CHAPTER 2 Probability

Chapter 1: Probability

1

CHAPTER 1 : PROBABILITY

Sub - Topic

Introduction.

The properties of probability.

Conditional probability.

Independent events.

Bayes’ theorem.

Chapter Learning Outcome

Compute the probability for random variables.

Learning Objective

By the end of this chapter, students should be able to:

Determine the sample space and events.

Determine the properties of probability.

Compute the probability for random variables using conditional probability.

Compute the probability of independent events.

Apply Bayes’ theorem.

Key Term (English to Bahasa Melayu)

No. English Bahasa Melayu

1. Bayes’ rule → Petua Bayes

2. Complement → Penggenap / pelengkap

3. Conditional probability → Kebarangkalian bersyarat

4. Contingency table → Jadual kontigensi

5. Event → Peristiwa

6. Independent → Merdeka

7. Intersection → Persilangan

8. Mutually exclusive → Tak bersandar / saling eksklusif

9. Probability → Kebarangkalian

10. Sample space → Ruang sampel

11. Tree diagram → Gambarajah pokok

12. Union → Kesatuan


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1.1 Introduction

In this world there are a lot of uncertainties and many decisions had to make that

involve this situation. Probability and statistics can provide the models top help

people make decisions. In the study of probability, experiments for which the

outcome cannot be predicted with certainty will be considered. Such experiments are

called random experiments where each experiment ends in an outcome that cannot be

determined with certainty before the performance of the experiment. However, the

experiment is such that the collection of every possible outcome can be described and

perhaps listed. The collection of all outcomes is called the sample space, S.

Definition 1

The set of all possible outcomes of a statistical experiment is called the sample space

and is represented by the symbol S.

Definition 2

Each outcome in a sample space is called an element of the sample space. The

number of the elements is denoted as n(S).

Definition 3

An experiment is the process by which an observation (or measurement) is obtained.

There are three types of presentation of outcomes which are:

(a) Contingency table.

(b) Tree diagram.

(c) Venn diagram.

Example 1

Find the sample space of the following experiment:

(a) Tossing a coin. (Show the presentation in a tree diagram).

(b) Tossing two coins. (Show the presentation in a table).


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Answer Example 1

Let H and T represents “heads” and “tails” respectively.

(a) Tree diagram

Outcome Sample point

S = {H, T } and

n(S) = 2

(b) Table

1st Coin H H T T

2nd Coin H T H T S = { HH, HT, TH, TT } and n(S) = 2

Definition 4

An event is a subset of a sample space.

Example 2

Which one from the following definition is false?

(a) An experiment is a situation involving chance or probability that leads to

results called outcomes.

(b) An outcome is the result of a single trial of an experiment.

(c) An event is one or more outcomes of an experiment.

(d) Probability is the measure of how likely a sample space is.

Answer Example 2

(a) True. (b) True.

(c) True. (d) False.

T

H H

T


4

Example 3

Which of the following is not the example of an experiment?

(a) Tossing a coin.

(b) Rolling a single 6-sided die.

(c) Choosing a ball from a box.

(d) Tossing a cat.

Answer Example 3

(a) An experiment.

(b) An experiment.

(c) An experiment.

(d) Not an experiment.

Example 4

Which of the following is an outcome?

(a) Rolling a pair of dice.

(b) Landing on red.

(c) Choosing two balls from a box.

Answer Example 4

(a) Not an outcome.

(b) An outcome.

(c) Not an outcome.

Example 5

Which of the following experiment does not have equally likely outcomes?

(a) Choose a number at random from 1 to 10.

(b) Toss a coin.

(c) Choose a letter at random from the word STATISTICS.


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Answer Example 5

(a) Equally likely outcomes.

(b) Equally likely outcomes.

(c) Not equally likely outcomes.

The set of events in different conditions can be shown in a Venn diagram.

Definition 5

The complement of an event A with respect to S is the subset of all elements of S that

are not on A. The complement of A is denoted by the symbol A’.

Definition 6

The intersection of two events A and B, denoted by the symbol AB, is the event

containing all elements that are common to A and B.

Definition 7

Two events A and B are mutually exclusive, if A B = , that is, if A and B have no

elements in common.

A

' A

S `S = A 'A

S A B

A B A B = {x x A and x B}

S A B

A B =


6

Definition 8

The union of two events A and B, denoted by the symbol A B, is the event

containing all the elements that belong to A and B or both.

Definition 9

The difference of two events A and B, is the set which consists of all elements which

are in A but not in B, and is denoted by A\B or BA .

Example 6

If S = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 }, let A be the event “even numbers”, B be the event

“odd numbers”, C be the event “prime numbers”. Find

(a) A, B and C

(b) A’

(c) A B

(d) B C

(e) B – C

(f) n (A C)

(g) n (A B C)

Answer Example 6

(a) A = {2, 4, 6, 8,10), B = {1, 3, 5, 7, 9} and C = {2, 3, 5, 7}

(b) A’ = {1, 3, 5, 7, 9}

A B A B = {x x A or x B}

S A B

A

A – B A – B = {x x A and x B}

S B


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(c) A B = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

(d) B C = {3, 5, 7}

(e) B – C = {1, 9}

(f) A C = {2, 3, 4, 5, 6, 7, 8,10}, n (A C) = 8

(g) A B C =, n (A B C) = 0

Note 1

Several results in the following can easily be verified by means of Venn diagrams.

(a) A =

(b) A = A

(c) A A’ =

(d) A A’= S

(e) S’ =

(f) ’ = S

(g) (A’)’ = A

(h) (A B)’ = A’ B’

(i) (A B)’ = A’ B’

Exercise 1.1

1. All statistical experiments have three things in common, except

(a) The experiment can have more than one possible outcome.

(b) Each possible outcome can be specified in advance.

(c) The outcome of the experiment depends on chance.

(d) The sample space can’t have more than one possible outcome.

2. Which of the following sets represent an event when you roll a die?

(a) {2, 4, 6}.

(b) {1, 3, 5}.

(c) {1, 2, 3, 4, 5, 6}.


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3. Find the sample space of the following experiment and draw the tree diagram

for each experiment.

(a) Tossing two dice.

(b) Tossing a die and a coin.

4. Let S = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, A = {2, 4, 6, 8, 10}, B = {3, 6, 9} and

C = {4, 8}. Find

(a) A B.

(b) A B.

(c) A – C.

5. Let S = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {2, 4, 6, 8}, B = {1, 3, 5, 7, 9} and

C = {1, 2, 3, 5, 8}. Find

(a) A’ B.

(b) A C’ B.

(c) B A C.

6. Let S = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15}, let A be the event of

even numbers, B be the event of odd numbers, C be the event of numbers that

are the multiple of 3, D the event of squared number.

(a) List the elements of A, B, C and D.

(b) Determine which of the events are mutually exclusive.

7. Let S = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {2, 4, 6, 8}, B = {1, 3, 5, 7, 9} and

C = {1, 2, 3, 5, 8}. Find

(a) n(A B).

(b) n(A C).

(c) n(C B’).

(d) n(C – B).


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8. Consider the sample space S = {FKAAS,FKEE,FKMP, FPTeK,FPT,FTMM }

and the events A = {FKAAS, FKEE, FKMP}, B = {FPT, FTMM}, C =

{FPTeK}. Find

(a) n(A B).

(b) n(A C).

(c) n(C B’).

(d) n(C B A).

9. Consider the sample space S = {Algebra, Calculus, Statistics, Differential

Equations, Numerical Method} and the events A = {Algebra}, B =

{Calculus, Differential Equations}, C = {Numerical Method}. Find

(a) A’.

(b) A C.

(c) (A B’) C’.

(d) (B’ C’).

(e) A B C.

(f) (A’ B’) (A’ C).

10. Let A, B and C are the three events for a given experiment. Use set notation to

express the following statements.

(a) At least one of the three events occurs.

(b) All of the events occur.

(c) Exactly one of the three events occurs.

(d) None of the three events occur.

(e) Events A and B occur but not C.


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Answer Exercise 1.1

1. (d) The sample space can’t have more than one possible outcome.

2. (c) {1, 2, 3, 4, 5, 6}.

3. (a) {(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), (5,5), (5,6), (6,1), (6,2), (6,3),

(6,4), (6,5), (6,6)}

(b) {(1,H), (1,T), (2,H), (2,T), (3,H), (3,T), (4,H), (4,T), (5,H), (5,T), (6,H),

(6,T)}

4. (a) {2, 3, 4, 6, 8, 9, 10}, (b) {6}, (c) {2, 6, 10}.

5. (a) {0,1,3,5,7,9}, (b) {1, 3, 4, 5, 6, 7, 9}, (c) {1, 2, 3, 5, 8}.

6. (a) A = {2, 4, 6, 8, 10, 12, 14}, B = {1, 3, 5, 7, 9, 11, 13, 15},

C = {3, 6, 9, 12, 15}, D = {4, 9}

(b) A and B.

7. (a) 9, (b) 2, (c) 7, (d) 2.

8. (a) 5, (b) 0, (c) 4, (d) 0.

9. (a) {Calculus, Statistics, Differential Equations, Numerical Method}.

(b) { }.

(c) {Algebra, Calculus, Statistics, Differential Equations}.

(d) {Algebra, Calculus, Statistics, Differential Equations, Numerical

Method}.

(e) {Algebra}.

(f) {Numerical Method}.

10. (a) A B C (b) A B C

(c) (A B’ C’) (A’ B C’) (A’ B’ ’)

(d) (A B C )’ (e) A B C’


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1.2 The properties of probability

Definition 10

Probability is a set function P that assigns to each event A in the sample space S a

number P(A), called the probability of equally likely outcome of the event A, is

defined as P(A) = )(

)(

Sn

An as such that 0 P(A) 1.

Theory 1

P() = 0.

Theory 2

P(S) = 1.

Example 7

What is the probability of choosing a

(a) vowel from the alphabet?

(b) consonant from the alphabet?

Answer Example 7

Alphabet :

{ A, B, C, D, E, F, G ,H, I, J, K, L, M, N, O, P, Q, R, S, T, U, V, W, X, Y, Z }

n (Alphabet) = 26

(a) P (vowel) = 26

5

(b) P (consonant) = 26

21


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Example 8

A number was randomly selected from S = { 1, 2, 3, … , 48, 49, 50 }. Find the

probability of selecting a number which is

(a) an even number.

(b) a prime number.

(c) a squared number.

(d) a cubic number.

(e) a multiple of 7.

Answer Example 8

(a) Even number = {2, 4, 6, … , 46, 48, 50}

P (Even) = 50

25

(b) Prime number = {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47}

P (Prime) = 50

15

(c) Squared number = {1, 4, 9, 16, 25, 36, 49}

P (Squared) = 50

7

(d) Cubic number = {1, 8, 27}

P (Cubic) = 50

3

(e) Multiple of 7 = {7, 14, 21, 28, 35, 49}

P (Multiple of 7) = 50

6

Theory 3

For each event A, P(A) = 1 – P(A’).


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Example 9

At the Parit Raja Primary School, 30 out of 153 students get 5As. What is the

probability that a student does not get 5As?

Answer Example 9

P (get 5As) = 153

30 and P (does not get 5As) =

153

123.

Theory 4

If A and B are any two events then P(A B) = P(A) + P(B) – P(A B).

Theory 5

If A, B and C are any three events then,

P(ABC) = P(A) + P(B) + P(C) – P(AB) – P(AC) – P(BC)+ P(AB C).

Example 10

The probability that Khalid passes Mathematics is 2/3, and the probability that he

passes Science is 5/6. If the probability of passing both subjects is 8/9, what is the

probability that Khalid will

(a) pass at least one of these subjects?

(b) neither pass Science nor Mathematics?

Answer Example 10

Let M = “Mathematics” and S = “Science”

(a) P(pass at least one of these subjects) = P(M S)

= P(M) + P(S) – P(M S)

= 2/3 + 5/6 – 8/9

= 11/18 or 0.6111


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(b) P(either pass Science nor Mathematics) = P(M S)’

= 1 – 11/9

= 7/18 or 0.3889

We can find the probability by using the tree diagram or Venn diagram.

Example 11

A pair of coins is tossed. Find the probability of getting

(a) two tails.

(b) one tail and one head.

(c) no tails.

Answer Example 11

(a) P(two tails) = P(TT) = 2

1

2

1 =

4

1.

(b) P(one tail and one head) = = P(TH) + P(HT) =

2

1

2

1+

2

1

2

1=

4

2.

(c) P(no tails) = P(HH) = 2

1

2

1 =

4

1.

} { TTTH,HT,HH,S

T

TH

TT

H

T

T

HT

HH H

H

1/2

1/2

1/2

1/2

1/2

1/2

Coin


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Example 12

A box contains 5 red marble and 3 blue marbles.

(a) If two marbles were taken from the box, with returning of the first ball before

taking the second ball. Find

(i) P(exactly two blue balls).

(ii) P(two balls with different colors).

(iii) P(at least one blue ball).

(b) If two marbles were taken from the box, without returning of the first ball

before taking the second ball. Find

(i) P(exactly two blue balls).

(ii) P(two balls with different colors).

(iii) P(at least one blue ball).

Answer Example 12

(a) With returning

(i) P(exactly two blue balls) = P(BB) = 8

3

8

3 =

64

9.

(ii) P(two balls with different colors) = P(BM) + P(MB)

=

8

5

8

3+

8

3

8

5=

64

30.

(iii) P(at least one blue ball) = P(BM) + P(MB) + P(BB)

=

8

5

8

3+

8

3

8

5+

8

3

8

3=

64

39.

},,,{ BBBMMBMMS

B

BM

BB

M

B

B

MB

MM M

M

5/8

5/8

5/8

3/8

3/8

3/8


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or P(at least one blue ball) = 1 – P(MM) = 1 –

8

5

8

5= 1 –

64

25=

64

39.

(b) Without Returning

(i) P(exactly two blue balls) = P(BB) = 7

2

8

3 =

56

6.

(ii) P(two balls with different colors) = P(BM) + P(MB)

=

7

5

8

3+

7

3

8

5=

56

30.

(iii) P(at least one blue ball) = P(BM) + P(MB) + P(BB)

=

7

5

8

3+

7

3

8

5+

7

2

8

3=

56

36.

or P(at least one blue ball) = 1 – P(MM) = 1 –

7

4

8

5= 1 –

56

20=

56

36.

Example 13

In the senior year of a high school graduating class of 100 students, 53 studied

Calculus, 56 studied Statistics, 49 studied Numerical Method, 21 studied both

Calculus and Numerical Method, 23 studied both Calculus and Statistics, 8 studied

Numerical Method but neither Calculus nor Statistics, 4 studied all three subjects.

(a) Draw the Venn diagram.

(b) If a student is selected at random, find the probability that a person

(i) takes all three subjects.

(ii) takes only Calculus and Statistics.

(iii) takes only Numerical Method and Statistics.

},,,{ BBBMMBMMS

B

BM

BB

M

B

B

MB

MM M

M

5/8

3/8

4/7

5/7

3/7

2/7


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(iv) takes only Calculus and Numerical Method.

(v) only takes Calculus.

(vi) only takes Numerical Method.

(vii) only takes Statistics.

(viii) does not take any of the three subjects.

Answer Example 13

(a)

(b) (i) P(C S N ) = 4/100

(ii) P(C S N’) = 19/100

(iii) P(C’ S N) = 20/100

(iv) P(C S’ N) = 17/100

(v) P(C S’ N’) = 13/100

(vi) P(C’ S’ N) = 8/100

(vii) P(C’ S N’) = 13/100

(viii) P(C S N’)’ = 6/100

Corollary 1

If A and B are mutually exclusive then P(A B ) = P(A) + P(B).

Example 14

Four friends, Adam, Badrul and Luqman apply for a job in a grocery store but only

one person will get the job. Find the probability that

(a) Badrul will get the job.

C S

N

13

6

4 20 17

19

8

13


18

(b) Adam or Luqman will get the job.

Answer Example 14

(a) P(Badrul will get the job) = 3

1.

(b) P(Adam or Luqman will get the job)= 3

1+

3

1 =

3

2.

Exercise 1.2

1. Given the word “statistics”, what is the probability of choosing

(a) “s” ?

(b) a consonant?

(c) a vowel?

2. In the club, there are 27 first year students, 31 second year students, 24 third

year students, and 16 fourth year students. A student is selected, find the

probability that the student

(a) is a second year student.

(b) is not a fourth year student.

3. A number was randomly selected from S = {25, 26, 26, …, 72, 73, 74}. Find

the probability of selecting a number which is

(a) an odd number.

(b) a prime number.

(c) a squared number.

(d) a multiple of 13.

4. Let S = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {2, 4, 6, 8}, B = {1, 3, 5, 7, 9} and

C = {1, 2, 3, 5, 8}. Find the probability of

(a) A B C.


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(b) B C.

(c) C B’.

5. In the mid-term exam, 7 out of 93 failed the exam. What is the probability that

a student pass the exam?

6. From the statistics class, 37 out of the 115 students wear spectacles. What is

the probability that a student

(a) wears spectacles?

(b) do not wear spectacles?

7. University A sends three female students Hazami, Laila and Mashitah to take

part in a badminton competition for female single event. The probabilities that

Hazami, Laila and Mashitah will become champion are 0.3, 0.35 and 0.25,

respectively. Find the probability that

(a) one of them will become champion.

(b) none of them will become champion.

8. After one senior staff retires, five other staffs, Haziq, Hafiz, Farhanah,

Syafinah and Khaleeda apply for the vacant post in that department but only

one person will get the post. Find the probability that

(a) Hafiz will get the post.

(b) Syafinah or Khaleeda will get the post.

9. The probability that Aman passes Engineering Mathematics is 0.962, and the

probability that he passes Differential Equations 0.873. If the probability of

passing both subjects is 0.789, what is the probability that Aman will

(a) pass at least one of these subjects?

(b) neither pass Engineering Mathematics nor Differential Equations?


20

10. Before going out of her house, Sarah filled up her bag with 3 oranges, 4

apples, and 5 pears. At the bus stop, she ate two fruit.

(a) Draw the tree diagram.

(b) Find the probability that

(i) exactly one apple was eaten.

(ii) both fruit are the same.

11. A pair of dice is tossed. Find the probability of getting

(a) two odd numbers.

(b) two prime numbers.

12. A fair die has 4 faces with number 1 to 4. Score of any toss is base on the top

number when throwing the die on a table. If 2 dice of this type is tossed, find

the probability that the multiplication of these numbers is

(a) an even number.

(b) a prime number.

13. A box contained nineteen transistors, four of which are defective. If three

transistors were selected without replacing after each selection, draw the tree

diagram and find the following probabilities that

(a) all are defective.

(b) at least one is defective.

14. A box contains eight green marble and seven blue marble. Two marbles are

randomly selected from the box. Find the probability that we get a green

marble and a blue if the marbles are selected

(a) without replacement.

(b) with replacement.


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15. A box contains 9 red marble and 13 blue marbles. If two marbles were taken

from the box, with returning of the first ball before taking the second ball, find

(a) P(exactly two blue balls).

(b) P(two balls with different colors).

(c) P(at least one blue ball).

16. Based on the previous questions. If two marbles were taken from the box,

without returning of the first ball before taking the second ball, find

(a) P(exactly two red balls).

(b) P(two balls with different colors).

(c) P(at least red blue ball).

17. In the senior year of a high school graduating class of 100 students, 61 studied

history, 56 studied geography, 36 studied both geography and history.



takes

(i) all two subjects.

(ii) only history.

(iii) only geography.

18. In the year five of a secondary school of 119 students, 57 studied biology, 59

studied chemistry, 44 studied physics, 26 studied both mathematics and

physics, 33 studied both mathematics and chemistry, 13 studied chemistry but

neither mathematics nor physics, 14 studied all three subjects.



(i) takes all three subjects.

(ii) takes only biology and chemistry.

(iii) takes only physics and biology.


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(iv) takes only physics and chemistry.

(v) only takes physics.

(vi) only takes chemistry.

(vii) only takes physics.

(viii) does not take any of the three subjects.

Answer Exercise 1.2

1. (a) 10

3 (b)

10

7 (c)

10

3

2. (a) 98

31 (b)

98

82

3. (a) 50

25 (b)

50

12 (c)

50

4 (d)

50

4

4. (a) 10

9 (b)

10

3 (c)

10

7

5. 93

86

6. (a) 115

37 (b)

115

78

7. (a) 0.9 (b) 0.1

8. (a) 5

1 (b)

5

2

9. (a) 0.946 (b) 0.054

10. (a) 64/132 (b) 28/132

11. (a) 9/36 (b) 9/36

12. (a) 12/16 (b) 4/16

13. (a) 4/969 (b) 514/969

14. (a) 56/210 (b) 56/225

15. (a) 269/484 (b) 234/484 (c) 403/484

16. (a) 156/462 (b) 228/462 (c) 390/462


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17. (a)

(b) (i) 36/100 (ii) 25/100 (iii) 19/100 (iv) 20/100

18. (a)

(b) (i) 14/119 (ii) 19/119 (iii) 12/119 (iv) 13/119

(v) 54/119 (vi) 59/119 (vii) 57/119 (viii) 21/119

1.3 Conditional probability

Definition 11

The conditional probability of B, given A, denoted by P(B | A), is defined by

P(B | A) = )(

)(

AP

ABP if P(A) 0 and P(B) 0.

The equation above can also be written in the form: )( ABP = P(B | A) P(A)

In the same way, it can be shown P(A | B) = )(

)(

BP

ABP .

Thus, the result can be written as )( ABP = P(B | A) P(A) = P(A | B) P(B).

If A and B are mutually exclusive events, then as P(A) 0 and P(B) 0 and

)( ABP = 0, it follows that )(

)(

AP

ABP = 0.

B C

P

13

21

14 13 12

19

H G

19

20

36 15

25


24

Example 15

From Statistics section 1 in a certain university, 15% of the students failed statistics,

26% of the students failed calculus, and 7 % failed both statistics and calculus. A

student is selected randomly.

(a) If the student failed statistics, what is the probability that he failed calculus?

(b) If the student failed calculus, what is the probability that he failed statistics?

(c) What is the probability that the student failed calculus or statistics?

(d) What is the probability that the student failed neither calculus nor statistics?

Answer Example 15

Let C = “calculus” and S = “statistics”

(a)15

7

15.0

07.0

)(

)()|(

SP

SCPSCP

(b)26

7

26.0

07.0

)(

)()|(

SP

SCPCSP

(c) )()()()( SCPSPCPSCP = 0.26 + 0.15 – 0.07 = 0.34

(d) )(1)'( SCPSCP = 0.26 + 0.15 – 0.07 = 0.66

Example 16

A random sample of 150 customers is classified below by their gender and their

favorite food.

Favorite food Male Female

Pizza 34 35

Burger 48 16

Spaghetti 23 34

If a customer is picked at random from this group, find the probability that

(a) the customer is a male, given that the customer’s favorite food is pizza.

(b) the customer’s favorite food is not burger, given that the customer is a female.


25

Answer Example 16

Let M = the person is male

F = the person is female

P = pizza,

B = burger,

S = spaghetti.

Using the contingency table as below:

Favorite food Male Female Total

Pizza 34 35 69

Burger 48 16 64

Spaghetti 23 34 57

Total 105 85 190

(a) The customer is a male, given that the customer’s favorite food is pizza.

P(M | P) = .6934)(

)(

PP

PMP

(b) The customer’s favorite food is not burger, given that the customer is a female

P(B’ | F) = .8569)(

)'(

FP

FBP

Definition 12

If in an experiment, the events A and B can both occur, then

P(A B) = P(A) P(B | A).

This is the multiplicative rule.

Example 17

Given that a fuse box containing 25 fuses, of which 4 are defective. If 2 fuses are

selected at random and removed from the box in succession without replacing the

first, what is the probability that both fuses are defective?


26

Answer Example 17

Let D = defective and D’ = not defective

From the tree diagram, P(D) = 25

4and )|( DDP =

24

3

)|()()( DDPDPDDP =24

3

25

4 =

600

12

Example 18

The probability that a married man watches a certain television show is 0.65 and the

probability that a married woman watches the show is 0.72. The probability that a

man watches the show, given that his wife does, is 0.87. Find the probability that

(a) a married couple watches the show.

(b) a wife watches the show given that her husband does.

(c) at least one person of a married couple will watch the show.

Answer Example 18

Let M = a married man watches a certain television show and

W = a married woman watches a certain television show.

Given P(M) = 0.65, P(W) = 0.72 and )|( WMP = 0.87.

(a) )|()()( WMPWPWMP

= 0.72 0.87 = 0.576

D’

D’D

D’D’

D

D’ DD’

DD D

D

25/4

25/21

243

H

24/21

24/20

24/4

12

Sub

ject

Lea

rnin

g

Out

com

e

D’


27

(b) )|( MWP = )(

)(

MP

WMP

=65.0

576.0= 0.886

(c) P(M W ) = P(M) + P(W) )( WMP

= 0.65 + 0.72 – 0.576 = 0.794

1.4 Independent events

Definition 13

Two events A and B are independent if and only if

(a) P(B | A) = P(B)

(b) P(A | B) = P(A)

(c) P(A B) = P(A) P(B)

Otherwise, A and B are dependent.

Theory 6

If two events A and B are mutually exclusive events with if P(A) 0 and P(B) 0,

then P(A B) = 0 (because A and B cannot occur together). Since P(B | A) =

)(

)(

AP

ABP , then P(B | A) = 0. Thus, P(B | A) = 0 if A and B are mutually exclusive

and P(B | A) P(A). In other words, if events A and B are mutually exclusive, the A

and B are not independent. If two events A and B are mutually exclusive events with

if P(A) 0 and P(B) 0, then P(A B) = 0 (because A and B cannot occur

together). Since P(B | A) = )(

)(

AP

ABP , then P(B | A) = 0. Thus, P(B | A) = 0 if A and

B are mutually exclusive and P(B | A) P(A). In other words, if events A and B are

mutually exclusive, the A and B are not independent.


28

Example 19

A town has 2 fire engines operating independently. The probability that a specific

engine is available when needed is 0.95.

(a) What is the probability that both fire engines are available when needed?

(b) What is the probability that neither is available when needed?

(c) What is the probability that a fire engine is available when needed?

Answer Example 19

Let A = available, and A’ = not available

(a) )()()( APAPAAP = 0.95 0.95 = 0.9025

(b) )'()'()''( APAPAAP = 0.05 0.05 = 0.0025

(c) )()'( )'()( APAPAPAP = (0.95 0.05) + (0.05 0.95) = 0.095

A’

A’A

A’A’

A

A’

A’

AA’

AA A

A

0.95

1st fire engine

0.05

0.05

0.05

0.95

0.95

2nd fire engine


29

Exercise 1.4

1. The probability that a regularly scheduled flight departs on time is P(D) =

0.97, it arrives on time is P(A) = 0.76 and it departs and arrives on time is

.68.0)( ADP Find the probability that a plane

(a) arrives on time given that it has departed on time.

(b) departs on time given that it has arrived on time.

2. Given P(A) = 0.45, P(B) = 0.5 and P(A B) = 0.2, find

(a) P(A’ ).

(b) P (A B).

(c) P(A’ B).

(d) P(A’ B).

3. A certain student goes to class in a college either by the college bus or a

public bus. Usually 70% of the time he takes the college bus. If he takes the

public bus, the probability that he will be early is 65% while the probability he

will be early if he takes the college bus is 80%.


(b) What is the probability that he takes the public bus?

(c) What is the probability that he takes the public bus, given he was late?

(d) What is the probability that he was early, given that he took the college

bus?

4. The 2nd year Technical Education students were divided into 3 sections which

comprised of 18 ex-polytechnic, 155 ex-matriculation and 12 ex-form six

students. Half of the ex-polytechnic students, 99 of the ex-matriculation and

three quarter of the ex-form six are male students.

(a) Draw the contingency table.

(b) If a student is selected at random, find the probability that the


30

(i) student is female.

(ii) student is an ex-matriculation student, given that the student is

a male.

5. Let box 1 has two red ribbons and three blue ribbons while box 2 has one red

ribbon and three blue ribbons while box. One fair die with 6 faces is tossed. If

the number on the die is a multiple of 3, box 1 is selected and if not box 2 is

selected.

(a) Sketch the tree diagram.

(b) Find the probability of selecting a red ribbon.

(c) Find P(red ribbon box 1).

(d) Find P(box 2blue ribbon).

6. A stationary company performed a survey of customers entering a

supermarket over a period of one week. The firm recorded 117 adult females,

73 adult males and 23 teenagers purchased goods above RM 100. From the

adult female, 25% bought stationary products while only 10% of the adult

males and 5% of the teenagers bought stationary products. If a customer is

selected at random, find the probability that the customer is a teenager who

did not buy the stationary product.

7. The probability that Zaid will be alive in 30 years is 0.8, and the probability

that Ahmad will be alive in 30 years is 0.75. If we assume independence for

both, what is the probability that neither will be alive in 30 years?

8. One medicine bag contains 2 bottles of panadol and 3 bottles of vitamins

tablets. A second medicine bag contains 1 bottle of panadol, 4 bottles of

vitamins and 2 bottles of painkiller. If a bottle is taken at random from each

bag, find the probability that


31

(a) both bottles contain panadol.

(b) the two bottle contain different tablets.

(c) neither bottle contains vitamins.

9. Find the probability of randomly selecting a good quarts of milk in succession

from a store who have 25 quarts of milk, of which 5 is spoiled and another

store who have 35 quarts of milk, of which that 10 is spoiled.

Answer Exercise 1.4

1. (a) 68/97 (b) 68/76

2. (a) 0.55 (b) 0.75 (c) 0.40 (d) 0.25

3. (a)

(b) 0.495 (c) 0.105 (d) 0.56

4. Let T = Matriculation, P = Polytechnic and F = Form Six

(a)

M F Total

T 9 9 18

P 99 56 155

F 9 3 12

Total 117 68 185

(b) (i) 68/185 (ii) 99/117

5. (a)

B2

B1

B

R

R

2/6

4/6

2/5

3/5

1/4

B 3/4

C’

C E’

E

E

70%

30%

80%

20%

35%

65%

E’


32

(b) 9/30 (c) 4/10 (d) 5/7

6. 0.103 7. 0.05

8. (a) 2/35 (b) 21/35 (c) 6/35

9. 0.686

1.5 Bayes’ theorem

Definition 14

If the events B1, B2, B3, …, Bk, constitute a partition of the sample space S such that

P(Bi) 0 for i = 1, 2, …, k, then for any event A of S,

P(A) =

k

i

k

i

iii BAPBPABP1 1

).|()()(

Example 20

In a certain company, three machines, B1, B2, and B3, make 25%, 35%, and 30%,

respectively, of the products. It is known from the past experience that 1%, 3%, and

2% of the products made by each machine, respectively, are defective. Now, suppose

that a finished product is randomly selected. What is the probability that it is

defective?

Answer Example 20

Let D = the product is defective,

B1 = the product is made by machine B1.



Applying the rule of elimination, we can write

P(D) = P(B1)P(D | B1) + P(B2)P(D | B2) + P(B3)P(D | B3).


33

Referring to the tree diagram, we find that the three branches give the probabilities

P(B1)P(D | B1) = (0.25)(0.01) = 0.0025.

P(B2)P(D | B2) = (0.35)(0.03) = 0.0105.

P(B3)P(D | B3) = (0.30)(0.02) = 0.0060.

Thus, P(D) = 0.0025 + 0.0105 + 0.006 = 0.0195.

Suppose we want to find the conditional probability P(Bi | A) instead of P(A) in

Example 20. In other words, we want to find the probability that the product was

made by machine Bi. We can find the answer by using Bayes’s Rule which is as

follow:

Theory 7

If the events B1, B2, B3, …, Bk, constitute a partition of the sample space S, where

P(Bi) 0 for i = 1, 2, …, k, then for any event A in S such that P(A) 0,

k

i

ii

rr

k

i

i

rr

BAPBP

BAPBP

ABP

ABPABP

11

)|()(

)|()(

)(

)()|( for r = 1, 2, …, k.

Example 21

With reference to Example 20, if a product were chosen randomly and found to be

defective, what is the probability that it was made by machine B1?

P(D | B1) = 0.01

P(D | B2) = 0.03

P(D | B3) = 0.02

B1

B3

B2

D

D

D

P(B1) = 0.25

P(B2) = 0.35

P(B3) = 0.30


34

Answer Example 21

Using Bayes’s rule,

P(B1 | D) = )(

)()( 11

DP

BDPBP=

0195.0

01.025.0 = 0.1282

In view of the fact that a defective product was selected, this result suggests that it

probably was not made by machine B1.

Example 22

With reference to Example 20, if a product were chosen randomly and found to be

defective, what is the probability that it was made by machine B3?

Answer Example 22

Using Bayes’s rule,

.3077.0

)|()(

)|()(

)(

)()|(

3

1

33

3

1

3

3

i

ii

i

i BAPBP

BAPBP

ABP

ABPABP

In view of the fact that a defective product was selected, this result suggests that it

probably was not made by machine B3.

Example 23

From a research, 119 patients were categorized as follows:

Habit Hypertension

(H)

No Hypertension

(H’)

Smoker (S) 58 9

Non smoker (S’) 15 37

Base on the information above, find

(a) P(patients who are smokers).

(b) P(S |H’).

Condition


35

(c) P(hypertension smoker).

(d) P(smoker hypertension).

Answer Example 23

Using contingency table.

Habit

Hypertension (H) No Hypertension (H’) Total

Smoker (S) 58 9 67

Non smoker (S’) 15 37 52

Total 73 46 119

(a) P(patient is a smoker) = P(S )

= P(S C ) + P(S 'C )

= 58/119+ 9/119

= 67/119

(b) P(S H’) = )'(

)'(

HP

HSP

=)''()'(

)'(

HSPHSP

HSP

=11946

1199

= 9/46

(c) P(hypertension smoker) = P(H S)

=)(

)(

SP

HSP

=11967

11958

= 58/67

Condition


36

(d) P(smoker hypertension) = P(S H)

= )(

)(

HP

HSP

= )'()(

)(

HSPHSP

HSP

=1191511958

11958

= 58/73

Exercise 1.5

1. A random sample of 166 adults is classified below by gender and their marital

status.

Male Female

Married 29 47

Single 37 53

If a person is picked at random from this group, find the probability that

(a) the person is a male,

(b) the person is a female and single,

(c) the person is still single, given that the person is a female.

2. From a research, 100 patients were categorized as follows.

High Blood (H) Not High Blood ( 'H )

Overweight (O) 138 23

Not Overweight ( 'O ) 54 121

Based on the information above, find

(a) P(patients who are overweight).

(b) P(overweight | not a high blood patient).

(c) P(not high blood patient overweight).

(d) P(overweight high blood patient).

Health Body


37

3. Three girls had been nominated for the presidential post. The probability that

Murni would be elected was 0.3, Asma would be elected is 0.33 and Hannah

would be elected is 0.4. If Murni was elected, the probability the membership

fee increased was 0.6 while if Asma or Hannah was elected, the probability

the membership fee increased were 0.4 and 0.5, respectively. A few weeks

later, Wan planned to join the club but realized the fee was increased. What is

the probability that Hannah was elected as the president of the club?

4. In the certain region of the country it is known from the past experience that

given the probability of a doctor correctly diagnosing a person over 40 years

old with cancer as having the disease is 0.67 and the probability of incorrectly

diagnosing a person without cancer as having the disease is 0.05. The

probability of selecting an adult over 40 years of age with cancer is 0.07.

What is the probability that a person is diagnosed incorrectly as having

cancer?

5. Suppose a student dormitory in a college consists of the following:

30% are first year students of whom 10% own a car

40% are second year students of whom 20% own a car

20% are third year students of whom 40% own a car

10% are fourth year students of whom 60% own a car

A student is randomly selected from the dormitory.


(b) Find the probability that the student owns a car.

(c) If the student owns a car, find the probability that the student is a third

year student.

6. Bowl B1 contains two red and five white chips and bowl B2 contains four red

and three white chips. A fair die is cast. If the outcome is a multiple of 3

(namely 3 and 6), a chip is taken from bowl B1, otherwise a chip is taken from


38

bowl B2.

(a) Compute the probability that a red chip is taken.

(b) Given that the selected chip is red, compute the conditional probability

that it was taken from bowl B1.

Answer Example 1.5

1. (a) 66/166 (b) 53/166 (c) 53/100

2. (a) 161/336 (b) 123/144 (c) 123/161 (d) 138/192

3. 0.4 4. 0.0716

5. (a) 0.25 (b) 0.32 (c) 0.5

6. (a) 0.096 (b) 0.06

EXERCISE CHAPTER 1

1. A random sample of 190 students is classified below by gender and their

qualifications.

Male Female

UPSR 41 47

PMR 29 19

SPM 37 17

If a student is picked at random from this group, find the probability that

(a) the student is a male.

(b) the student is a female with PMR.

(c) the student has UPSR or PMR.

2. The probability that a married man watches Discovery programme 0.85 and

the probability that the wife watches the programme is 0.65. The programme

that a man watches the programme, given that his wife watches the

programme is 0.8. Find the probability that

(a) a married couple watches the Discovery programme.


39

(b) a wife watches Discovery programme given that her husband does.

(c) at least one person of a married couple will watch Discovery

programme.

3. An opened box contains 9 white kittens and 10 black kittens. Two kittens

were taken from the box, with returning of the first kitten before taking the

second kitten.


(b) Find P(exactly two black kittens).

(c) Find P(two kittens with different colours).

4. In a research, 57 students from subject DSM2163 were selected randomly for

migraine research. The result of this research is shows in the table below.

Habit

Have migraine, M Does not have migraine,

M’

Sleep after 12 midnight, S 25 4

Sleep before 12 midnight, S’ 7 21

Find

(a) P(student with migraine).

(b) P(student who sleep after 12 midnight | have migraine).

(c) P(does not have migraine| sleep before 12 midnight).

5. There are 100 form five students in a secondary school. Each student can take

either one or two or three subjects. After the registration, there are 40 students

in the mathematics class, 30 students in the chemistry class, 30 students in the

physics class. Given that 15 students study all three subjects, 10 students study

both mathematics and physics, 5 students studied both mathematics and

chemistry and 10 students studied chemistry but neither mathematics nor

physics.


(b) If a student is selected at random, find the probability that the student

studied physics but neither mathematics nor chemistry.

Health


40

(c) If a student is selected from the physics class, find the probability that

the student take all the subjects.

6. A random sample of 166 adults is classified below by gender and their marital

status.

Male Female

Married 29 47

Single 37 53

If a person is picked at random from this group, find the probability that

(a) the person is a male.

(b) the person is a male and single.

(c) the person is still single, given that the person is a female.

7. A girl uses a metal detector to look for the valuable metal objects on a beach

in Pulau Langkawi. There is a fault in the machine causes to signal only 95%

of the metallic objects over which it passes, and to signal 60% of the non-

metallic objects over which it passes. Twenty percents of the objects over

which the machine passes are metallic.

Find the probability that:

(a) a given object over which the machine passes is metallic and the

machine gives a signal.

(b) a signal being received by the girl for any given object over which the

machine passes.

(c) the girl has found a metal object when she receives a signal.

8. A survey indicates that 60% of tourists in Malaysia are from United States,

30% from Australia and 10% from the other countries. The result shows that

50% tourists from United States, 60% from Australia and 90% from the other

countries, are really happy staying in Malaysia. One tourist is selected at

random.

(a) Draw a tree diagram to show all the possible events.


41

(b) Find the probability a tourist are happy staying in Malaysia.

(c) Find the probability that he/she is from United States, given that he/she

is not happy staying in Malaysia.

(d) Find the probability a tourist from Australia and happy staying in

Malaysia.

9. A municipal bond service has three rating categories (A, B and C) in cities,

suburbs and rural areas. Suppose that in the past year of the municipal bond

issued throughout Malaysia, 70% were rated A, 20% were rated B and 10%

were rated C. Of the municipal bonds rated A, 50% were issued by cities and

40% by suburbs. Of the municipal bonds rated B, 60% were issued by cities

and 20% by rural areas. Of the municipal bonds rated C, 5% were issued by

suburbs and 5% by rural areas. Draw a tree diagram form the situation.

(a) What is the probability that municipal bond is issued by cities?

(b) If a new municipal bond is to be issued by a city, what is the

probability that it will receive an A rating?

10. (a) If A and B are mutually exclusive, ,37.0)( AP and ,44.0)( BP

find:

(i) )( BAP

(ii) )''( BAP

(b) There are 90 applicants for a job with the news department of a

television station. Some of them are college graduates and some are

not, some of them have at least three years’ experience and some have

not, with the exact breakdown being

College graduates Not college

graduates

At least 3 years’ experience 18 9

Less than 3 years’ experience 36 27


42

If the applicants are interviewed randomly, what is the probability that,

(i) the applicants is college graduates?

(ii) the applicants is college graduates and have less than three

years’ experience?

(iii) the applicants is not college graduates given that they have at

least three years’ experience?

11. During an epidemic of a certain disease, a doctor is consulted by 100 people

suffering from symptoms commonly associated with the disease. Of the 100

people, 40 are females of whom 15 actually have the disease and 25 do not. 20

males have the disease and the rest do not. A person is selected at random. Let

M denote the event that this person is a male and D denote the event that this

person is suffering the disease.

(a) Construct the contingency table

(b) Evaluate

(i) P ( D )

(ii) P ( M D )

(iii) P ( M | D )

ANSWER EXERCISE CHAPTER 1

1. (a) 107/190 (b) 19/190 (c) 136/190

2. (a) 0.52 (b) 0.6118 (c) 0.98

3. (b) 100/361 (c) 180/361

4. (a) 32/57 (b) 25/32 (c) 21/28

5. (b) 10/100 (c) 10/100

6. (a) 66/166 (b) 37/166 (c) 53/166

7. (a) 0.19 (b) 0.67 (c) 0.2836

8. (b) 0.57 (c) 0.6976 (d) 0.18

9. (a) 0.56 (b) 0.35


43

10. (a) (i) 0.81 (ii) 0.3528 (b) (i) 54/90 (ii) 36/90 (iii) 9/90

11. (b) (i) 7/20 (ii) 1/5 (iii) 4/7

SUMMARY CHAPTER 1

P(A) = )(

)(

Sn

An as such that 0 P(A) 1.

P() = 0.

P(S) = 1.

For each event A, P(A) = 1 – P(A’).

If A and B are any two events then: P(A B) = P(A) + P(B) – P(A B).

If A, B and C are any three events then:

P(ABC) = P(A) + P(B) + P(C) – P(AB) – P(AC) – P(BC)+ P(AB C).

If A and B are mutually exclusive then: P(A B ) = P(A) + P(B).

The conditional probability of B, given A, denoted by P(B | A), is defined by :

P(B | A) = )(

)(

AP

ABP if P(A) 0 and P(B) 0, it can be shown as

P(A | B) = )(

)(

BP

ABP .

If in an experiment, the events A and B can both occur, then:

P(A B) = P(A) P(B | A), this is the multiplicative rule.

Two events A and B are independent if and only if

(a) P(B | A) = P(B)

(b) P(A | B) = P(A)

(c) P(A B) = P(A) P(B)

Otherwise, A and B are dependent.


44

Bayes’ Theorem : If the events B1, B2, B3, …, Bk, constitute a partition of the sample

space S, where P(Bi) 0 for i = 1, 2, …, k, then for any event A in S such that

P(A) 0,

k

i

ii

rr

k

i

i

rr

BAPBP

BAPBP

ABP

ABPABP

11

)|()(

)|()(

)(

)()|( for r = 1, 2, …, k.


45

CORRECTION PAGE CHAPTER 1

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