Lecture Notes

CHAPTER 2 PROBABILITY

Objectives

1. To be able to describe the concept of set, event and probability.

2. To be able to apply multiplication rule and addition rule to calculate

the probability.

3. To be able to apply Bayes’ Theorem.

2.1 Concept of Set

Definition of Experiment

A process that when performed, results in one and only one of many

observations.

Definition of Outcome

The observations of an experiment.

Definition of Sample Space

The set of all possible outcomes of a statistical experiment and presented

by the symbol, S.

Definition of Sample Points

The elements of a sample space.

Example 2.1

Experiment Outcomes Sample space

(a) Roll a die 1,2,3,4,5,6 S = {1,2,3,4,5,6}

(b) Select a chip Defective, Non-defective S = {Defective, Non-

Defective}

The sample space for an experiment can be illustrated by:

32 Intro to Statistics & Probability

1. Venn Diagram: a picture (closed geometric shape such as circle) that

Consists of all possible outcomes for an experiment.

2. Tree Diagram: Each outcome is represented by a branch of a tree.

Definition of Event

Event: any subset of a sample space.

Simple event: an event that includes one and only one of the final

outcomes for an experiment.

Example 2.2

Roll a die three times and observe whether you will get “no.6” or not for

each roll.

(a) Find the event that you will get “no.6” at most twice.

(b) Find the event that you will not get “no.6” at least twice.

(c) Find the event that you will not get “no.6” exactly once.

Solution

Let G as get ‘no. 6’.

G as do not get ‘no. 6’.

Then,

.H

.T

H

T

Chapter 2 Probability 3 3

{ } 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 S = G G G ,G G G ,G G G ,G G G ,G G G ,G G G ,G G G ,G G G

Let A be the event that get ‘no. 6’ at most twice.

B be the event that will not get ‘no. 6’ at least twice.

C be the event that will not get ‘no. 6’ exactly once.

Then,

(a) { } 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 A = G G G ,G G G ,G G G ,G G G ,G G G ,G G G ,G G G

(b) { } 1 2 3 1 2 3 1 2 3 1 2 3 B = G G G ,G G G ,G G G ,G G G

(c) { } 1 2 3 1 2 3 1 2 3 C = G G G ,G G G ,G G G

Definition of Complement Event

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

elements of S that are not in A and is denoted by the symbol A or A' .

Definition of Intersection Event

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

the event containing all elements that are common to A and B.

Definition of Union of Event

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

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

Definition of Mutually Exclusive Events

Two events A and B are mutually exclusive or disjoint if A∩ B =φ or

{ } that is, if A and B have no element in common.

Exercise 2.1

If S = {p, q, r, s,t,u, v,w, x, y} and A = {q, s,t,v, x}, B = {p, r,v,w, y},

C = {r,u,w, y} and D = {p,t, x}, list the elements of the sets

corresponding to the following events:

(a) A∪C

34 Intro to Statistics & Probability

(b) A∩ B

(c) C'

(d) (A'∪B)∩ D

(e) (B ∩C)'

(f) (A∩ B'∩D)∪C

2.2 Concept of Probability

Definition of Probability

The likelihood of the occurrence of an event that is measured by using

numerical value.

Definition of Equally Likely Outcomes

Two or more outcomes that have the same probability of occurrence.

Example 2.3

(a) Roll a die once:

The probability of getting no. 1 = probability of getting no. 2 =

probability of getting no. 3 = probability of getting no. 4 =

probability of getting no. 5 = probability of getting no. 6 =

6

1 .

(b) Toss a coin once:

The probability of getting a head = probability of getting a tail =

2

1 .

Theorem

If an experiment can result in any one of N different equally likely

outcomes, and if exactly n of these outcomes correspond to event A, then

the probability of event A is

N

P(A) = n

Example 2.4

Chapter 2 Probability 3 5

A mixture of marbles contains 5 red marbles, 7 green marbles and 8

yellow marbles. If a person makes a random selection of one of these

marbles, find the probability of getting

(a) a green marble

(b) a yellow marble

Solution

Total of marbles = 20

(a)

20

P(G) = 7

(b)

5

2

20

P(Y) = 8 =

Definition

The probability of an event A is the sum of the weights of all sample

points in A. Therefore,

0 ≤ P(A) ≤ 1, and P(S) = 1

For an impossible event, M: P(M) = 0

For a sure event, C: P(C) = 1

2.3 Marginal and Conditional Probability

Definition of Marginal Probability

The probability of a single event without consideration of any other

event.

Definition of Conditional Probability

The probability that an event will occur given that another event has

already occurred.

If A and B are two events, then

36 Intro to Statistics & Probability

Marginal Probability: P(A) or P(B)

Conditional Probability:

P(A| B) - read as “The probability of A given that B has

already occurred”.

P(B | A) - read as “The probability of B given that A has

already occurred”.

Example 2.5

Samples of bottles from three companies are classified for conformance

to proper filling heights. The results from 115 samples are summarized as

follows:

Conforms

Yes (Y) No (N) Total

1 (C1) 27 6 33

2 (C2) 28 9 37

3 (C3) 35 10 45

Company

Total 90 25 115

Marginal Probability:

(a) P(sample of bottles from company 1) = P(C1) =

115

33

(b) P(C2) =

115

37

(c)

115

P(C3) = 45

(d)

115

P(Y) = 90

(e)

115

P(N) = 25

Conditional Probability:

(f) P(sample of bottles from company 1 given that it conforms to

proper filling heights)

Chapter 2 Probability 3 7

= P(C1| Y)

no.of samples that conform to proper filling heights

= no.of samples fromcompany 1and they conform to proper filling heights

90

= 27

(g)

90

P(C2 | Y) = 28

(h)

90

P(C3 | Y) = 35

(i)

25

P(C1| N) = 6

(j)

25

P(C2 | N) = 9

(k)

25

P(C3 | N) = 10

(l)

33

P(Y | C1) = 27

(m)

37

P(Y | C2) = 28

(n)

45

P(Y | C3) = 35

(o)

33

P(N | C1) = 6

(p)

37

P(N | C2) = 9

(q)

45

P(N | C3) = 10

2.4 Mutually Exclusive Events

Two events that cannot occur together are called mutually exclusive

events.

38 Intro to Statistics & Probability

Example 2.6

A statistical experiment has 15 equally likely outcomes that are denoted

by a, b, c, d, e, f, g, h, i, j, k, l, m, n and o. Let event A = {d, f, g, j, k, m, n,

o}, B = {c, d, e, f, o}, C = {k, a, c, j} and D = {k, n, m, a, g}.

Are events A & B mutually exclusive?

Are events A & C mutually exclusive?

Are events B & C mutually exclusive?

Are events D & B mutually exclusive?

Solution

A and B are not mutually exclusive since A∩ B = {d, f ,o}

A and C are not mutually exclusive since A∩C = {k, j}

B and C are not mutually exclusive since B ∩C = {c}

D and B are mutually exclusive since D ∩ B = { } or φ

2.5 Independent and Dependent Events

Two events are said to be independent if the occurrence of one does not

affect the probability of the occurrence of the other.

A and B are independent if either

P(A | B) = P(A) or P(B | A) = P(B)

Otherwise two events are said to be dependent.

Example 2.7

A company had checked causes of defects in 40 computer keyboards last

week. Of these, 16 have scratches and 15 have spray marks. Of the 16

computer keyboards with scratches, 5 have spray marks. Are the events

“scratches” and “spray marks” independent? Are they mutually

exclusive?

Chapter 2 Probability 3 9

Solution

Scratches

S S Total

M 5 10 15

M 11 14 25

Spray

Marks

Total 16 24 40

If M and S are independent, then P(M | S) = P(M) or P(S | M) = P(S) .

We show only one, that is P(M | S) = P(M) .

,

16

P(M | S) = 5 while

8

3

40

P(M) = 15 =

∴P(M | S) ≠ P(M)

Then, M and S are not independent.

M and S are not mutually exclusive events since they both can occur

together.

Exercise 2.2

A certain national car comes equipped with either an automatic or a

manual transmission, and the car is available in one of three colors.

Relevant frequency for various combinations of transmission type and

color are given in the accompanying table.

Color

White Blue Red

Transmission

Type

Automatic

Manual

425

345

460

278

321

433

Are the events “Automatic” and “Red” mutually exclusive?

40 Intro to Statistics & Probability

Are the events “Manual” and “Blue” independent?

Exercise 2.3

A case contains a total of 150 batteries that were manufactured on three

machines. Of them, 40 were manufactured on machine 1. Of the total

batteries, 100 meet company’s specifications. Of the 40 batteries that

were manufactured on machine 1, 20 meet company’s specifications

while of the 80 batteries that were manufactured on machine 3, 60 meet

company’s specifications. Are the events “do not meet company’s

specifications” and “machine 2” independent?

2.6 Complementary Events

The complement of event A, denoted by A or A' is the event that

includes all the outcomes for an experiment that are not in A. Two

complementary events are always mutually exclusive.

Take note that

P(A) + P(A) = 1 or P(A) + P(A') = 1

and

P(A) = 1− P(A) = 1− P(A')

2.7 Multiplication Rule

The probability of the intersection of two events is called their joint

probability. It is written as P(A∩ B).

The probability will be:

( ) ( ) ( | )

( ) ( ) ( | ) or

P A B P B P A B

P A B P A P B A

∩ =

∩ =

In other words, conditional probability will be

Chapter 2 Probability 4 1

( )

( | ) ( )

( )

( | ) ( )

P A

P B A P A B

P B

P A B P A B

∩

=

∩

=

given that P(A) ≠ 0 and P(B) ≠ 0 .

Multiplication Rule for Mutually Exclusive Events

If A and B are two mutually exclusive events, then

P(A∩ B) = 0

Example 2.8

Disks from 3 suppliers are analyzed for surface finish.

Surface Finish

Excellent Good Bad

Supplier 1 140 450 80

Supplier 2 60 250 65

Supplier 3 200 300 70

Suppose one disk is selected at random. Find the following probabilities:

(a) P(supplier 1 and good)

(b) P(bad and supplier 3)

(c) P(supplier 1 and supplier 3)

Solution

Surface Finish

Excellent

(E)

Good

(G)

Bad

(B)

Total

Supplier 1

(S1)

140 450 80 670

42 Intro to Statistics & Probability

Supplier 2

(S2)

60 250 65 375

Supplier 3

(S3)

200 300 70 570

Supplier

Total 400 1000 215 1615

(a)

323

90

1615

450

1615

1000

1000

( 1 ) ( 1| ) ( ) 450 = = ⎟⎠

⎞

⎜⎝

⎛

⎟⎠

⎞

⎜⎝

P S ∩G = P S G P G = ⎛

(b)

323

14

1615

P(B ∩ S3) = 70 =

(c) P(S1∩ S3) = 0

Example 2.9

The probability that a sample of fly ash contains sulfate is 0.80; the

probability that this sample contains calcium is 0.78; and the probability

that this sample contains both components is 0.75. Find the probability

that this sample of fly ash will contain

a. sulfate given that this sample contains calcium.

b. calcium given that this sample contains sulfate.

Solution

P(S) = 0.8, P(C) = 0.78, P(S ∩C) = 0.75

(a) 0.9615

0.78

0.75

( )

( | ) ( ) = =

∩

=

P C

P S C P S C

(b) 0.9375

0.8

0.75

( )

( | ) ( ) = =

∩

=

P S

P C S P C S

Exercise 2.4

A box contains 30 computer disks, 10 of which are defective. If 3

computer disks are selected at random and without replacement from this

box, what is the probability that exactly 2 non-defective computer disks

are chosen?

Chapter 2 Probability 4 3

Exercise 2.5

One bag contains 6 white marbles, 7 black marbles and 7 green marbles.

A second bag contains 8 white marbles, 5 black marbles and 4 green

marbles. Ahmad selects a marble from the first bag and places it in the

second bag. A marble is now drawn from the second bag by Azli. What

is the probability that the marble drawn by Azli is green?

Multiplication Rule for Independent Events

The probability of the intersection of two independent events A and B

is

P(A∩ B) = P(A)P(B)

Example 2.10

Azila will be having 2 mid-term exams in a week. The probability that

she will pass an Islamic Civilization subject is 0.92, and the probability

that she will pass an IT subject is 0.88. Assume that the events she will

pass any of these two subjects are independent. Find the probability that

(a) she will pass both subjects.

(b) she will pass neither Islamic Civilization nor IT subject.

Solution

Let I be the event that Azila will pass an Islamic Civilization subject.

T be the event that Azila will pass an IT subject.

P(I ) = 0.92, P(T) = 0.88

(a) P(I ∩T) = P(I )P(T) = (0.92)(0.88) = 0.8096

(b) P(I ∩T) = P(I )P(T ) = (0.08)(0.12) = 0.0096

Exercise 2.6

If P(C) = 0.65, P(D) = 0.4 and P(C ∩ D) = 0.24 , are the events C and

D independent?

44 Intro to Statistics & Probability

Exercise 2.7

The probability that a new car needs a service within 6 months of

purchase is 0.4. Three independent cars are randomly selected.

(a) What is the probability that all cars selected will need a service

within 6 months of purchase?

(b) What is the probability that exactly 2 cars selected will need a

service within 6 months of purchase?

(c) What is the probability that two or less cars selected will not need

a service within 6 months of purchase?

Exercise 2.8

A coin is biased so that a tail is 5 times as likely to occur as a head. If the

coin is tossed 3 times, what is the probability of getting 1 tail and 2

heads?

2.8 Addition Rule

The probability of the union of two events A and B is

P(A∪ B) = P(A) + P(B) − P(A∩ B)

Addition Rule for Mutually Exclusive Events

The probability of the union of two mutually exclusive events A and B is

P(A∪ B) = P(A) + P(B)

Example 2.11

All 200 cars were rented by a consulting firm from 3 agencies A, B and

C. The cars were checked whether they have bad tyres or not. Based on

the information given, the following two-way classification table was

prepared.

Agency Have bad tyres Do not have bad

tyres

A 10 38

B 17 80

Chapter 2 Probability 4 5

C 15 40

Suppose one car is selected at random from this firm. Find the following

probabilities:

a. P(a car selected comes from agency A or does not have bad tyres)

b. P(a car selected has bad tyres or comes from agency C)

c. P(a car selected comes from Agency A or B)

Solution

Agency Have bad tyres

(T)

Do not have

bad tyres

(T )

Total

A 10 38 48

B 17 80 97

C 15 40 55

Total 42 158 200

(a)

25

21

200

38

200

158

200

P(A∪T) = P(A) + P(T) − P(A∩T) = 48 + − =

(b)

100

41

200

15

200

55

200

P(T ∪C) = P(T) + P(C) − P(T ∩C) = 42 + − =

(c)

40

29

200

97

200

P(A∪ B) = P(A) + P(B) = 48 + =

Exercise 2.9

A pair of dice is thrown twice. What is the probability of getting totals of

6 and 9?

2.9 Bayes’ Theorem

In general, a collection of sets E1, E2 , E3 ,..., Ek such that

E1 ∪ E2 ∪ E3...∪ Ek = S is said to be exhaustive.

Total Probability Rule

46 Intro to Statistics & Probability

Assume E1, E2 , E3 ,..., Ek are k mutually exclusive and exhaustive sets.

Then,

P(B) = P(B ∩ E1 ) + P(B ∩ E2 ) + ... + P(B ∩ Ek )

= P(B | E1 )P(E1 ) + P(B | E2 )P(E2 ) + ... + P(B | Ek )P(Ek )

Example 2.12

In a certain assembly plant, four machines, A, B, C and D make 25%,

20%, 35% and 20% respectively, of the products. It is known from past

experience that 2%, 4%, 1% and 0.5% 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?

Solution

P(A) = 0.25, P(B) = 0.2, P(C) = 0.35, P(D) = 0.2

P(DF | A) = 0.02, P(DF | B) = 0.04, P(DF | C) = 0.01,

P(DF | D) = 0.005

Then,

0.0175

(0.02)(0.25) (0.04)(0.2) (0.01)(0.35) (0.005)(0.2)

( ) ( | ) ( ) ( | ) ( ) ( | ) ( ) ( | ) ( )

=

= + + +

P DF = P DF A P A + P DF B P B + P DF C P C + P DF D P D

Bayes’ Theorem

From previous, we know that,

( )

( | ) ( )

P B

P A B P A B ∩

= and

( )

( | ) ( )

P A

P B A P B A ∩

=

then P(A∩ B) = P(B ∩ A) = P(A | B)P(B) = P(B | A)P(A)

and we also can write

Chapter 2 Probability 4 7

( )

( | ) ( | ) ( )

P B

P A B = P B A P A

In general, we obtain the following result, which is known as Bayes’

Theorem

If E1, E2, …, Ek are k mutually exclusive and exhaustive events and B is

any event, then

( | ) ( ) ( | ) ( ) ... ( | ) ( )

( | ) ( | ) ( )

1 1 2 2

1 1

1

P B E P E P B E P E P B Ek P Ek

P E B P B E P E

+ + +

=

Exercise 2.10

A firm rents cars from three companies - 20% from company 1, 20%

from company 2 and 60% from company 3. If 10% of the cars from

company 1, 12% of the cars from company 2 and 4% of the cars from

company 3 have bad conditions, what is the probability that

(a) the firm will get a car with bad conditions?

(b) a car with bad conditions rented by the firm came from company 2?

(c) a car without bad conditions rented by the firm came from

company 3?

Exercise 2.11

Marketing managers must estimate the sales of a new model of one

telephone. The records of one major telecommunication company

indicate that 10% of all new telephones sell more than projected, 30%

sell close to projected, and 60% sell less than projected. Of those that sell

more than projected, 70% are proposed for additional new features, as

are 50% of those that sell close to projected, and 20% of those that sell

less than projected.

(a) What percentage of telephones produced by this company will

be proposed for additional new features?

(b) What percentage of telephones produced by this company that

are proposed for additional features sold less than projected in their

first edition?

Exercise 2.12

48 Intro to Statistics & Probability

It is known from past experience that the probability of selecting an adult

over 60 years of age who are smokers is 0.35. Of those adults over 60

years of age who are smokers, 55% of them have heart attack. Of those

adults over 60 years of age who are nonsmokers, 12% of them have heart

attack. What is the probability of selecting one of these adults with heart

attack is found to be a nonsmoker? What is the probability of selecting

one of these adults without heart attack is found to be a smoker?

Review Exercises

1. List the elements of each of the following sample spaces:

(a) the set of integers between 1 and 50 divisible by 3

(b) the set S = {x | x3 − x2 − 4x + 4 = 0}

(c) the set 9}

2

{ | 1 3 5 ≤

−

S = x − ≤ x

2. Consider the sample space S = {Toyota, Honda, Mercedes, BMW,

Naza, Ferrari, Renault} and the events

A = {Toyota, Honda, Renault}

B = {Honda, Mercedes, BMW}

C = {Ferrari}

List the elements of the sets corresponding to the following events:

(a) B'

(b) (A'∩B')∪C

(c) A∩ B ∩C'

(d) (B ∪C)'

(e) (B ∩C)∪ A'

(f) (A'∩B')∪(B ∩C')

3. When a machine goes down, there is 60% chance that it is due to a

technical problem and a 50% chance that it is due to a human error

problem. There is a 40% chance that it is due to a technical and a

human error problem.

(a) What is the probability that at least one of these problems are at

fault?

(b) What is the probability that there is a technical problem but no

human error problem?

Chapter 2 Probability 4 9

4. Samples of banners produced by a printing company are classified on

the basis of colour intensity and print quality. The results of five

hundred banners are summarized as follows:

Print Quality

Good Bad

Colour Good 359 57

Intensity Bad 77 7

Let X denotes the event that a banner has good colour intensity, and

let Y denotes the event that a banner has good print quality. If a

banner is selected at random, determine the following probabilities:

(a) P(X )

(b) P(Y)

(c) P(X ')

(d) P(X ∩Y)

(e) P(X ∪Y)

(f) P(X ∪Y')

(g) P(X | Y)

(h) P(Y | X )

(i) If the selected banner has good colour intensity, what is the

probability that the print quality is good?

(j) If the selected banner has bad print quality, what is the probability

that the colour intensity is good?

5. Assume that 2% of all unacceptable cereal cartons taken from a

production are due to unreadable label and assume that 1% of all these

unacceptable cereal cartons are due to improper package weight. Also,

2.5% of this unacceptable cereal cartons will experience at least one of

these problems. What is the probability that for a randomly selected

unacceptable cereal cartons, improper package weight will be

identified as the cause but there will be no problem due to unreadable

label?

6. Suppose that in an assembly of 500 defective computers, it is found

that 230 are due to overload, 245 are due to software problem, 208 are

due to hardware problem, 132 are due to overload and software

problems, 67 are due to hardware and software problems, 75 are due

to overload and hardware problems and 45 defective computers are

due to all 3 problems. If a defective computer is selected at random

50 Intro to Statistics & Probability

from this assembly, find the probability that this computer is

defective due to:

(a) overload but not due to software problems.

(b) overload and hardware problems but not due to software

problems.

(c) neither software nor hardware problems.

7. If P(A) = 0.6, P(B) = 0.3 and P(A∩ B) = 0.05 , determine

the

following probabilities:

(a) P(A')

(b) P(A∪ B)

(c) P(A'∩B)

(d) P(A∩ B')

(e) P[(A∪ B)']

(f) P(A'∪B)

8. In an inspection of cans of babies’ food, it was found that 8% showed

signs of cracks on top of cans and surface defects on side of cans,

30% showed cracks on top of cans, and 40% showed surface defects

on side of cans.

(a) If a can of baby’s food has surface defects on side of it, what is

the probability that this can will also have cracks on top of it?

(b) If a can of baby’s food does not have cracks on top of it, what is

the probability that this can has surface defects on side it?

9. Suppose that for a given computer program, there is a 40% chance

that computational errors will occur and a 28% chance that

configuration errors will occur. If the computational errors occur,

there is an 80% chance that configuration errors will occur.

(a) Find the probability that for given computer programs,

both computational and configuration errors will not occur.

(b) Find the probability that the computational errors will occur given

that the configuration errors do not occur.

10. A batch of 500 soft drink bottles from a bottling company contains

10 bottles that are defective. Three bottles are selected, at random,

without replacement, from the batch.

(a) What is the probability that the third one selected is

defective given that the first was okay and second one

selected was defective?

Chapter 2 Probability 5 1

(b) What is the probability that the third one selected is okay given

that the first one selected was defective and the second one

selected was okay?

(c) What is the probability that exactly 1 defective bottle is selected?

11. Two ordinary dice are thrown. Find the probability that the sum of

the scores obtained is

(a) a multiple of 5.

(b) greater than 9.

(c) multiple of 5 or is greater than 9.

12. One bag contains 8 white and 9 black marbles. A second bag

contains 11 white, 10 black and 12 green marbles. If 1 marble is

selected at random from each bag, find the probability that

(a) all marbles are black.

(b) neither marbles are white.

(c) there are 2 marbles with different colours.

13. A first year student has final exams for three consecutive days. The

probability that he will fail Social Ethnic subject is 0.05%, the

probability that he will fail Chemistry is 2% while the probability he

will fail an IT subject is 1%. Assume that the events that he will fail

any of these three subjects are independent.

(a) What is the probability that this student will not fail any of these

subjects?

(b) What is the probability that this student will fail either Chemistry

or IT subject?

14. A pharmaceutical manufacturing process uses a high speed

compressing machine as part of the process of tablet production.

The company will use machine A, B, C and D with probabilities

0.23, 0.45, 0.19 and 0.13 respectively. From past experience it is

known that the probability of defective tablet produced by the

machines are 0.11, 0.2, 0.07 and 0.05 respectively. Suppose a

defective tablet is produced:

(a) What is the probability that it is produced by machine D?

(b) What is the probability that it is produced by machine B?

15. An experiment was conducted to assess the effect of using magnets

at the filler point in the manufacturing of tea filter packs. Usage of

magnets at the filler point produces 80% of the tea filter packs.

From past experience, the technician who controls the filler

52 Intro to Statistics & Probability

point discovers that the usage of magnet will affect the weights of

filter packs where 5% of the filter packs with weight less than 20g

are produced while without using magnets, 3% of filter packs with

weight less than 20g are produced. Suppose a filter pack weighing

more than 20g is produced. Which of the two methods (with

magnet and without magnet) is more likely to have supplied this

filter pack weighing more than 20g?

16. Suppose that five workers at a soft drink company are supposed

to paste expiration date on each can at the end of the assembly line.

A, who pastes 30% of the cans, fails to paste the expiration date

once in every 300 cans, B, who pastes 15% of the cans, fails to

paste the expiration date once in every 250 cans, C, who pastes

35% of the cans fails to paste the expiration date once in every 100

cans, D, who pastes 7% of the cans, fails to paste the expiration

date once in every 150 cans and E, who pastes 13%, fails to paste

the expiration date once in every 200 cans. If a quality control

inspector finds a can without an expiration date, what is the

probability that C fails to paste it?

17. Computer centre had three printers that print at different speeds.

Programs are routed to the first available printer.

Let A – printer A

B – printer B

C – printer C

J – printer jam

D – destroy the quality of print out

Assume:

P(A) = 0.6, P(C) = 0.1

P(J | A) = 0.2, P(J '| B) = 0.7, P(J | C) = 0.15

P(D | J ∩ A) = 0.02, P(D'| J '∩ A) = 0.97, P(D'| J ∩ B) =0.95

P(D | J '∩B) = 0.01, P(D | J ∩C) = 0.01, P(D | J '∩C) =0.015

(a) Find P(A∩ J '∩D) .

(b) Find P(D') .

(c) Find P(J ∩ D) .

Chapter 2 Probability 5 3

(d) Find the probability that a print out came from printer C given

that the quality of print out is not destroyed but the printer is jam.