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PROGRAM DIDIK CEMERLANG AKADEMIK
SPM
ADDITIONAL MATHEMATICS
FORM 5
MODULE 12
PROBABILITY
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PROBABILITY
Contents
12.0 Concept Map
12.1 Probability Of An Event
Page
2
3
12.2 Probability Of Two Events.
4
12.3 Probability of Mutually Exclusive Events
5
12.4 Probability Of Independent Events.
6-7
SPM Questions
8
Assessment test
8
Answers
10
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12.0 CONCEPT MAP.
Experiment
The result of
an experiment
A process to obtain
observations
Possible outcomes
The set of all
possible outcomes
Sample space, S
An Event, A
_______________
_______________
_______________
______________
______________
______________
Mutually Exclusive
Events
Independent Event
PROBABILITY
P(A) =
P(A  B) = P(A) + P(B) – P(A  B)
The complement of the event A.
P(A’) = __________
P(two events A and B that are
mutually exclusive) is
P(A  B) = P(A) + P(B)
A  B= 
P(two or three independent
events)
P(A  B) = P(A).P(B)
P(A  B) = P(A).P(B).P(C)
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12.1
PROBABILITY OF AN EVENT
Example 1.
Question
Box A contains 3 black balls, 2 green
balls and 5 red balls. A ball is drawn at
random from box A. Calculate the
probability that the colour of the ball is
(a) black
(b) not black
(c) yellow
Answer
(a) Let R represent the event that a black ball is drawn.
n(R) 3
P(R) =
 .
n(S) 10
(b) P(the ball is not black)
3
7
=P(R’) = 1 – P(R) = 1 –
=
10 10
(c) P(the ball drawn is yellow) = 0.
An event that is impossible to occur
because there are no yellow balls in
the box
Exercises 1.
No.
1.
Questions
A box contains 8 cards where each card is marked with an
alphabet from the word ‘TAMBAHAN’. If a card is chosen
at random, calculate the probability that
(a) the card with the alphabet B is chosen.
(b) a card with vowel is chosen.
2.
There are 24 red marbles and y green marbles in a bag. A
marble is drawn at random from the box. Given that the
1
probability of drawing a green is , calculate the value of y.
3
3.
A bag contain x green marbles, y blue marbles and 5 brown
marbles. A marbles is drawn at random and the probability
1
of getting a brown marble is . Write down the equation
3
relating x and y.
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3
Answers
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12.2
PROBABILITY OF THE TWO EVENTS
Example 2.
Question
Answer
Let
A represent the event that the number on the chosen card is a
The above figure shows six
multiple of 3, and
numbered cards. A card is chosen
B represent the event that the number on the chosen card is a
at random. Calculate the probability factor of 12.
that the number on the chosen card A = {3, 6, 9}, n(A)= 3
(a) is a multiple of 3 and a factor B = {2, 3, 4, 6}, n(B) = 4
of 12
A  B = {3, 6}
(b) is a multiple of 3 or a factor of A  B = {2, 3, 4, 6, 9}
12.
2 1
(a) P(A  B) =  .
6 3
5
(b) P(A  B) =
6
2
3
4
6
8
9
Alternative method
P(A  B) = P(A) + P(B) – P(A  B)
3 4 2
=  
6 6 6
5
= .
6
Exercises 2
No.
1.
Questions
A dice is thrown once. Calculate the probability that the
score on the dice either an odd number or a prime number.
2.
A card is chosen at random from a bag which contains the
26 different letters of the alphabet. Find the probability that
(a) the card chosen has a letter from the word ‘BAHASA’
(b) the card chosen is not a letter from the word ‘KACA’
3.
The set X and Y are given as follows:
X = {1, 3, 5, 7, 9}
Y = {2, 4, 6}
A number is chosen at random from set X and another
number from set Y. Calculate the probability that the sum of
the number is 9 or the product of the number is 18.
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4
Answers
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12.3
PROBABILITY OF MUTUALLY EXCLUSIVE EVENTS
Example 3.
Question
A box contains 5 red balls, 3 yellow balls
and 4 green balls. A ball is chosen at
random from the box. Calculate the
probability that the balls drawn neither a
yellow nor a green.
Solution
3
.
12
4
P(green) =
12
P (yellow) =
P(yellow or green) =
3 4 7
+ = .
12 12 12
Exercises 3
No.
1.
Questions
A fair dice is thrown. Let x be the event when
the dice shows 5 and Y be the event when the
dice shows an even number.
(a) Are the two events mutually exclusive?
(b) Find the probability that 5 or even
number is the outcome.
2.
A marble is drawn at random from a box
containing 3 black marbles, 4 green marbles
and 5 white marbles.
(a) What is the probability of drawing a
black or a green marble?
(b) What is the probability of drawing
neither a black nor a white marble?
3.
Box T contains three card numbered 1, 2 and
3. Box U contains three card numbered 1, 3
and 4. A card is drawn at random from box T
and at the same time, another card is drawn
from box U. Calculate the probability that the
two numbers drawn have the same value or a
sum of 3.
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5
Answers
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12.4
PROBABILITY OF INDEPENDENT EVENTS
Example 4
No
Questions
1.
Box C contains 4 black marbles and 6
yellow marbles. A marbles is chosen
at random from box C, its colour is
noted and the marbles is noted and the
marbles is returned to the box. Then a
second marbles is chosen. Determine
the probability that
(a) both the marbles are black.
(b) the two balls are of different
colours.
(c) at least one of the balls chosen is
yellow.
2.
The probability that participants K, L
and M will win a dancing contest are
1 1
1
, and respectively. If the events
6 7
8
of each participant winning are
independent, calculate the probability
that
(a) only L wins,
(b) two participants win.
Solutions
4
10
6
10
4
10
Black
Black
6
10
4
10
Yellow
Yellow
Black
6
10
(a) P(black  black)=
Yellow
4 4
4

=
10 10 25
(b) P(same colours)
= P(black  black) + P(yellow  yellow)
 4   6 6  13
=   +   =
.
 25   10 10  25
4
21
(c) 1 – P(both blacks) = 1 –
=
25 25
K
L
1
7
Win
k
1
6
6
7
Win
ck
Lose
7
8
1
8
7
8
1
8
5
6
lose
1
7
6
7
Win
k
Lose
7
8
1
8
7
8
(a)
M
1
8
Win
ck
lose
Win
ck
lose
Win
ck
lose
Win
ck
lose
P(only L wins)
= P(K’  L  M’)
= P(K’)  P(L)  P(M’).
5 1 7
 
6 7 8
5
=
48
=
(b) P(2 participants win)
= P(K  L  M’) + P(K  L’  M) +
P(K’  L  M).
1 1 7 1 6 7 5 1 1
6 7 8 6 7 8 6 7 8
=          =
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3
56
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Exercises 4
No.
Questions
1.
A bag has 8 green cubes and 3 red
cubes. Two cubes are drawn from the
bag at random, one after the other,
without replacement. Calculate the
probability that the green cube and a
red cube are drawn.
2.
Hasan competes with John in the
finals of a squash competition. The
competition will end when a player
wins three sets. The probability that
2
Hasan will win any set is . Calculate
3
the probability that
(a) the competition ends after only
three sets.
(b) Hashim is the winner after
playing four sets.
3.
Bag B contains 6 red balls and 5
yellow balls. A ball is drawn at
random from bag B. The ball is then
put into bag D that contains 4 red balls
and 3 yellow balls. After that, another
ball is drawn at random from bag D.
Calculate the probability that the ball
drawn from bag B is of the same
colour as the ball drawn from bag D.
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Solutions
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PAST YEAR QUESTIONS.
No.
1.
Questions
A box contains 6 white marbles and k black
marbles. If a marble is picked randomly from
the box, the probability of getting a black
3
marble is . Find the value of k.
5
SPM 04(No.24 / Paper 1).
2.
Table 1 shows the number of coloured cards
in a box
Colour
Black
Blue
Yellow
Solutions
Number of cards
5
4
3
Two cards are drawn at random from the box.
Find the probability that both cards are of the
same colour.
SPM.05(No. 24 / Paper 1)
ASSESSMENT TEST
No.
Questions
1.
There are 48 blue balls and y red balls in the
box. A ball is drawn at random from the box.
Given that the probability of drawing a red
1
ball is , calculate the value of y.
3
2.
Two dice, one white and one black, are
thrown together. Calculate the probability that
the score on the white dice is twice the score
on the black dice.
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Solutions
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No.
3.
Questions
A box contains 40 marbles. The colours of
the marbles are green and red. If a marble is
drawn at random from the box, the probability
2
that a green marble is drawn is . Calculate
5
(a) the number of red marbles in the box,
(b) the number of red marbles that have to be
added to the box such that the probability
3
to drawn a red marble becomes .
5
4.
Bag I contains 2 blue marbles and 6 black
marbles while bag II contains 3 blue marbles
and 4 black marbles. If a marble is chosen at
random from each bag, calculate the
probability that
(a) both the marbles are black,
(b) the marble from bag I is blue and the
marble from bag II is black.
(c) At least one of the marbles chosen is
black.
5.
Two six-faced unbiased dice are thrown
together. Calculate the probability that
(a) the sum of two numbers is 8.
(b) The difference of two numbers is 5,
(c) The sum of two numbers is 8 or The
difference of two numbers is 5.
6.
In a soccer match between team B and team
D, the result can be a draw or a win for team
B or a win for team D. The probability that
1
1
team B and team D will win are and . In
3
2
two matches, calculates the probability that
team B wins once and draw once.
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Solutions
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ANSWERS
Exercises 1
1
1. (a)
8
2. 12
3. x + y = 10
Exercises 2
2
1.
3
2
2. (a)
13
4
3.
15
3
(b)
8
2. (a)
3.
7
12
25
44
PAST YEAR QUESTION
1. 9
19
2.
66
(b)
23
26
ASSESSMENT TEST
1.
3.
24
1
12
24
4.
(a)
2.
Exercise 3
1. (a) Yes
3.
2
3
2
(b)
3
(b)
1
3
5.
6.
3
7
5
(a)
36
7
(c)
36
1
9
Exercise 4
1.
24
55
2. (a)
1
3
(b)
8
27
.
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10
1
7
1
(b)
18
(b)