The document discusses probability and provides examples and solutions. It defines probability as the number of favorable outcomes divided by the total number of possible outcomes. It gives examples of calculating probabilities of events such as choosing balls of different colors from a bag. It also discusses combined events and finding probabilities of "or" and "and" events.

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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