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5 / 6 MARKS
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SAMPLE SPACE
An experiment is a process or an operationwith an outcomes.
Toss a balanced dice once and observe
its uppermost face.
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When toss the coin, we can get only2 results:
1. Head2. Tail
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The set of all possible outcomes of anexperiment is called the sample space.It usually denoted by S.
Example 1:
En. Adam has a fruit stall that sells bananas, apples,watermelons, papayas and durians. Students of class5M are asked to select their favorite fruit from thefruits at En. Adams stall.
Solution:S = { banana, apple, watermelon, papaya, durian}
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Example 2:
A month is randomly selected from ayear. Describe the sample space ofthis experiment by using set notation.
TOPIC MENU
Solution:
S= { January, February, March, April, May,
June, July, August, September, October,November, December}
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Sspace,samplein theoutcomespossibleofnumber
AeventforoutcomesofnumberP(A)
P(A) = 1 Event A is sure to happen
P(A) = 0 Event A will not to happen
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EXAMPLE 3 :
2 135 6 10 15
The above diagram shows six number cards in a box. Acard is picked at random from the box, find theprobability of obtaining
a) A prime number
b) A whole number
c) A multiple of 7
d) A number between 3 and 7
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Solution: Sample space , S = { 2, 5, 6, 10, 13, 15}
n(S) = 6
a) A prime number
(S)n(A)nP(A)
A = { 2, 5,13}
n(A) = 3
6
3P(A)
2
1
b) A whole number
(S)n
(A)nP(A)
A = { 2, 5, 6, 10, 13, 15}
n(A) = 6
6
6P(A)
1
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c) A multiple of 7 d) A number between 3and 7
(S)n
(A)nP(A)
A = { }
n(A) = 0
6
0
P(A)
0
Sample space ,
S = { 2, 5, 6, 10, 13, 15}n(S) = 6
A = { 5, 6 }n(A) = 2
(S)n
(A)nP(A)
6
2P(A)
3
1
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Calculating the probability of a combined event by listingthe outcomes.
The steps to calculate the probability of the combined events of
a) A or B b) A and B
Step 1 : List out all outcomes of the sample spaces.
Step 2 : List out all outcomes of the combined events.
a) A or B
Determine n (A B)
a) A and B
Determine n (A B)
Step 3 : Calculate.
)(
)(
Sn
BAn
)( BAP
)(
)(
Sn
BAn
)( BAP
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Calculating the probability of a combined event involvingthe sum of probability
A B
S
P (A B) = P(A) + P(B)
A B
S
P (A B) = P(A) + P(B) P(A B)
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Example 4 (SPM 2007)
The diagram shows ten labelled cards in two boxes.
A 2 B C D G3 E 4 F
A card is picked at random from each of the box.
By listing the outcomes, find the probability that
a) Both cards are labelled with a number. [ 2 marks ]
b) One card is labeled with a number and the other is labelled
with a letter. [ 2 marks ]
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Solution:
Step 1: List all the sample space
S = {
A 2 B C
D G3 E 4 F
(A,3), (A,E), (A,F), (A,G),(A,4),(A,D),
(2,3), (2,E), (2,F), (2,G),(2,4),(2,D),
(B,3), (B,E), (B,F), (B,G),(B,4),(B,D),
(C,3), (C,E), (C,F), (C,G),(C,4),(C,D),
1 mark
}
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}
a) Both cards are labelled with a number. [ 2 marks ]
S = { (A,3), (A,E), (A,F), (A,G),(A,4),(A,D),
(2,E), (2,F), (2,G),(2,4),(2,D),
(B,3), (B,E), (B,F), (B,G),(B,4),(B,D),
(C,3), (C,E), (C,F), (C,G),(C,4),(C,D),
(2,3),
(2,3)(2,4),={ }
24
2
12
1
A 2 B C
Method II
4
1
D G3 E 4 F 62
4
1
6
2
12
1
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S = { (A,3), (A,E), (A,F), (A,G),(A,4),(A,D),
(2,E), (2,F), (2,G),(2,4),(2,D),
(B,3), (B,E), (B,F), (B,G),(B,4),(B,D),
(C,3), (C,E), (C,F), (C,G),(C,4),(C,D),
(2,3),
b) One card is labeled with a number and the other is labelled
with a letter. [ 3 marks ]
(2,G),
(A,3),
(A,4), (2,E),
(2,F),(2,D),
(B,3), (B,4), (C,3),(C,4),
={
}
24
10
12
5 A 2 B C
Method II
D G3 E 4 F
6
4
4
1 +
6
2
4
3
12
5
}
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Example 5 (SPM 2008)
The diagram shows three numbered cards in box P andtwo cards in box Q.
2 3 6 Y R
A card is picked at random from box P and then a card ispicked at random from box Q
By listing the outcomes, find the probability that
a) A card with an even number and the card labelled Y are
picked. [ 2 marks ]
b) a card with a number which is multiple of 3 or the card
labelled R are picked. [ 3 marks ]
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Solution:
Step 1: List all the sample space
2 3 6 Y R
S = { (2, Y), (2, R), (3, Y), (3, R), (6, Y), (6, R)}
a) A card with an even number and the card labelled Y are picked.[ 2 marks ]
(2, Y), (6, Y),=
6
2
3
2
Method II
2 3 6 Y R
3
1
2
1
2
1
3
2
3
1
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S = { (2, Y), (2, R), (3, Y), (3, R), (6, Y), (6, R)}
b) a card with a number which is multiple of 3 or
the card labelled R are picked. [ 3 marks ]
)(
)(
Sn
BAn
)(or)( BPAP
Event A
Event B
)( BAP
6
5
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Method II
2 3 6 Y R3
2
2
1
)(or)( BPAP
)( BAP )()()( BAPBPAP
3
2 +
2
1
3
2x
2
1
3
1
6
34
6
5
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Example 6 (SPM 2009)
The diagram shows five cards labelled with letter.
S M I L E
All these cards are put into a box. A two-letter code is to beformed by using any of these cards. Two cards are picked atrandom, one after another, without replacement
a) List all the sample space [ 2 marks ]
b) List all the outcomes of the events and find the probability
that
i. The code begins with letter M,
ii. The code consists of two vowels or two consonents.
[ 4 marks ]
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Solution:
S = { ( ),
( ),
( ),
( ),
( ),
( ),
( ),
( ),
( ),
( ),
( ),
( ),
( ),
( ),
( ),
( ),
( ),
( ),
( ),
( ),
M
S,
M,
L,
E,
I,
S,
M,
L,
E,
I,
S,
M,
L,
E,
I,
S,
M,
L,
E,
I,
I L E
S
S
S
S
I
M
M
M
M
I L
L E
L E
E
}
a) List all the sample space [ 2 marks ]
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b) List all the outcomes of the events and find the probability
that
i. The code begins with letter M [ 4 marks ]
S = { ( ),
( ),
( ),
( ),
( ),
( ),
( ),
( ),
( ),
( ),
( ),
( ),
( ),
( ),
( ),
( ),
( ),
( ),
( ),
( ),
M
S,
M,
L,
E,
I,
S,
M,
L,
E,
I,
S,
M,
L,
E,
I,
S,
M,
L,
E,
I,
I L E
S
S
S
S
I
M
M
M
I
M M
L E
L E
E
}
20
4
5
1
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ii. The code consists of two vowels or two consonents.
S = { ( ),
( ),( ),
( ),
( ),
( ),
( ),( ),
( ),
( ),
( ),
( ),( ),
( ),
( ),
( ),
( ),( ),
( ),
( ),
M
S,
M,
L,
E,
I,
S,
M,
L,
E,
I,
S,
M,
L,
E,
I,
S,
M,
L,
E,
I,
I L E
SS
S
S
I
M
M
M
I
I M
L EL E
E
}
20
8
5
2
Method II
S M I L E
4
2
5
3+
4
1
5
2
5
2