }$$            ∴ A and B are independent.                    [ Q 1.3 ]

$$\hspace*{6 mm}\mathrm{1.4.1\kern3mm\ P(boy) = \frac{110}{250}\kern2mm\ }$$ $$\hspace*{31 mm}\mathrm{= 0,44\kern2mm\ }$$
[ Q 1.4.1 ]
$$\hspace*{6 mm}\mathrm{1.4.2\kern3mm\ P(boy\ ∩\ R) = \frac{34}{250}\kern2mm\ }$$ $$\hspace*{40 mm}\mathrm{= 0,136\kern2mm\ }$$
[ Q 1.4.2 ]

$$\hspace*{6 mm}\mathrm{1.4.3\kern3mm\ P(T) = \frac{70}{250}\kern2mm\ }$$ $$\hspace*{25 mm}\mathrm{= 0,280\kern2mm\ }$$
[ Q 1.4.3 ]

$$\hspace*{6 mm}\mathrm{1.4.4\kern3mm\ P(girl\ ∩\ T) = \frac{38}{250}\kern2mm\ }$$ $$\hspace*{36 mm}\mathrm{= 0,152\kern2mm\ }$$
[ Q 1.4.4 ]

$$\hspace*{6 mm}\mathrm{2.1\kern3mm\ a = 15 − 12 = 3\kern2mm\ }$$ $$\hspace*{15 mm}\mathrm{b = 25 − 9 = 16\kern2mm\ }$$ $$\hspace*{15 mm}\mathrm{c = 100 − 60 − 25 = 15\kern2mm\ }$$ $$\hspace*{15 mm}\mathrm{d = 100 − 70 = 30\kern2mm\ }$$
[ Q 2.1 ]

2.2    Event A = "Employee is a man"
Event B = "Highest qualification is a
Senior Certificate".
A and B are mutually exclusive if
P(A or B) = P(A) + P(B)
i.e. P(A ∩ B) = 0
$$\hspace*{10 mm}\mathrm{P(A\ ∩\ B)\ = \frac{9}{100}\kern2mm\ }$$                                         ≠ 0
A and B are not mutually exclusive.
[ Q 2.2 ]

2.3    Event A = "Employee is a lady"
Event B = "Highest qualification is
a Diploma".
A and B are independent if
P(A ∩ B) = P(A) × P(B)
$$\hspace*{1 mm}\mathrm{P(A)\ \times P(B) = \frac{70}{100} \times \frac{62}{100}\kern2mm\ }$$ $$\hspace*{20 mm}\mathrm{= 0,434\kern2mm\ }$$ $$\hspace*{1 mm}\mathrm{P(A\ ∩\ B) = \frac{42}{100} = 0,42\kern2mm\ }$$          P(A ∩ B) ≠ P(A) × P(B)
A and B are not independent.                    [ Q 2.3 ]

$$\hspace*{1 mm}\mathrm{2.4.1\kern3mm\ P(no\ degree) = \frac{25 + 60}{100}\kern2mm\ }$$ $$\hspace*{35 mm}\mathrm{= 0,85\kern2mm\ }$$ $$\hspace*{36 mm}\mathrm{\bold{OR}\kern2mm\ }$$ $$\hspace*{11 mm}\mathrm{P(degree \bold{'})= 1 − P(degree)\kern2mm\ }$$ $$\hspace*{29 mm}\mathrm{= 1 − \frac{15}{100}\kern2mm\ }$$ $$\hspace*{29 mm}\mathrm{= 0,85\kern2mm\ }$$
[ Q 2.4.1 ]

$$\hspace*{1 mm}\mathrm{2.4.2\kern3mm\ P(only\ SC) = \frac{25}{100}\kern2mm\ }$$ $$\hspace*{31 mm}\mathrm{= 0,25\kern2mm\ }$$
[ Q 2.4.2 ]

$$\hspace*{1 mm}\mathrm{2.4.3\kern3mm\ P(F\ ∩\ SC) = \frac{16}{100}\kern2mm\ }$$ $$\hspace*{31 mm}\mathrm{= 0,16\kern2mm\ }$$
[ Q 2.4.3 ]

$$\hspace*{1 mm}\mathrm{2.4.4\kern3mm\ P(M\ ∩\ Degree) = \frac{3}{100}\kern2mm\ }$$ $$\hspace*{38 mm}\mathrm{= 0,03\kern2mm\ }$$
[ Q 2.4.4 ]

$$\hspace*{1 mm}\mathrm{3.1\kern3mm\ a = 20 − 12 = 8\kern2mm\ }$$ $$\hspace*{10 mm}\mathrm{b = 6 + 22 = 28\kern2mm\ }$$ $$\hspace*{10 mm}\mathrm{c = 26 − 12 = 14\kern2mm\ }$$ $$\hspace*{10 mm}\mathrm{d = 80 − 50 = 30\kern2mm\ }$$
[ Q 3.1 ]

3.2    Event A = "Male M"
Event B = "Prefers mutton SM".
A and B are mutually exclusive if
P(A or B) = P(A) + P(B)
i.e. P(A ∩ B) = 0
$$\hspace*{9 mm}\mathrm{P(A ∩ B) = \frac{8}{80}\kern2mm\ }$$                                    ≠ 0
A and B are not mutually exclusive.
[ Q 3.2 ]

3.3    Event A = "Female L"
Event B = "Prefers chicken HC".
A and B are independent if
P(A ∩ B) = P(A) × P(B)
$$\hspace*{4 mm}\mathrm{P(A)\ \times P(B) = \frac{50}{80}\ \times \frac{28}{80}\kern2mm\ }$$ $$\hspace*{23 mm}\mathrm{= 0,21875\kern2mm\ }$$ $$\hspace*{7 mm}\mathrm{P(A\ ∩ B) = \frac{22}{100}\ = 0,220\kern2mm\ }$$              P(A ∩ B) ≠ P(A) × P(B)
A and B are not independent.              [ Q 3.3 ]

$$\hspace*{1 mm}\mathrm{3.4.1\kern3mm\ P(M) = \frac{30}{80}\kern2mm\ }$$ $$\hspace*{21 mm}\mathrm{= 0,38\kern2mm\ }$$
[ Q 3.4.1 ]

$$\hspace*{1 mm}\mathrm{3.4.2\kern3mm\ P(VP) = \frac{25}{80}\kern2mm\ }$$ $$\hspace*{23 mm}\mathrm{= 0,31\kern2mm\ }$$
[ Q 3.4.2 ]

$$\hspace*{1 mm}\mathrm{3.4.3\kern3mm\ P(L\ ∩\ SM) = \frac{16}{80}\kern2mm\ }$$ $$\hspace*{32 mm}\mathrm{= 0,20\kern2mm\ }$$
[ Q 3.4.3 ]

$$\hspace*{6 mm}\mathrm{3.4.4\kern3mm\ P(B\bold{'}) = 1 − P(B)\kern2mm\ }$$ $$\hspace*{27 mm}\mathrm{= 1 − \frac{26}{80}\kern2mm\ }$$ $$\hspace*{27 mm}\mathrm{= 0,68\kern2mm\ }$$
[ Q 3.4.4 ]

 Like ice-cream (L) Do not like ice-cream (D) Total Boys (B) 65 30 95 Girls (G) 70 55 125 Total 135 85 220

$$\hspace*{6 mm}\mathrm{4.1\kern3mm\ Percentage\ of\ boys\ that\ like\ ice-cream\kern2mm\ }$$ $$\hspace*{27 mm}\mathrm{= \frac{65}{95} \times \frac{100}{1}\%\kern2mm\ }$$ $$\hspace*{27 mm}\mathrm{= 68,42\%\kern2mm\ }$$
[ Q 4.1 ]

$$\hspace*{6 mm}\mathrm{4.2\kern3mm\ P(BL) = \frac{65}{220}\kern2mm\ }$$ $$\hspace*{25 mm}\mathrm{= \frac{13}{44} = 0,2955\kern2mm\ }$$
[ Q 4.2 ]

$$\hspace*{6 mm}\mathrm{4.3\kern3mm\ P(B) = \frac{95}{220}\kern2mm\ }$$ $$\hspace*{23 mm}\mathrm{= \frac{19}{44} = 0,4318\kern2mm\ }$$ $$\hspace*{13 mm}\mathrm{P(B) \times P(L) = \frac{95}{220} \times \frac{135}{220}\kern2mm\ }$$ $$\hspace*{31 mm}\mathrm{= \frac{513}{1936} = 0,2650\kern2mm\ }$$              P(B) × P(L) ≠ P(BL)
∴ Events are not independent.
[ Q 4.3 ]

$$\hspace*{1 mm}\mathrm{5.1.1\kern3mm\ P(person\ from\ Area\ 1) = \frac{1832}{2646}\kern2mm\ }$$ $$\hspace*{51 mm}\mathrm{= 69,24\%\kern2mm\ }$$
[ Q 5.1.1 ]

$$\hspace*{1 mm}\mathrm{5.1.2\kern3mm\ P(person\ from\ Area\ 2\ and\ < R3\ 200)\kern2mm\ }$$ $$\hspace*{41 mm}\mathrm{= \frac{460}{2646}\%\kern2mm\ }$$ $$\hspace*{41 mm}\mathrm{= 17,38\%\kern2mm\ }$$
[ Q 5.1.2 ]

$$\hspace*{1 mm}\mathrm{5.1.3\kern3mm\ P(person\ from\ Area\ 2\ and\ > R3\ 200)\kern2mm\ }$$ $$\hspace*{32 mm}\mathrm{= \frac{340 + 14}{2646}\%\kern2mm\ }$$ $$\hspace*{32 mm}\mathrm{= \frac{59}{441}\%\ = 13,38\%\kern2mm\ }$$
[ Q 5.1.3 ]

5.2  Let the event of a randomly selected
person living in Area 1, be A
Let the event of a randomly selected
person earning less than R3 200 be B.
$$\hspace*{8 mm}\mathrm{P(A\ and\ B) = \frac{500}{2646} = 18,90\%\kern2mm\ }$$ $$\hspace*{8 mm}\mathrm{P(A) \times P(B) = \frac{1832}{2646} \times \frac{960}{2646}\kern2mm\ }$$ $$\hspace*{27 mm}\mathrm{= 25,12\%\kern2mm\ }$$           Clearly P(A and B) ≠ P(A) × P(B)
Hence A and B are not independent.
OR
Let the event of a randomly selected
person living in Area 2 be C.
Let the event of a randomly selected
person earning less than R3 200 be D.
$$\hspace*{8 mm}\mathrm{P(C\ and\ D) = \frac{460}{2646} = 17,38\%\kern2mm\ }$$ $$\hspace*{8 mm}\mathrm{P(C) \times P(D) = \frac{814}{2646} \times \frac{960}{2646}\kern2mm\ }$$ $$\hspace*{27 mm}\mathrm{= 11,16\%\kern2mm\ }$$           Clearly P(C and D) ≠ P(C) × P(D)
Hence C and D are not independent.
[ Q 5.2 ]

$$\hspace*{1 mm}\mathrm{5.3\kern3mm\ P(Area\ 1\ person\ earns\ less\ than\ R3\ 200)\kern2mm\ }$$ $$\hspace*{21 mm}\mathrm{= \frac{500}{1832} = 27,29\%\kern2mm\ }$$ $$\hspace*{1 mm}\mathrm{P(Area\ 2\ person\ earns\ less\ than\ R3\ 200)\kern2mm\ }$$ $$\hspace*{21 mm}\mathrm{= \frac{460}{814} = 56,51\%\kern2mm\ }$$           A person from Area 2 is more likely
to earn less than R3 200.      [ Q 5.3 ]

$$\hspace*{1 mm}\mathrm{6.\kern3mm\ P(V) \times P(M) = \frac{32}{150} \times \frac{67}{150}\kern2mm\ }$$ $$\hspace*{27 mm}\mathrm{= 0,095\kern2mm\ }$$ $$\hspace*{8 mm}\mathrm{P(V\ and\ M) = \frac{12}{150}\kern2mm\ }$$ $$\hspace*{25 mm}\mathrm{= 0,08\kern2mm\ }$$                 P(V and M) ≠ P(V) × P(M)
The events are not independent.
$$\hspace*{38 mm}\mathrm{\bold{OR}\kern2mm\ }$$ $$\hspace*{13 mm}\mathrm{P(V) \times P(F) = \frac{32}{150} \times \frac{83}{150}\kern2mm\ }$$ $$\hspace*{32 mm}\mathrm{= 0,118\kern2mm\ }$$ $$\hspace*{13 mm}\mathrm{P(V\ and\ F) = \frac{20}{150}\kern2mm\ }$$ $$\hspace*{29 mm}\mathrm{= 0,113\kern2mm\ }$$                 P(V and F) ≠ P(V) × P(F)
The events are not independent.
[ Q 6. ]