1.
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PROBABILITY II
Notes:
Probability of an event is the likelihood an event to occur.
1. The probability that event A occurs = Number of outcomes of A
Total number of possible outcomes
P (A) = n(A)
n(S)
and the value of probality A is between 0 and 1 = 0 ≤ P(A) ≤ 1
2. S - is the sample space ( the set of possible outcomes)
3. (a) If A is an impossible event , then P(A) = 0
(b) If A is a confirmed event , then P(A) = 1
4. example
5 7 8 10 13 16
Probability of an odd number is chosen
S = { 5, 7 ,8, 10, 13, 16 }
Let A be the event of getting an odd number
n (S )
3
=
6
1
=
2
1
2.
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5. If A is an event , the A’ is the complementary event of A , that is
P ( A’) = 1 − P(A)
example,
A box contains a total of 100 red and green marbles. The probability of
3
choosing a red marble is , find the probability of choosing a green marble.
5
Let A be the event of choosing a red marble and A’ be the event of
choosing a green marble.
P( A’) = 1 − P(A)
3
= 1 −
5
2
=
5
2
Therefore , the probability of getting a green marble is
5
6. Probability of a Combined Event.
A combined event is made up of two or more events that happen in either an “or”
or and “and” condition.
Outcomes of a Combined Events, ( 1) Event A or Event B = A U B
(2) Event A and Event B = A I B
Example ,
Two dice are rolled at the same time .
Let A = Event of obtaining two even numbers in the two dice
B = Event that the sum of the numbers from the dice is less than 10
2
3.
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Solution
All possible outcomes when two dice are rolled
S ={(1,1), (2,1), (3,1), (4,1),(5,1),(6,1), (1,2), (2,2), (3,2), (4,2), (5,2),(6,2),
(1,3), (2,3), (3,3), (4,3), (5,3), (6,3), (1,4), (2,4), (3,4), (4,4), (5,4), (6,4)
(1,5), (2,5), (3,5), (4,5), (5,5), (6,5),(1,6), (2,6), (3,6), (4,6), (5,6), (6,6) }
n(S) = 36
A = Event of obtaining two even numbers in the two dice
A = { (2,2), (2,4), (2,6),(4,2), (4,4), (4,6), (6,2), (6,4), (6,6) }
n(A) = 9
B = Event that the sum of the numbers from the two dice is less than 10
B = { (1,1), (1,2), 1,3), (1,4),(1,5), (1,6), (2,1), (2,2), (2,3), (2,4), (2,5), (2,6)
(3,1), (3,2), (3,3), (3,4), (3,5), (3,6), (4,1),(4,2),(4,3), (4,4), (4,5), (5,1),
(5,2), (5,3), (5,4), (6,1), (6,2), (6,3) }
n(B) = 30
The outcomes of combined events
(a) A or B = A U B
= {(1,1), (1,2), (1,3),(1,4),(1,5),(1,6),(2,1),(2,2),(2,3),(2,4),(2,5),(2,6),
(3,1),(3,2),(3,3),(3,4),(3,5),(3,6),(4,1),(4,2),(4,3),(4,4),(4,5),(4,6),
(5,1),(5,2),(5,3),(5,4),(6,1),(6,2),(6,3),(6,4),(6,6) }
n ( A U B ) = 33
P ( AU B ) = n ( AU B )
n(S)
33 11
= =
36 12
3
4.
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(b) A and B = A I B
= {(2,2), (2,4), (2,6), (4,2),(4,4),(6,2)
n( A I B)= 6
n( A I B) = n( A I B)
n (S)
6
=
36
1
=
6
7. If A I B ≠ Ø , therefore P(A U B ) = P(A) + P(B) − P( A I B )
If A I B = Ø , therefore P(A U B ) = P(A) + P(B)
example
If a dice is rolled, the possible outcomes will be
S = {1, 2, 3, 4, 5, 6 } and find the probability that
( i). number 4 or odd number is obtained
(ii). number 2 or even number is obtained
(iii). even number or prime number is obtained
( i). Let A = event of obtaining number 4. Let B = event of obtaining odd number
A= {4} B = { 1, 3, 5 }
n( A) = 1 n( B ) = 3
P(A U B ) = P(A) + P(B)
= n( A) + n( B )
n(S) n(S)
1 3
= +
6 6
4
=
6
2
=
3
4
5.
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(ii). Let A = event of obtaining number 2 . Let B = event of obtaining even number
A={2} B = { 2, 4, 6 }
n( A ) = 1 n(B) = 3
P(A U B ) = P(A) + P(B) − P( A I B )
= n( A) + n( B ) − P ( A I B )
n(S) n(S) n(S)
1 3 1
= + −
6 6 6
3
=
6
1
=
2
(iii). Let A = event of obtaining even number. Let B = event of obtaining prime
= { 2, 4, 6 } number
n(A) = 3 = { 2, 3, 5 }
(B) = 3
P(A U B ) = P(A) + P(B) − P( A I B )
= n( A) + n( B ) − P ( A I B )
n(S) n(S) n(S)
3 3 1
= + −
6 6 6
5
=
6
5
6.
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Exercise 1 Paper 2
1. A box contains a total of 42 yellow and green marbles. 14 of them are yellow.
A marble is picked randomly from the box. Find the probability of picking
a green marble
2. A tray contains some soft-boiled eggs and some hard-boiled eggs. The probability
2
of choosing a soft –boiled egg is . Find the probability of choosing a hard-boiled
5
egg.
3. There are 5 English books, 4 Mathematics books and 3 Science books on a table.
A book is choosen at random from the table, find the probability of choosing a
Mathematics or Science book.
3
4. The probability of winning a chess competition between team P and team Q are
7
7
and respectively . Find the probability that
15
(a) team P loses ,
(b) team P wins but team Q loses,
(c) both teams win,
(d) at least one team wins.
5. Bag A contains six red balls and two purple balls. Bag B contains eight red balls
and four purple balls. A ball is taken out randomly from
bag A followed by another ball from bag B .
Write the probability in fraction form of
(a) both balls being red ,
(b) both balls being purple ,
(c) both balls being of the same colour ,
(d) both balls being different colours.
6. A parcel consists of five cards numbered 1 to 5. A card is taken out randomly and its
number is recorded. After replacement, another card is taken out from the parcel and
its colour is recorded. Find the probability that
( a ) the sum of the numbers of the two cards selected is an even number.
( b ) the first card is ‘ 1 ‘ and the second card is an odd number,
( c ) the number of the first card divided by the number of the second card is less than
1
6
7.
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7. A fair coin is tossed thrice. State , as a fraction , the probability of
(a) getting 3 tails,
(b) getting 2 heads and 1 tail,
(c) getting 2 heads and 1 tail,
(d) getting at least 1 head.
8. Two students are going to be selected from a group 24 boys and 10 girls. Find the
pobability that
( a ) both students are girls,
( b ) both students are of a different gender,
( c ) both students are boys.
9. A box contains 50 electronic chips and 20 of them are damaged. Two chips are
selected randomly from the box. Calculate the probability that
( a ) two chips are in good condition,
( b ) at least one of the chips are in good condition,
( c ) both chips are damaged.
10. Figure (1) is a Venn diagram showing the involvement of 60 students in three types
of games ,badminton,basket ball and volley ball.
Badminton (B) Basket ball (K)
16 y
10
4
x 12
fig: 1 8
Volley ball (V)
1
Given the probability of choosing a badminton player and a volleyball is . Find
10
the probability of
( a ) choosing a player who plays both badminton and basket ball,
( b ) choosing two students playing badminton.
7
9.
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TOPIC 7 : PROBABILITY II
EXERCISE 2 (Paper 2 )
1. A bag contains 2 black and 2 white balls. Two balls are taken from the bag at
random. Find the probability that
a) both balls are black.
b) at least one ball is white .
c) the balls are of the same colour.
2. A bag contains three cards, one marked with the letter A and two with the letters
B. A card is drawn from the bag and then replaced before a card is drawn again.
Find the probability that
a) both cards have the letter A
b) one card has the letter A and another card has the letter B.
3. A box X contains three cards bearing the numbers 1, 2 and 3. Another box
contains four cards bearing the numbers 2, 3, 4 and 5. A card is chosen at random
from each box. Calculate the probability that
a) the cards bear the same number,
b) the sum of the two numbers on the cards is less than 7,
c) the product of the two numbers on the card is at least 8.
10.
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4. A bag contains 2 red sweets and a green sweet. A sweet is taken from the bag and
its colour noted. Without replacing the first sweet, another sweet is taken from the
bag. Find the probability that
a) both the sweets are red,
b) the first sweet is green,
c) at least one sweet is green.
5. X = {7, 8, 9, 10} and Y = { 1, 3 }. An element x is selected randomly from X and
placed in Y. A second element y is selected from Y.
a) Draw a tree diagram to show the possible outcomes. Then list the sample
space for the experiment.
b) Find the probability that the sum x + y is
( i) a prime number (ii) an even number and is greater than 10
c) Find the probability that the product x y is
( i) an odd number ( ii) an even number or a number greater than 12
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6. A bag contains 4 red balls and 8 purple balls. A ball is selected randomly and its
colour is recorded before being replaced into the same bag. Another ball is collected
and its colour is recorded.
a) State the probability of selecting a red ball at the first trial.
b) Find the probability of getting a red ball followed by a purple ball.
7. The probability of winning a carom competition between team A and team B are
3 7
and respectively. Find the probability that
7 15
a) team A loses.
b) team A wins but team B loses
c) both teams win
d) at least one team wins
12.
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8. 1 1 2 3 2 4 2
7 labelled marbles are kept in a tin.
a) If a marbles is taken out randomly from the tin, state the probability that the
marble is number 2.
b) If two marbles are taken out one after another , calculate the probability
i) that both marbles are labelled ‘1’,
ii) of getting at least one marble labelled ‘2’
9. 15 cards are kept in two bags, P and Q, as shown in Figure 1.
Bag P B E R B U D I
B E R B A K T I
Bag Q
Figure 1
A card is taken out randomly from the bag P and put into bag Q. Then a card is taken
out randomly from bag Q. Calculate the probability that
a) a card put into bag Q is labeled ‘B’ and a card taken out from bag Q is ‘B’ too.
b) a card taken out from bag P is not consonant and a card taken out from bag Q is a
consonant
13.
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10. The table shows the number of pens and markers placed in different
compartments in a cabinet. A pen and a marker are picked at random from the
cabinet.
Item
Colour Pen marker
Black 13 10
Blue 25 5
Red 10 7
Table 1
Calculate the probability that
a) a red pen is picked,
b) a black pen and a blue marker are picked ,
c) a pen and a marker of the same colour are picked.
14.
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TOPIC 7: PROBABILITY 11
DIAGNOSTIC TEST
1. A box contains 1 yellow and 2 red balls which are identical. A ball is picked at
random from the box with its colour being noted and is returned to the box. Then a
second ball is picked.
(a) Draw a tree diagram to represent the possible outcomes using Y to represent the
yellow ball and R1 and R2 to represent the red balls.
(b) Find the probability that a yellow and a red ball are picked.
Answer :
2. Four cards which are labeled from 1 to 4 are placed in a box. Two cards are drawn at
random. Write down the sample space by listing the possible outcomes. Find the
probability that
(a) the sum of the two numbers drawn is 4.
(b) the first number drawn is even and the second number is greater than 2.
Answer :
15.
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3. A code is formed by a letter and a number selected at random from the word ‘P E N’
and the set { 11, 12, 13, 14 } respectively.
(a) Draw a tree diagram to show all possible outcomes .
(b) Find the probability of choosing a code that contains
(i) the number 12.
(ii) a vowel or a odd number.
Answer :
4. F O R M
DIAGRAM 2
The four cards in Diagram 2 are placed into a box. Two cards are selected one by one
at random from the box. The first card selected is not replaced in the box before the
second card is selected. Write down the sample space by listing all possible outcomes
Find the probability that
(a) the first card is a vowel or the second card is the letter R
(b) the first card and the second card selected are consonants.
Answer :
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5. Diagram 1 shows two spinners I and II
I II
3 4
5 10 2 15
4 1 17 8
15 3
DIAGRAM 1
If the dials are spun, calculate the probability that
(a) The pointer on spinner I points at a number that is smaller than 10.
(b) Both pointers on spinners I and II stop point at prime numbers.
(c) Both pointers on spinners I and II point at the same number.
Answer :
TOPIC 7 : PROBABILITY II
EXERCISE 2
ANSWERS
1 5 1
1. a) b) c)
6 6 3
1 4
2 a) b)
9 9
1 3 5
3. a) b) c)
6 4 12
1 1 2
4. a) b) c)
3 3 3
5. a) S= { (7,7) , (7,1) , (7,3) , (8,8) , (8,1) , (8,3) , (9,9) , (9,1) ,
(9,3) , (10,10) , (10,1) , (10,3) }
1 5
b) i) ii)
4 12
1 5
c) i) ii)
2 6
17.
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1 2
6. a) b)
3 9
4 8 1 73
7. a) b) c) d)
7 35 5 105
3 1 5
8. a) b) i) ii)
7 21 7
2 5
9. a) b)
21 21
5 65 325
10. a) b) c)
24 1056 1056
TOPIC 7: PROBABILITY 11
DIAGNOSTIC TEST
ANSWERS
1.
(a)
Y YY
Y R1 Y R1
R2 Y R2
Y R1 Y
R1 R1 R1 R1
R2 R1 R2
Y R2 Y
R2 R1 R2 R1
R2 R2 R2
(b) A = {(Y, R1), (Y, R2), (R1, Y), (R2, Y)}
4
P(A) =
9
2.
S = {(1, 2), (1, 3), (1, 4), (2, 1), (2, 3), (2, 4), (3, 1), (3, 2), (3, 4), (4, 1), (4, 2), (4, 3)}
(a) F = {(1, 3), (3, 1)}
2 1
P(F) = =
12 6
(b) G = {(2,3) , (2, 4), (4, 3), (4 ,4)}
18.
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4 1
P(G) = =
12 3
3.
(a)
11
12
P 13
14
11
12
E 13
14
11
12
N 13
14
(b) (i) T = {(P, 12), (E, 12), (N, 12)}
3 1
P(T) = =
12 4
(ii) V = {(P, 11), (P, 13), (E, 11), (E, 12), (E, 13), (E, 14), (N, 11), (N, 13)}
8 2
P(V) = =
12 3
4.
S = {(F, O), (F, R), (F, M), (O, F), (O, R), (O, M), (R, F), (R, O), (R, M), (M, F), (M, O),
(M, R)}
(a) W = {(F, R), (O, F), (O, R), (O, M), (M, R)}
5
P(W) =
12
(b) C = {(F, R), (F, M), (R, F), (R, M), (M, F), (M, R)}
6 1
P(C) = =
12 2
5.
4 2
(a) =
6 3
(b) n(S) = 36
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