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Probability - Venn diagrams.

Question  1

The Venn diagram above represents the chosen
sport of 100 Grade 11 boys :
43 boys play cricket(C), 54 play soccer(S)
and 45 play tennis(T).

1.1  How many boys participate in all three
games? [ A 1.1 ]

1.2  How many boys do not participate in
any sport? [ A 1.2 ]

1.3  If a boy is selcted at random, calculate
the probability, correct to two decimal
places, that the boy will

1.3.1  not participate in any sport. [ A 1.3.1 ]

1.3.2  play soccer.   [ A 1.3.2 ]

1.3.3  only play cricket. [ A 1.3.3 ]

1.3.4  participate in only one kind of sport.
[ A 1.3.4 ]

1.3.5  participate in two sports.
[ A 1.3.5 ]

1.3.6  participate in all three.
[ A 1.3.6 ]

1.3.7  play tennis but not soccer.
[ A 1.3.7 ]

1.3.8  not play tennis. [ A 1.3.8 ]

1.3.9  play soccer and cricket but
not tennis. [ A 1.3.9 ]

Question  2

The Venn diagram above represents three
events, A, B and C.
Event A has 40 elements,
event B has 30 elements and
event C has 43 elements.

2.1  Show that x = 12.   [ A 2.1 ]

2.2  Calculate y and z and also the total
number of elements. [ A 2.2 ]

2.3  Calculate the following probabilities,
in the simplest fraction form :

2.3.1  P(A) [ A 2.3.1 ]

2.3.2  P(A ∪ C)  [ A 2.3.2 ]

2.3.3  P(B')    [ A 2.3.3 ]

2.3.4  P(A ∩ B ∩ C)  [ A 2.3.4 ]

2.3.5  P(A ∩ B) [ A 2.3.5 ]

2.3.6  P(B ∩ C') [ A 2.3.6 ]

2.3.7  P(A and B)'  [ A 2.3.7]

2.3.8  P(B or C)  [ A 2.3.8 ]

2.3.9  P(only B) [ A 2.3.9 ]

Question  3

Sample space S has 200 elements
and A, B and C are 3 events in S.

3.1  Calculate x if P(B ∩ C) = 0,17 [ A 3.1 ]

3.2  Calculate the number of elements in
A, B and C.   [ A 3.2 ]

3.3  Calculate the following probabilities,
correct to 3 decimals :

3.3.1  P(A) [ A 3.3.1 ]

3.3.2  P(B) [ A 3.3.2 ]

3.3.3  P(C')    [ A 3.3.3 ]

3.3.4  P(A ∪ B) [ A 3.3.4 ]

3.3.5  P(A ∩ B) [ A 3.3.5 ]

3.3.6  P(A ∩ C ∩ B') [ A 3.3.6 ]

Question  4

The numbers 1 to and including 15, are
arranged in 3 events, A, B and C, as shown
in the Venn diagram above.
Calculate the following probabilities,
in simplest fraction form :
4.1  P(A) [ A 4.1 ]

4.2  P(C) [ A 4.2 ]

4.3  P(B') [ A 4.3 ]

4.4  P(A ∩ B)   [ A 4.4 ]

4.5  P(B ∪ C)   [ A 4.5 ]

4.6  P(A ∩ C)   [ A 4.6 ]

4.7  P(A ∪ B ∪ C)'   [ A 4.7 ]

4.8  P(A' ∩ B ∩ C'[ A 4.8 ]

Question  5

The Venn diagram above represents the
chosen sport of Grade 11 boys.
The sports are Hockey, H, Tennis, T,
and Soccer, S.
16 boys do all three sports and
23 boys do not do any sports.

5.1  Show that x = 13 if 58 boys each partake
only in one sport.
[ A 5.1 ]

5.2  Calculate the number of boys who only
play Hockey.    [ A 5.2 ]

5.3  18 boys play at least 2 of Hockey, Tennis
and or Soccer. How many boys play Hockey
and Tennis but not Soccer?
[ A 5.3 ]

5.4  If 4 boys play Hockey and Soccer but
not Tennis, how many boys play
Hockey?    [ A 5.4 ]

5.5  How many Grade 11 boys are there
if P(no sports) = 0,2?  [ A 5.5 ]

Question  6

The Venn diagram above represents the
method of transport used by workers.
Taxis, T, busses, B, and
motor cars, C, are used.

6.1  Show that x = 5 if 44 workers use
only two methods of transport.                  [ A 6.1 ]

6.2  How many workers use Taxis and at least
one other form of transport?
[ A 6.2 ]

6.3  How many workers use both Taxis and
busses but not motor cars? [ A 6.3 ]

6.4  If in total 150 workers were interviewed
and 80 used Taxis, 59 used motor cars and
31 used no transport at all, what is the
probability, in simplified fraction form, that
a worker will

6.4.1  use busses?     [ A 6.4.1 ]

6.4.2  travel by motor cars only? [ A 6.4.2 ]

6.4.3  use all three methods of transport?
[ A 6.4.3 ]

6.4.4  use some form of transport?
[ A 6.4.4 ]

Question  7
A, B and C are three events.

The probability that these events (or
any combination thereof) will occur, is given
by the Venn diagram below.

7.1  If it is given that the probability that
at least one of the events will occur
is 0,893, calculate the value of
7.1.1  y, the probability that not one of
the events will occur. [ A 7.1.1 ]

7.1.2  x, the probability that all three
events will occur.   [ A 7.1.2 ]

7.2  Determine the probability that at least
two of the events will ocur.  [ A 7.2 ]

7.3  Are the events B and C independent?

These questions are derived from
Mathematics Paper 1, November 2022. DBE.

Question  8
A survey was conducted amongst 100
learners at a school to establish their
involvement in three codes of sport,
soccer, netball and volleyball.
The results are shown below :
55 learners play soccer (S)
21 learners play netball (N)
7 learners play volleyball (V)
3 learners play netbal only
2 learners play volleyball and netball
1 learner plays all 3 sports.
The Venn diagram below shows the
information above.

8.1  Determine the values of a, b, c, d and e
[ A 8.1 ]

8.2  One of the learners is chosen at random
from this group.
Calculate the probability that this learner
8.2.1  plays netball or volleyball.     [ A 8.2.1 ]

8.2.2  plays soccer and volleyball.   [ A 8.2.2 ]

8.2.3  plays at least two sports. [ A 8.2.3 ]

8.2.4  plays only soccer or only volleyball.
[ A 8.2.4 ]

1.1    The number of boys who play all
three sports                                                = x.
Calculate x : Number who play cricket = 43
∴ 16 + 6 + 10 + x = 43
∴ x = 11
11 boys play all three sports.
[ Q 1.1 ]
1.2    Number of boys that play
= 16 + 6 + 10 + 11 + 25 + 12 + 12
= 92
Number of boys that do not play = 100 - 92
= 8
[ Q 1.2 ]
$$\text{1.3.1\hspace{2 mm}Number of boys that do not play = 8}\\ \text{\hspace{9 mm}P(non-participant) = \frac{8}{100}}\\ \text{\hspace{39 mm}= 0,08}\\$$
[ Q 1.3.1 ]
$$\hspace*{0 mm}\mathrm{1.3.2\kern2mm\ P(S) = \frac{54}{100}\kern2mm\ }$$ $$\hspace*{18 mm}\mathrm{= 0,54\kern2mm\ }$$
[ Q 1.3.2 ]
$$\hspace*{2 mm}\mathrm{1.3.3\kern2mm\ P(only\ C) = \frac{16}{100}\kern2mm\ }$$ $$\hspace*{27 mm}\mathrm{= 0,16\kern2mm\ }$$
[ Q 1.3.3 ]

$$\hspace*{2 mm}\mathrm{1.3.4\kern2mm\ P(only\ 1) \ = \frac{16 + 25 + 12}{100}\kern2mm\ }$$ $$\hspace*{27 mm}\mathrm{= 0,53\kern2mm\ }$$
[ Q 1.3.4 ]

$$\hspace*{2 mm}\mathrm{1.3.5\kern2mm\ n(2\ sports) = 16+6+25+10+12+12\kern2mm\ }$$ $$\hspace*{30 mm}\mathrm{= 81\kern2mm\ }$$
$$\hspace*{28 mm}\mathrm{\bold{OR}}$$ $$\hspace*{12 mm}\mathrm{n(2\ codes) = 100 - 8 - 11\kern2mm\ }$$ $$\hspace*{30 mm}\mathrm{= 81\kern2mm\ }$$ $$\hspace*{12 mm}\mathrm{P(2\ codes) = \frac{81}{100}\kern2mm\ }$$ $$\hspace*{31 mm}\mathrm{= 0,81\kern2mm\ }$$
[ Q 1.3.5 ]

$$\hspace*{2 mm}\mathrm{1.3.6\kern2mm\ n(all\ 3) = 100 - n(non-participants)\kern2mm\ }$$ $$\hspace*{23 mm}\mathrm{= 92\kern2mm\ }$$ $$\hspace*{12 mm}\mathrm{P(all\ 3) = \frac{92}{100}\kern2mm\ }$$ $$\hspace*{24 mm}\mathrm{= 0,92\kern2mm\ }$$
[ Q 1.3.6 ]

$$\hspace*{2 mm}\mathrm{1.3.7\kern2mm\ n(Tennis,\ not\ soccer) = 12 + 10 + 11\kern2mm\ }$$ $$\hspace*{47 mm}\mathrm{= 33\kern2mm\ }$$ $$\hspace*{12 mm}\mathrm{P(T,\ not\ S) = \frac{33}{100}\kern2mm\ }$$ $$\hspace*{30 mm}\mathrm{= 0,33\kern2mm\ }$$
[ Q 1.3.7 ]

$$\hspace*{2 mm}\mathrm{1.3.8\kern2mm\ n(not\ Tennis) = n(all\ boys) - n(T)\kern2mm\ }$$ $$\hspace*{34 mm}\mathrm{= 100 - 45\kern2mm\ }$$ $$\hspace*{34 mm}\mathrm{= 55\kern2mm\ }$$ $$\hspace*{32 mm}\mathrm{\bold{OR}\kern2mm\ }$$ $$\hspace*{12 mm}\mathrm{n(not\ Tennis) = 16+6+25+8\kern2mm\ }$$ $$\hspace*{33 mm}\mathrm{= 55\kern2mm\ }$$ $$\hspace*{12 mm}\mathrm{P(not\ T) = \frac{55}{100}\kern2mm\ }$$ $$\hspace*{25 mm}\mathrm{= 0,55\kern2mm\ }$$ $$\hspace*{32 mm}\mathrm{\bold{OR}\kern2mm\ }$$ $$\hspace*{12 mm}\mathrm{P(not\ T) = 1 - p(T)\kern2mm\ }$$ $$\hspace*{26 mm}\mathrm{= 1 - 0,45\kern2mm\ }$$ $$\hspace*{26 mm}\mathrm{= 0,55\kern2mm\ }$$
[ Q 1.3.8 ]

$$\hspace*{2 mm}\mathrm{1.3.9\kern2mm\ n(S + C,\ not\ T) = n(all\ 3) - n(T)\kern2mm\ }$$ $$\hspace*{37 mm}\mathrm{= 92 - 45\kern2mm\ }$$ $$\hspace*{37 mm}\mathrm{= 47\kern2mm\ }$$ $$\hspace*{29 mm}\mathrm{\bold{OR}\kern2mm\ }$$ $$\hspace*{12 mm}\mathrm{n(S + C,\ not\ T) = 16+6+25\kern2mm\ }$$ $$\hspace*{36 mm}\mathrm{= 47\kern2mm\ }$$ $$\hspace*{12 mm}\mathrm{P(S+C,\ not\ T) = \frac{47}{100}\kern2mm\ }$$ $$\hspace*{37 mm}\mathrm{= 0,47\kern2mm\ }$$
[ Q 1.3.9 ]

2.1    A contains 40 elements.
$$\hspace*{12 mm}\mathrm{∴ \ \ 13 + 8 + x + 7 = 40\kern2mm\ }$$ $$\hspace*{36mm}\mathrm{x = 40 - 28\kern2mm\ }$$ $$\hspace*{39mm}\mathrm{= 12\kern2mm\ }$$
[ Q 2.1 ]

2.2    B contains 30 elements.
$$\hspace*{12 mm}\mathrm{∴ \ \ 8 + 6 + y + 12 = 30\kern2mm\ }$$ $$\hspace*{36 mm}\mathrm{y = 30 - 26\kern2mm\ }$$ $$\hspace*{39mm}\mathrm{= 4\kern2mm\ }$$            C contains 43 elements.
$$\hspace*{12 mm}\mathrm{∴ \ \ 7 + 12 + 4 + z = 43\kern2mm\ }$$ $$\hspace*{36 mm}\mathrm{z = 43 - 23\kern2mm\ }$$ $$\hspace*{39mm}\mathrm{= 20\kern2mm\ }$$ $$\hspace*{8 mm}\mathrm{∴ \ Total\ number\ = 40 + 6 + 4 + 20\kern2mm\ }$$ $$\hspace*{34mm}\mathrm{= 70\kern2mm\ }$$
[ Q 2.2 ]

$$\hspace*{2 mm}\mathrm{2.3.1\kern2mm\ P(A) = \frac{40}{70}\kern2mm\ }$$ $$\hspace*{21 mm}\mathrm{= \frac{4}{7}\kern2mm\ }$$
[ Q 2.3.1 ]

$$\hspace*{2 mm}\mathrm{2.3.2\kern2mm\ P(A\ ∪\ C) = \frac{40 + 4 + 20}{70}\kern2mm\ }$$ $$\hspace*{29 mm}\mathrm{= \frac{32}{35}\kern2mm\ }$$
[ Q 2.3.2 ]

$$\hspace*{2 mm}\mathrm{2.3.3\kern2mm\ P(B\bold{'}) = \frac{13 + 7 + 20}{70}\kern2mm\ }$$ $$\hspace*{22 mm}\mathrm{= \frac{4}{7}\kern2mm\ }$$
$$\hspace*{34 mm}\mathrm{\bold{OR}\kern2mm\ }$$
$$\hspace*{13 mm}\mathrm{P(B\bold{'}) = 1 − P(B)\kern2mm\ }$$ $$\hspace*{22 mm}\mathrm{= 1 − \frac{30}{70}\kern2mm\ }$$ $$\hspace*{22 mm}\mathrm{= \frac{4}{7}\kern2mm\ }$$
[ Q 2.3.3 ]

$$\hspace*{2 mm}\mathrm{2.3.4\kern2mm\ P(A\ ∩\ B\ ∩\ C) = \frac{12}{70}\kern2mm\ }$$ $$\hspace*{38 mm}\mathrm{= \frac{6}{35}\kern2mm\ }$$
[ Q 2.3.4 ]

$$\hspace*{2 mm}\mathrm{2.3.5\kern2mm\ P(A\ ∩\ B) = \frac{8 + 12}{70}\kern2mm\ }$$ $$\hspace*{29 mm}\mathrm{= \frac{2}{7}\kern2mm\ }$$
[ Q 2.3.5 ]

$$\hspace*{2 mm}\mathrm{2.3.6\kern2mm\ P(B\ ∩\ C\bold{'}) = \frac{8 + 6}{70}\kern2mm\ }$$ $$\hspace*{31 mm}\mathrm{= \frac{1}{5}\kern2mm\ }$$
[ Q 2.3.6 ]

$$\hspace*{2 mm}\mathrm{2.3.7\kern2mm\ P((A\ and\ B)\bold{'}) = \frac{13 + 7 + 6 + 4 + 20}{70}\kern2mm\ }$$ $$\hspace*{32 mm}\mathrm{= \frac{5}{7}\kern2mm\ }$$ $$\hspace*{37 mm}\mathrm{\bold{OR}\kern2mm\ }$$ $$\hspace*{11 mm}\mathrm{P((A\ and\ B)\bold{'}) = 1 − P(A\ and\ B)\kern2mm\ }$$ $$\hspace*{30 mm}\mathrm{= 1 − \frac{8 + 12}{70}\kern2mm\ }$$ $$\hspace*{30 mm}\mathrm{= \frac{5}{7}\kern2mm\ }$$
[ Q 2.3.7 ]

$$\hspace*{2 mm}\mathrm{2.3.8\kern2mm\ P(B\ or\ C) = \frac{8 + 6 + 12 + 4 + 7 + 20}{70}\kern2mm\ }$$ $$\hspace*{30 mm}\mathrm{= \frac{57}{70}\kern2mm\ }$$ $$\hspace*{36 mm}\mathrm{\bold{OR}\kern2mm\ }$$ $$\hspace*{12 mm}\mathrm{Remembeer :\kern2mm\ }$$ $$\hspace*{12 mm}\mathrm{P(A\ or\ B) = P(A\ ∪\ B)\ = P(A) + P(B) − P(A\ ∩\ B)\kern2mm\ }$$ $$\hspace*{12 mm}\mathrm{P(B\ or\ C) = P(B\ ∪\ C)\ = P(B) + P(C) − P(B\ ∩\ C)\kern2mm\ }$$ $$\hspace*{48 mm}\mathrm{= \frac{30}{70} + \frac{43}{70}\ −\ \frac{12 + 4}{70}\kern2mm\ }$$ $$\hspace*{48 mm}\mathrm{= \frac{57}{70}\kern2mm\ }$$
[ Q 2.3.8 ]

$$\hspace*{2 mm}\mathrm{2.3.9\kern2mm\ P(only\ B) = \frac{6}{70}\kern2mm\ }$$ $$\hspace*{27 mm}\mathrm{= \frac{3}{35}\kern2mm\ }$$
[ Q 2.3.9 ]

$$\hspace*{2 mm}\mathrm{3.1\kern3mm\ P(B\ ∩\ C) = 0,17\kern2mm\ }$$ $$\hspace*{16 mm}\mathrm{\frac{x + 21}{200} = 0,17\kern2mm\ }$$ $$\hspace*{17 mm}\mathrm{x + 21 = 34\kern2mm\ }$$ $$\hspace*{24 mm}\mathrm{x = 13\kern2mm\ }$$
[ Q 3.1 ]

$$\hspace*{2 mm}\mathrm{3.2\kern3mm\ n(A) = 53 + 28 + 21 + 10\kern2mm\ }$$ $$\hspace*{18 mm}\mathrm{= 112\kern2mm\ }$$ $$\hspace*{10 mm}\mathrm{n(B) = 28 + 30 + 13 + 21\kern2mm\ }$$ $$\hspace*{18 mm}\mathrm{= 92\kern2mm\ }$$ $$\hspace*{10 mm}\mathrm{n(C) = 10 + 21 + 13 + 45\kern2mm\ }$$ $$\hspace*{18 mm}\mathrm{= 89\kern2mm\ }$$
[ Q 3.2 ]

$$\hspace*{2 mm}\mathrm{3.3.1\kern3mm\ P(A) = \frac{112}{200}\kern2mm\ }$$ $$\hspace*{22 mm}\mathrm{= 0,560\kern2mm\ }$$
[ Q 3.3.1 ]

$$\hspace*{2 mm}\mathrm{3.3.2\kern3mm\ P(B) = \frac{92}{200}\kern2mm\ }$$ $$\hspace*{22 mm}\mathrm{= 0,460\kern2mm\ }$$
[ Q 3.3.2 ]

$$\hspace*{2 mm}\mathrm{3.3.3\kern3mm\ P(C\bold{'}) = \frac{53 + 28 + 30}{200}\kern2mm\ }$$ $$\hspace*{23 mm}\mathrm{= 0,555\kern2mm\ }$$ $$\hspace*{18 mm}\mathrm{\bold{OR}\kern2mm\ }$$ $$\hspace*{14 mm}\mathrm{P(C\bold{'}) = 1 − P(C)\kern2mm\ }$$ $$\hspace*{23 mm}\mathrm{= 1 − \frac{89}{200}\kern2mm\ }$$ $$\hspace*{23 mm}\mathrm{= 0,555\kern2mm\ }$$
[ Q 3.3.3 ]

$$\hspace*{2 mm}\mathrm{3.3.4\kern3mm\ P(A\ ∪\ B) = \frac{112 + 30 + 13}{200}\kern2mm\ }$$ $$\hspace*{30 mm}\mathrm{= 0,775\kern2mm\ }$$ $$\hspace*{28 mm}\mathrm{\bold{OR}\kern2mm\ }$$ $$\hspace*{14 mm}\mathrm{P(A\ ∪\ B) = P(A) + P(B) − P(A\ ∩\ B)\kern2mm\ }$$ $$\hspace*{30 mm}\mathrm{= \frac{112}{200} + \frac{92}{200} − \frac{49}{200}\kern2mm\ }$$ $$\hspace*{30 mm}\mathrm{= 0,775\kern2mm\ }$$
[ Q 3.3.4 ]

$$\hspace*{2 mm}\mathrm{3.3.5\kern3mm\ P(A\ ∩\ B) = \frac{28 + 21}{200}\kern2mm\ }$$ $$\hspace*{30 mm}\mathrm{= 0,245\kern2mm\ }$$
[ Q 3.3.5 ]

$$\hspace*{2 mm}\mathrm{3.3.6\kern3mm\ P(A\ ∩\ C\ ∩\ B\bold{'}) = \frac{10 + 21 - 21}{200}\kern2mm\ }$$ $$\hspace*{40 mm}\mathrm{= 0,050\kern2mm\ }$$
[ Q 3.3.6 ]

$$\hspace*{2 mm}\mathrm{4.1\kern3mm\ P(A) = \frac{6}{15} = \frac{2}{5}\kern2mm\ }$$
[ Q 4.1 ]

$$\hspace*{2 mm}\mathrm{4.2\kern3mm\ P(C) = \frac{4}{15}\kern2mm\ }$$
[ Q 4.2 ]

$$\hspace*{2 mm}\mathrm{4.3\kern3mm\ P(B\bold{'}) = \frac{9}{15} = \frac{3}{5}\kern2mm\ }$$
[ Q 4.3 ]

$$\hspace*{2 mm}\mathrm{4.4\kern3mm\ P(A\ ∩\ B) = \frac{2}{15}\kern2mm\ }$$
[ Q 4.4 ]

$$\hspace*{2 mm}\mathrm{4.5\kern3mm\ P(B\ ∪\ C) = \frac{9}{15} = \frac{3}{5}\kern2mm\ }$$
[ Q 4.5 ]

$$\hspace*{2 mm}\mathrm{4.6\kern3mm\ P(A\ ∩\ C) = 0\kern2mm\ }$$
[ Q 4.6 ]

$$\hspace*{2 mm}\mathrm{4.7\kern3mm\ P(A\ ∪\ B\ ∪\ C)\bold{'} = \frac{2}{15}\kern2mm\ }$$
[ Q 4.7 ]

$$\hspace*{2 mm}\mathrm{4.8\kern3mm\ P(A\bold{'} ∩\ B\ ∩\ C\bold{'}) = \frac{3}{15} = \frac{1}{5}\kern2mm\ }$$
[ Q 4.8 ]

$$\hspace*{0 mm}\mathrm{5.1\kern3mm\ Number\ of\ boys\ that\ play\ oen\ sport\ only\ = 58\kern2mm\ }$$ $$\hspace*{15 mm}\mathrm{\therefore (x - 2) + (x + 1) + (2x + 7) = 58\kern2mm\ }$$ $$\hspace*{47 mm}\mathrm{4x + 6 = 58\kern2mm\ }$$ $$\hspace*{55 mm}\mathrm{x = 13\kern2mm\ }$$
[ Q 5.1 ]

$$\hspace*{0 mm}\mathrm{5.2\kern3mm\ Number\ of\ boys\ that\ only\ play\ Hockey\ = x - 2\kern2mm\ }$$ $$\hspace*{51 mm}\mathrm{\therefore 16 - 2 = 14\kern2mm\ }$$ $$\hspace*{7 mm}\mathrm{14\ boys\ play\ only\ Hockey\kern2mm\ }$$
[ Q 5.2 ]

$$\hspace*{0 mm}\mathrm{5.3\kern3mm\ Number\ of\ boys\ (H,\ T,\ S)\ = 18\kern2mm\ }$$ $$\hspace*{9 mm}\mathrm{But\ 16\ boys\ play\ all\ three.\kern2mm\ }$$ $$\hspace*{17 mm}\mathrm{\therefore 18 - 16 = 2\kern2mm\ }$$ $$\hspace*{7 mm}\mathrm{\therefore 2\ boys\ play\ Hockey\ and\ Tennis\kern2mm\ }$$ $$\hspace*{11 mm}\mathrm{but\ not\ Soccer\kern2mm\ }$$
[ Q 5.3 ]

$$\hspace*{0 mm}\mathrm{5.4\kern3mm\ Number\ of\ boys\ that\ play\ Hockey\kern2mm\ }$$ $$\hspace*{39 mm}\mathrm{=\ 14 + 2 + 16 + 4\kern2mm\ }$$ $$\hspace*{39 mm}\mathrm{=\ 36\kern2mm\ }$$
[ Q 5.4 ]

$$\hspace*{2 mm}\mathrm{5.5\kern3mm\ P(no\ sport) = \frac{numberl\ that\ do\ not\ play }{total\ number\ of\ boys}\kern2mm\ }$$ $$\hspace*{27 mm}\mathrm{\frac{24}{y} = 0,2\kern2mm\ }$$ $$\hspace*{30 mm}\mathrm{y = \frac{24}{0,2}\kern2mm\ }$$ $$\hspace*{32 mm}\mathrm{= 120\kern2mm\ }$$
[ Q 5.5 ]

$$\hspace*{2 mm}\mathrm{6.1\kern3mm\ P(2\ transports) = (4x + 1) + (3x + 3) + (x)\kern2mm\ }$$ $$\hspace*{29 mm}\mathrm{= 44\kern2mm\ }$$ $$\hspace*{19 mm}\mathrm{8x + 4 = 44\kern2mm\ }$$ $$\hspace*{27 mm}\mathrm{x = 5\kern2mm\ }$$
[ Q 6.1 ]

$$\hspace*{0 mm}\mathrm{6.2\kern3mm\ P(T\ and\ 2\ transports) = (4x + 1) + (3x + 3) + 5x \kern2mm\ }$$ $$\hspace*{35 mm}\mathrm{= 12x + 4\kern2mm\ }$$ $$\hspace*{35 mm}\mathrm{= 12(5) + 4\kern2mm\ }$$ $$\hspace*{35 mm}\mathrm{= 64\kern2mm\ }$$
[ Q 6.2 ]

$$\hspace*{0 mm}\mathrm{6.3\kern3mm\ P(both\ T\ and\ B) = 4(5) + 1 \kern2mm\ }$$ $$\hspace*{35 mm}\mathrm{= 21\kern2mm\ }$$
[ Q 6.3 ]

$$\hspace*{2 mm}\mathrm{6.4.1\kern3mm\ T ∪ B = (4(5) + 1) + 5(5)) \kern2mm\ }$$ $$\hspace*{23 mm}\mathrm{= 46\kern2mm\ }$$ $$\hspace*{12 mm}\mathrm{Taxis\ but\ no\ Busses = 80 - 46 \kern2mm\ }$$ $$\hspace*{50 mm}\mathrm{ = 34 \kern2mm\ }$$ $$\hspace*{12 mm}\mathrm{C ∪ B = 5(5) + (5) \kern2mm\ }$$ $$\hspace*{22 mm}\mathrm{= 30\kern2mm\ }$$ $$\hspace*{12 mm}\mathrm{Cars\ but\ no\ Busses = 59 - 30 \kern2mm\ }$$ $$\hspace*{52 mm}\mathrm{= 29 \kern2mm\ }$$ $$\hspace*{12 mm}\mathrm{Busses\ no\ Taxis\ and\ no\ Cars = 150 - (34 + (29 - (3(5) +3)) + 31 )\kern2mm\ }$$ $$\hspace*{65 mm}\mathrm{= 74\kern2mm\ }$$ $$\hspace*{25 mm}\mathrm{P(B) = \frac{74}{150}\kern2mm\ }$$ $$\hspace*{33 mm}\mathrm{= \frac{37}{75}\kern2mm\ }$$
[ Q 6.4.1 ]

$$\hspace*{2 mm}\mathrm{6.4.2\kern3mm\ Motor\ cars\ only = 59 - (3(5)+3 +5(5)+(5))\kern2mm\ }$$ $$\hspace*{32 mm}\mathrm{= 11\kern2mm\ }$$ $$\hspace*{17 mm}\mathrm{P(C\ only) = \frac{11}{150}\kern2mm\ }$$
[ Q 6.4.2 ]

$$\hspace*{2 mm}\mathrm{6.4.3\kern3mm\ P(All\ 3) = \frac{5(5)}{150}\kern2mm\ }$$ $$\hspace*{26 mm}\mathrm{= \frac{1}{6}\kern2mm\ }$$
[ Q 6.4.3 ]

$$\hspace*{2 mm}\mathrm{6.4.4\kern3mm\ P(some\ form) = \frac{150 - 31}{150}\kern2mm\ }$$ $$\hspace*{37 mm}\mathrm{= \frac{119}{150}\kern2mm\ }$$
[ Q 6.4.4 ]

$$\hspace*{2 mm}\mathrm{7.1.1\kern3mm\ y = 1 − 0,893 \kern2mm\ }$$ $$\hspace*{17 mm}\mathrm{= 0,11 \kern2mm\ }$$
[ Q 7.1.1 ]

$$\hspace*{2 mm}\mathrm{7.1.2\kern3mm\ x = 0,893 − 0,733 \kern2mm\ }$$ $$\hspace*{16 mm}\mathrm{= 0,16 \kern2mm\ }$$
[ Q 7.1.2 ]

$$\hspace*{2 mm}\mathrm{7.2\kern3mm\ P(at\ least\ 2\ events) \kern2mm\ }$$ $$\hspace*{17 mm}\mathrm{= 0,1 + 0,15 + 0,16 + 0,2\kern2mm\ }$$ $$\hspace*{17 mm}\mathrm{= 0,61 \kern2mm\ }$$
[ Q 7.2 ]

$$\hspace*{2 mm}\mathrm{7.3\kern3mm\ P(B) = 0,1 + 0,16 + 0,2 + 0,183\kern2mm\ }$$ $$\hspace*{19 mm}\mathrm{= 0,643 \kern2mm\ }$$ $$\hspace*{8 mm}\mathrm{\kern3 mm}\mathrm{P(C) = 0,15 + 0,16 + 0,2 + 0,05\kern2mm\ }$$ $$\hspace*{19 mm}\mathrm{= 0,56 \kern2mm\ }$$ $$\hspace*{9 mm}\mathrm{\kern3 mm}\mathrm{P(B\ and\ C) = 0,16 + 0,2\kern2mm\ }$$ $$\hspace*{28 mm}\mathrm{= 0,36 \kern2mm\ }$$ $$\hspace*{8 mm}\mathrm{\kern3 mm}\mathrm{P(B) \times P(C) = 0,643 \times 0,56\kern2mm\ }$$ $$\hspace*{30 mm}\mathrm{= 0,36 \kern2mm\ }$$ $$\hspace*{8 mm}\mathrm{\kern3 mm}\mathrm{∴ P(B\ and\ C) = P(B) \times P(C)\kern2mm\ }$$ $$\hspace*{8 mm}\mathrm{\kern3 mm}\mathrm{∴ B\ and\ C\ are\ inpendnt.\kern2mm\ }$$
[ Q 7.3 ]

8.1  a = 15,    b = 1,    c = 38,     d = 3,    e = 17
[ Q 8.1 ]
$$\hspace*{0 mm}\mathrm{8.2.1\kern3mm\ P(N\ or\ V) = P(N) + P(V) − P(N\ and\ V)\kern2mm\ }$$ $$\hspace*{28 mm}\mathrm{= \frac{21}{100} + \frac{7}{100} − \frac{3}{100}\kern2mm\ }$$ $$\hspace*{28 mm}\mathrm{= \frac{25}{100} = 0,25\kern2mm\ }$$
[ Q 8.2.1 ]

$$\hspace*{2 mm}\mathrm{8.2.2\kern3mm\ P(S\ and\ V) = \frac{2}{100}\kern2mm\ }$$ $$\hspace*{30 mm}\mathrm{= 0,02\kern2mm\ }$$
[ Q 8.2.2 ]

$$\hspace*{2 mm}\mathrm{8.2.3\kern3mm\ P(at\ least\ 2) = \frac{15 + 1 + 1 + 2}{100}\kern2mm\ }$$ $$\hspace*{39 mm}\mathrm{= 0,19\kern2mm\ }$$
[ Q 8.2.3 ]

$$\hspace*{2 mm}\mathrm{8.2.4\kern3mm\ P(only\ S\ or\ only\ V) = \frac{38 + 3}{100}\kern2mm\ }$$ $$\hspace*{41 mm}\mathrm{= 0,41\kern2mm\ }$$
[ Q 8.2.4 ]