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7 . PROBABILITY 7.1 Probability of an event

a. The probability of an event A is defined by

P(A) = )()(

SnAn and 0 ≤ P(A) ≤ 1

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

P(A) = )()(

SnAn = 5

2104=

ii. Let B be the event of chosing a black ball. n(B) = 6

P(B) = 53

106

)()(

==SnBn

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iii. Let C be the event of chosing a red ball. n(C) = 0

P (C) = 0100

)()(

==SnCn

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

In an archery training, the probability of an archer hitting the target is 83 . Find the

probability that he misses the target. Solution Let A be the event of hitting the target.

Then, P(A) = 83

Thus, P(A’) = 1 - 85

83=

7.3 Probability of a combined event

a. Probability of the combined event A or B

P( A or B) = )(

)(Sn

BAn U

b. Probability of the combined event A and B

P(A and B) = )(

)(Sn

BAn I

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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. AU 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(AU B) = 15 Probability that one of the numbers obtained is 5 or the sum of the numbers is 7

= P(A125

3615

)()() ===

SnBAnB U

U

ii. A BI = {(2,5),(5,2)}

n(A BI )= 2 Probability that one of the numbers obtained is 5 and the sum of the numbers is7

= P(A BI ) = n181

362

)(==

SnBAI

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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.

3. In a lucky draw , the probability of a participant getting a free gift is 83 .If

276 participants got free gift , find the total number of participants.

4. The probability of a football team winning is 75 . Find the probability of the

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,

the probability of picking a green card is 95 . If 12 more green cards are added

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

random , the probability of obtaining a red ball is 103 . Then, if 10 green balls are

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

P is 83 . Find

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.

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

T A U F A N

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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. 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

Brand X Y Z Probability 11/24 3/8 1/6

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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