MATHEMATICS
MORE EXERCISES
Probability - calculations.

Question  1
Study the following sets:
S, the sample set, represents all the pupils at
a certain high school.
A is the set of all Grade 11 learners in this school.
B is the set of all the pupils in this school that
study Mathematics.
C is the set of all the boys in this school.

Describe in words and with reference to the above
mentioned sets what is meant by each of the
following :
Example : B' represents all the pupils in the school
that do not take Mathematics.

1.1  A'    [ A 1.1 ]

1.2  A ∪ B [ A 1.2 ]

1.3  B ∩ C [ A 1.3 ]

1.4  A ∪ C' [ A 1.4 ]

1.5  A ∩ B ∩ C   [ A 1.5 ]

1.6  A' ∪ C' [ A 1.6 ]

1.7  A' ∩ B' [ A 1.7 ]

Question  2
A and B are two mutally exclusive events.
Determine

2.1  P(A or B) if P(A) = 0,6 and
P(B) = 0,37  [ A 2.1 ]

2.2  P(A) if P(A or B) = 0,17 and
P(B) = 0,23 [ A 2.2 ]

Question  3
A and B are two independent events.
Determine

3.1  P(A and B) if P(A) = 0,13 and
P(B) = 0,2 [ A 3.1]

3.2  P(B) if P(A and B) = 0,06 and
P(A) = 0,2 [ A 3.2 ]

Question  4
The probability that event A will occur is
0,45 and the probability that event B will
occur is 0,35. The probability that event A
or event B will occur is 0,55.

4.1  Calculate the probability that both events
A and B will occur. [ A 4.1 ]

4.2  Are the events A and B independent?

4.3  Are the events A and B mutually exclusive?

Question  5
The probability that event A will occur is 0,3
and the probability that event B will occur
is 0,4. The probability that both events A and B
will occur is 0,12.

5.1  Are the events A and B mutually exclusive?

5.2  Are the events A and B independent?

5.3  Calculate the probability that event A or
event B will occur. [ A 5.3 ]

Question  6
The probabilitty that event A will occur is 0,4
and the probability that event B will occur
is 0,6. The probability that event A or
event B will occur is 0,7.

6.1  Calculate the probability that both
events A and B will occur. [ A 6.1 ]
6.2  Are the events A and B mutually
[ A 6.2 ]
6.3  Are the events A and B independent?

Question  7
If P(A) = 0,35, P(B) = 0,2 and P(A or B) = 0,4
determine

7.1  P(A and B) [ A 7.1 ]

7.2  P(A'[ A 7.2 ]

7.3  P(A' or B)   [ A 7.3 ]

Question  8
8.1  Given P(A or B) = 0,4;    P(A) = 3P(B)
and events A and B are independent.
Determine P(B)    (4) [ A 8.1 ]

Free State Gr. 12 Sept. 2020

8.2  Events A and B are mutually exclusive.
It is given that P(B) = 2P(A) and
P(A or B) = 0,57. Calculate P(B)        (3)
[ A 8.2 ]
Mathematics Paper 1 Gauteng 2022

8.3  A and B are independent events. It is further
given that P(A and B) = 0,3
and P(only B) = 0,2.

8.3.1   Are A and B mutually exclusive?
[ A 8.3.1 ]

8.3.2   Determine
8.3.2.1 P(only A)          (4) [ A 8.3.2.1 ]
8.3.2.2 P(not A or not B)     (2) [ A 8.3.2.2 ]

Mathematics Paper 1 November 2021 NSS

Question  9
9.1  Jan goes to the cinema or a club on
a Friday night. He goes to a club 60%
of the time and then sleeps late on a
Saturday morning 70% of the time.
If he goes to the cinema, he has a 40%
probability of sleeping late on a
Saturday morning.
Determine the probability that Jan
sleeps late on a randomly selected
Saturday.          (4) [ A 9.1 ]

From Freestate Gr. 12 Sept. 2020

9.2  A box of 40 calculators is sent to a store
by a supplier. The owner of the store is
not aware that 5 of the calculators
are defective.
Two calculators are selected at random
from the box, the first one not being
replaced before the second one
is selected.

9.2.1  What is the probability that the
first calculator chosen is
NOT defective?      (1) [ A 9.2.1 ]

9.2.2  What is the probability that
if two calculators are selected
ONE calculator is defective and
the other not?        (3) [ A 9.2.2 ]

9.2.3  What is the probability that if
two calculators are selected,
BOTH are defective?      (3) [ A 9.2.3 ]
Mathematics Paper 1 Gauteng 2022

Question  10
10.  Given : P(A) = 0,2, P(B) = 0,5 and
P(A or B) = 0,6 where A and B
are two different events.

10.1  Calculate P(A and B)      (2) [ A 10.1]
10.2  Are the two events A and B independent?
Show your calculations.    (3) [ A 10.2]

From Gr. 11 Paper 1 November 2016 DBE

Antwoorde  1
1.1  A' is the set of all the pupils that
[ Q 1.1 ]

1.2  A ∪ B  is the set of all the pupils that are
in Grade 11 or all the pupils that
study Mathematics.
[ Q 1.2 ]

1.3  B ∩ C  is the set of all the pupils that study
Mathematics and are boys
OR all the boys who study Mathematics.
[ Q 1.3 ]

1.4  A ∪ C'  is the set of all Grade 11 pupils
and who are not boys.
OR all the girls in Grade 11.
[ Q 1.4 ]

a     1.5  A ∩ B ∩ C  : is the set of all pupils in Grade 11
who study Mathematics and are boys.
OF all the Grade 11 boys who
study Mathematics.
[ Q 1.5 ]

1.6  A' ∪ C'  :  all the pupils who are not in Grade 11
and who are not boys.
OR all the girls who are not in Grade 11
is nie.
[ Q 1.6 ]

1.7  A' ∩ B'  :  all the pupils who are not in Grade 11
and who do not study Mathematics.
Wiskunde neem nie.
[ Q 1.7 ]

A and B are mutually exclusive.
∴ P(A or B) = P(A) + P(B)
$$\text{2.1\hspace{2 mm}P(A or B) = P(A) + P(B)}\\ \text{\hspace{22 mm}= 0,17 + 0,23}\\ \text{\hspace{22 mm}= 0,40}\\$$
[ Q 2.1 ]
$$\text{2.2\hspace{2 mm}P(A or B) = P(A) + P(B)}\\ \text{\hspace{16 mm}0,6 = 0,17 + 0,23}\\ \text{\hspace{14 mm}P(A) = 0,6 - 0,37}\\ \text{\hspace{22 mm}= 0,23}\\$$
[ Q 2.2 ]

A and B are independent.
∴ P(A and B) = P(A) × P(B)
$$\text{3.1\hspace{2 mm}P(A and B) = P(A) × P(B)}\\ \text{\hspace{22 mm}= 0,13 × 0,25}\\ \text{\hspace{22 mm}= 0,0325}\\$$
[ Q 3.1 ]
$$\text{3.2\hspace{2 mm}P(A and B) = P(A) × P(B)}\\ \text{\hspace{16 mm}0,06 = 0,2 + P(B)}\\ \text{\hspace{14 mm}P(B) = 0,06 ÷ 0,2}\\ \text{\hspace{22 mm}= 0,3}\\$$
[ Q 3.2 ]

$$\text{4.1\hspace{2 mm}P(A or B) = P(A) + P(B) - P(A and B)}\\ \text{\hspace{14 mm}0,55 = 0,45 + 0,35 - P(A and B)}\\ \text{\hspace{6 mm}P(A and B) = 0,45 + 0,35 - 0,55}\\ \text{\hspace{22 mm}= 0,25}\\$$
[ Q 4.1 ]
$$\text{4.2\hspace{2 mm}P(A) × P(B) = 0,45 × 0,34}\\ \text{\hspace{16 mm}= 0,153}\\ \text{\hspace{14 mm}\neq 0,25 . . . [P(A ∩ B)]}\\ \text{\hspace{7 mm}A and B are not independent.}\\$$
[ Q 4.2 ]
$$\text{4.2\hspace{2 mm}P(A) ∩ P(B) = 0,25}\\ \text{\hspace{16 mm}\neq 0}\\ \text{\hspace{7 mm}A and B are not mutually exclusive.}\\$$
[ Q 4.3 ]

$$\text{5.1\hspace{2 mm}P(A ∩ B) = 0,12}\\ \text{\hspace{14 mm}\neq 0}\\ \text{\hspace{6 mm}A and B are not mutually exclusive.}\\$$
[ Q 5.1 ]
$$\text{5.2\hspace{2 mm}P(A × B) = 0,3 × 0,4}\\ \text{\hspace{23 mm}= 0,12}\\ \text{\hspace{23 mm}= P(A ∩ B)}\\ \text{\hspace{6 mm}A and B are independent.}\\$$
[ Q 5.2 ]
$$\text{5.3\hspace{2 mm}P(A or B) = P(A) + P(B) - P(A and B)}\\ \text{\hspace{22 mm}= 0,3 + 0,4 - 0,12}\\ \text{\hspace{22 mm}= 0,58}\\$$
[ Q 5.3 ]

$$\text{6.1\hspace{2 mm}P(A or B) = P(A) + P(B) - P(A and B)}\\ \text{\hspace{16 mm}0,7 = 0,4 + 0,5 - P(A and B)}\\ \text{\hspace{6 mm}P(A and B) = 0,4 + 0,5 - 0,7}\\ \text{\hspace{22 mm}= 0,2}\\$$
[ Q 6.1 ]
$$\text{6.2\hspace{2 mm}A and B are mutually exclusive if}\\ \text{\hspace{7 mm}P(A and B) = 0}\\ \text{\hspace{7 mm}P(A and B) = 0,2}\\ \text{\hspace{23 mm}\neq 0}\\ \text{\hspace{7 mm}A and B are not mutually exclusive.}\\$$
[ Q 6.2 ]
$$\text{6.3\hspace{2 mm}A and B are independent if}\\ \text{\hspace{7 mm}P(A) × P(AB) = P(A and B)}\\ \text{\hspace{11 mm}P(A) × P(B) = 0,4 × 0,5}\\ \text{\hspace{30 mm}= 0,2}\\ \text{\hspace{30 mm}= P(A and B)}\\ \text{\hspace{11 mm}A and B are independent.}\\$$
[ Q 6.3 ]

$$\text{7.1\hspace{2 mm}P(A or B) = P(A) + P(B) - P(A and B)}\\ \text{\hspace{16 mm}0,4 = 0,35 + 0,2 - P(A and B)}\\ \text{\hspace{6 mm}P(A and B) = 0,35 + 0,2 - 0,4}\\ \text{\hspace{22 mm}= 0,15}\\$$
[ Q 7.1 ]
$$\text{7.2\hspace{2 mm}P(A') = 1 - P(A)}\\ \text{\hspace{15 mm}= 1 - 0,35}\\ \text{\hspace{15 mm}= 0,65}\\$$
[ Q 7.2 ]
7.3
$$\text{\hspace{5 mm}P(A' or B) = P(A') + P(B) - P(A' and B)}\\ \text{\hspace{21 mm}= 0,65 + 0,2 - 0,05}\\ \text{\hspace{21 mm}= 0,8}\\$$
[ Q 7.3 ]

\text{8.1\hspace{2 mm}P(A or B) = P(A) + P(B) - P(A and B)}\\ \text{\hspace{7 mm}P(A or B) = P(A) × P(B) . . . A and B independent}\\ \text{\hspace{7 mm}Let P(B) = x}\\ \text{\hspace{14 mm}3x^2 = 3x + x - 0,4}\\ \begin{align*}\\ 3x^2 - 4x + 0,4 &= 0\\ x &= \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\\ &= \frac{-(-4) \pm \sqrt{(-4)^2 - 4(3)(0,4)}}{2(3)}\\ &= \frac{4 \pm \sqrt{11,2}}{6}\\ &= 1,22 .\ .\ .\ n.a. \\ &= 0,11\\ \end{align*}\\ \text{\hspace{11 mm}\therefore P(B) = 0,11}\\
[ Q 8.1 ]
\text{8.2\hspace{2 mm}P(A or B) = P(A) + P(B)}\\ \begin{align*}\\ 0,57 &= P(A)\ + 2 P(A)\\ 3 P(A) &= 0,57\\ P(A) &= 0,19\\ P(B) &= 2 P(A)\\ &=2(0,19)\\ &= 0,38 \end{align*}\\
[ Q 8.2 ]
$$\text{8.3.1\hspace{2 mm}No, because P(A and B) = 0}\\$$
[ Q 8.3.1 ]
8.3.2.1
$$\text{\hspace{2 mm}P(A and B) = P(A) × P(B)}\\ \text{\hspace{12 mm}0,3 = P(A) × 0,5}\\ \text{\hspace{10 mm}P(A) = 0,6}\\ \text{\hspace{4 mm}P(only A) = 0,6 - 0,3}\\ \text{\hspace{18 mm}= 0,3}\\$$
[ Q 8.3.2.1 ]
$$\text{8.3.2.2\hspace{2 mm}P(not A or not B) = 0,2 + 0,2 + 0,3}\\ \text{\hspace{39 mm}= 0,7}\\ \text{\hspace{30 mm}\bold{OR}}\\ \text{\hspace{13 mm}P(not A or not B) = 1 - P(A and B)}\\ \text{\hspace{39 mm}= 1 - 0,3}\\ \text{\hspace{39 mm}= 0,7}\\ \text{\hspace{30 mm}\bold{OR}}\\ \text{\hspace{13 mm}P(A' or B') = P(A') + P(B') - P(A' and B')}\\ \text{\hspace{29 mm}= 0,4 + 0,5 - 0,2}\\ \text{\hspace{29 mm}= 0,7}\\$$
[ Q 8.3.2.2 ]

$$\text{9.1\hspace{2 mm}P(goes to the club) = 60\% = 0,6 en}\\ \text{\hspace{7 mm}P(sleeps late) = 70\% = 0,7}\\ \text{\hspace{7 mm}P(club and sleeps late) = 0,6 × 0,7}\\ \text{\hspace{40 mm}= 0,42}\\ \text{\hspace{7 mm}P(goes to the cinema) = 40\% = 0,4 en}\\ \text{\hspace{7 mm}P(sleeps late) = 40\% = 0,4}\\ \text{\hspace{7 mm}P(cinema and sleeps late) = 0,6 × 0,4}\\ \text{\hspace{40 mm}= 0,16}\\ \text{\hspace{7 mm}Club and cinema are mutually exclusive }\\ \text{\hspace{7 mm}thus P(club and cinema) = 0}\\ \text{\hspace{16 mm}\therefore P(sleeps late) = 0,42 + 0,16}\\ \text{\hspace{41 mm}= 0,58}\\$$
[ Q 9.1 ]
$$\text{9.2.1\hspace{2 mm}Not defective (ND)}\\ \text{\hspace{10 mm}P(ND) = \frac{35}{40}}\\ \text{\hspace{21 mm}= \frac{7}{8} = 0,88}\\$$
[ Q 9.2.1 ]
$$\text{9.2.2\hspace{2 mm}Not defective (ND) and defectice (D)}\\ \text{\hspace{10 mm}P(ND and D) + P(D and ND)}\\ \text{\hspace{22 mm}= (\frac{35}{40} × \frac{5}{39}) + (\frac{5}{40} × \frac{5}{39})}\\ \text{\hspace{22 mm}= 0,128}\\$$
[ Q 9.2.2 ]
$$\text{9.2.3\hspace{2 mm}Defektief (D)}\\ \text{\hspace{10 mm}P(D en D) = (\frac{5}{40} × \frac{4}{39})}\\ \text{\hspace{26 mm}= \frac{1}{78} = 0,0128}\\$$

$$\text{10.1\hspace{2 mm}P(A) = 0,2; P(B) = 0,5 and}\\ \text{\hspace{9 mm}P(A or B) = 0,6}\\ \text{\hspace{9 mm}P(A or B) = P(A) + P(B) - P(A and B)}\\ \text{\hspace{18 mm}0,6 = 0,2 + 0,5 - P(A and B)}\\ \text{\hspace{8 mm}P(A and B) = 0,1}\\$$
[ Q 10.1 ]
$$\text{10.2\hspace{2 mm}P(A and B) = 0,1}\\ \text{\hspace{9 mm}P(A) × P(B) = 0,2 × 0,5}\\ \text{\hspace{28 mm}= 0,1}\\ \text{\hspace{8 mm}\therefore P(A and B) = P(A) × P(B)}\\ \text{\hspace{8 mm}\therefore A and B are independent}\\$$
[ Q 10.2 ]