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7 . PROBABILITY
7.1 Probability of an event
a. The probability of an event A is defined by
n( A)
P(A) = and 0 ≤ P(A) ≤ 1
n( S )
Where S is the sample space which consists of all the possible outcomes, which
are equally likely to occur in an experiment and A subset S.
b. An event with probability 0 is an impossible event.
c. An event with probability 1 is a confirmed event.
Example 1
A bag contains 4 white and 6 black balls.A ball is chosen at random from the bag.Find
the probability of chosing
i. a white ball
ii. a black ball
iii. a red ball
Solution
n(S) = 10
i. Let A be the event of choosing a white ball.
n(A) = 4
n( A) 4 2
P(A) = = =
n( S ) 10 5
ii. Let B be the event of chosing a black ball.
n(B) = 6
n( B ) 6 3
P(B) = = =
n( S ) 10 5

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iii. Let C be the event of chosing a red ball.
n(C) = 0
n(C ) 0
P (C) = = =0
n( S ) 10
d. If an experiment is repeated for n times and P(A) is the probability that event A
occurs, then the expected number of times that event A occurs is n x P(A).
7.2 Probability of the compliment of an event
The compliment of an event A, that is A’ , is an event consisting of outcomes thar are not
in A but in S.
P(A’) = 1 – P(A)
P(A) + P(A’) = 1
Example
3
In an archery training, the probability of an archer hitting the target is . Find the
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probability that he misses the target.
Solution
Let A be the event of hitting the target.
3
Then, P(A) =
8
3 5
Thus, P(A’) = 1 - =
8 8
7.3 Probability of a combined event
a. Probability of the combined event A or B
n( AU B )
P( A or B) =
n( S )
b. Probability of the combined event A and B
n( A I B )
P(A and B) =
n( S )

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Example
Two fair dice are rolled simultaneously.Find the probability that
i. one of the numbers obtained is 5 or the sum of the numbers is 7.
ii. one of the numbers obtained is 5 and the sum of the numbers is 7.
Solution
N(S) = 36
Let A = Event that one of the numbers obtained is 5
= {(1,5),(2,5), (3,5),(4,5),,(6,5),(5,6),(5,5),(5,4),(5,3),(5,2),(5,1)}
And B= Event that the sum of the numbers is 7
= {(1,6),(2,5),(3,4),(4,3),(5,2),(6,1)}
i. A U B = {(1,5),(2,5), (3,5),(4,5),(5,5),(6,5),(5,6), (5,4),(5,3),
(5,2),(5,1),(1,6),(3,4),(4,3),(6,1)}
n(A U B) = 15
Probability that one of the numbers obtained is 5 or the sum of the numbers is 7
n( A U B) 15 5
= P(A U B) = = =
n( S ) 36 12
ii. A I B = {(2,5),(5,2)}
n(A I B )= 2
Probability that one of the numbers obtained is 5 and the sum of the numbers is7
AI B 2 1
= P(A I B ) = n = =
n( S ) 36 18

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EXERCISE 1
1. A letter is selected at random from the word GENEROSITY.Find the
probability of selecting a vowel.
2. A student is asked to write a two digit number on a piece of paper.Find the
probability that at least one digit of the number is 7.
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3. In a lucky draw , the probability of a participant getting a free gift is .If
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276 participants got free gift , find the total number of participants.
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4. The probability of a football team winning is . Find the probability of the
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football team losing a match.
5. A bag contains 80 balls .16 of them are red, 24 are yellow while the rest are blue.
If a ball is chosen at random from the bag, find the probability that the ball
chosen is neither red nor yellow.
6. A box contains 20 green cards and some red cards..If a card is picked at random,
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the probability of picking a green card is . If 12 more green cards are added
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into the box, find the probability of picking a green card.
7. An integer is selected at random from the numbers 1 to 10. Find the probability of
selecting the numbers 8 or an odd number.
8. A jar contains 1 yellow , 1 red and 2 blue marbles. 2 marbles are chosen at
random from the jar one after another with replacement . Find the probability that
the first marble chosen is red and the second marble is blue.
9. A box contains 15 red balls and some some white balls. If a ball is picked at
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random , the probability of obtaining a red ball is . Then, if 10 green balls are
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added into the box , and a ball is picked at random, find the probability of
obtaining a red ball.
10. An electrical shop has a number of radios of which 12 are of brand P. If a radio
is selected at random from the shop, the probability of selecting a radio of brand
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P is . Find
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a. the probability of selecting a radio which is not of brand P.
b. the number of radios that are not of brand P.

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PROBABILITY: Exercise 2
1. A bag contains 3 red balls, 4 blue balls and 5 green balls. If a ball is picked at
random from the bag, find the probability that the ball is
a) a red ball b) a blue ball
2. A fair dice is rolled. Find the probability that the number obtained is
a) an even number b) less than 3
3. A box contains a set of cards which are numbered from 1 to 15. If a card is drawn
at random from the box, what is the probability that the number drawn is
a) a prime number b) a multiple of 5
4. A letter is chosen at random from the word “STATISTICS”. Find the probability
of choosing
a) a letter S b) a consonant
5. Nine number cards 5, 6, 7, 8, 9, 10, 12, 15 and 16 are placed into a bag. A card is
chosen at random from the bag. Find the probability of choosing a card with
a) an even number b) a perfect square
6. A number is selected at random from set { x: 4 ≤ x ≤ 20, x is an integer}. Find
the probability of selecting
a) a number with a digit 5 b) a multiple of 5
7. Some Arts students and 24 Science students participated in a Science camp. A
Student is chosen at random from the group. The probabilityof choosing a Science
Student is 8/15. find
a) the probability of choosing an Arts student
b) the total number of participants in the camp.
8. The are 12 English books and x Mathematics books in a box. If a book is
selected at random from the box, the probability of selecting an English book is
3/5.
a) If a book is selected at random from the box, state the probability of selecting
a Mathematics book.
b) Find the value of x
9. A number is selected at random from the set { x: 1 ≤ x ≤ 60, x is an integer}.
Find the probability that the number selected
a) is a multiple of 5 or 7 b) has digits 4 and 5

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10. Six cards in the diagram are placed into an empty box.
T A U F A N
a) If a card is selected at random from the box, state the probability that the card
marked A is selected.
b) A number of cards marked A are added into the box. If a card is selected at
Random from the box, the probability of selecting a card marked A is 5/7.
How many of cards marked A are added into the box?
PROBABILITY: DIAGNOSTIC TEST
1.A factory employs 45 workers. Eighteen of them are men. A worker is picked at
random from the factory. What is the probability of picking a man?
A 2/9 B 4/15 C 2/5 D 3/5
2. Set M = {8, 12, 23, 39, 42, 51, 61, 73, 88}. An element is chosen at random from
set M. What is the probability of choosing a prime number?
A 1/9 B 3/10 C 1/3 D 3/7
3. A box contains 30 coloured marbles. 12 of them are green marbles. A marble is
picked at random from the box. What is the probability that it is not a green
marble?
A 5/8 B 3/8 C 3/5 D 1/4
4. P is the event of rolling a dice . Which of the following represents the possible
outcomes of the event of obtaining a prime number or a number that is greater
than 2?
A { 3, 5} C { 3, 4, 5, 6}
B { 2, 3, 5} D {2, 3, 4, 5, 6}
5. Given that box A contains 6 cards numbered from 2 to 7 and box B contains 2
white cards and 4 black cards. What is the probability of obtaining an even
number and a white card?
A 1/6 B 1/5 C 1/3 D 1/2

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6. There are eight boys in a group of students. If a student is picked at random
From the group, the probability of picking a boy is 1/5. It is given that the
Number of girls who wear glasses in the group is 4. If a student is picked at
random from the group, what is the probability of picking a girl who does not
wear glasses?
A 1/10 B 2/3 C 7/10 D 9/10
7. A box contains 25 blue balls and several yellow balls. If a ball is drawn at random
From the box, the probability of drawing a blue ball is 5/8. If x black balls are
Added and a ball is drawn, then the probability of drawing a black ball is 1/6.
The value of x is
A 5 B 8 C 15 D 20
8. Wan has 75 bags of graded eggs. 30 bags are grade P eggs and the rest are grade B
And grade C eggs. If a bag is picked at random, the probability of picking a bag of
Grade B eggs is 1/3. Calculate the probability of picking a bag of grade C eggs.
A 4/15 B 16/35 C ¾ D 2/5
9. A jar contains 1 yellow, 1 red and 2 blue marbles. 2 marbles are chosen at random
From the jar one after another with replacement. Finf the probability that the first
Marble chosen is red and the second one is blue.
A 1/9 B 1/8 C 1/6 D ¼
10..The table shows the probabilities of the customers choosing brands X, Y and Z.
Brand X Y Z
Probability 11/24 3/8 1/6
If a survey is carried out on 1200 customers, find the number of customers who
Choose brand Y or Z.
A 550 B 650 C 680 D 700

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ANSWERS: PROBABILITY
EXERCISE 1
1. 2/5
2. 4/5
3. 736
4. 2/7
5. ½
6. 2/3
7. 3/5
8. 1/8
9. ¼
10. a) 5/8 b) 20
EXERCISE 2
1. a) ¼ b) 1/3
2. a) ½ b) 1/3
3. a) 2/5 b) 1/5
4. a) 3/10 b) 7/10
5. a) 5/9 b) 2/9
6. a) 2/17 b) 4/17
7. a) 7/15 b) 45
8. a) 2/5 b) x = 8
9. a) 19/60 b) 1/30
10. a) 1/3 b) 8

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DIAGNOSTIC TEST
1. C
2 C
3 C
4 D
5 A
6 C
7 B
8 A
9 B
10 B