topic 5 probability permutation and combination

CHAPTER 4 - PROBABILITY

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STATISTICS 1 / QMT 110 / KOLEJ UNIKOP / MMF / RSZ / 2008 35

PROBABILITY

Experiment : A situation involving chance or probability

that leads to result called outcomes.

Outcome : The result of single trial of an experiment.

Event : One or more outcomes of an experiment.

Probability : The measure of how likely an event is.

Problem 1 : A spinner has 4 equal sectors colored

yellow, blue, green and red.

a. What are the chances of landing on

blue after spinning the spinner?

b. What are the chances of landing on

red?

Experiment : Spinning the spinner.

Outcomes : The possible outcomes are landing on

yellow, blue, green or red.

Event : a. Landing on blue.

b. Landing on red.

Probability/Solution : a. The chances of landing on blue are 1

in 4, or one fourth. Therefore, the

probability of landing on blue is one

fourth.

b. The chances of landing on red are 1

in 4, or one fourth. Therefore, the

probability of landing on red is one

fourth.

EXAMPLE:


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Problem 2 : Toss a dice and observe the number that

appears on the upper face.

a. What is probability of rolling an even

number?

b. What is the probability of rolling a

number less than 4?

Experiment : __________________________________________

Outcomes : __________________________________________

Event : a. ____________________________________

b. ____________________________________

Probability/Solution : a. ____________________________________

____________________________________

b. ____________________________________

____________________________________

Simple event : An outcome or an event that cannot be

further broken down into simpler

components. Usually denote by E.

Sample space : Set of all simple events

In terms of simple events; sample space is a

collection of one or more simple events.

Usually denote by S.

Probability : If the probability space S consist of a finite

number of equal likely outcomes, then the

probability of an event E , written ( )P E is

defined as:

( )( )( )Sn

EnEP ==

Outcomes Possible of Number Total The

occur can event waysof number The E


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Refer to Problem 1:

S = { Yellow (Y), Blue (B), Green (G), Red (R) }

Let Event A = { Landing on Blue } = { B }

Event B = { Landing on Red } = { R }

n (S) = 4; n (A) = 1; n (B) = 1

Therefore; a. P (A) = 4

1=

n(S)

n(A)

b. 4

1

n(S)

n(B)P(B) ==

Refer Problem 2:

S =

Let Event A =

Event B =

n(S) = ; n (A) = ; n (B) =

Therefore; a. P(A) =

b. P(B) =

TAKE NOTE:

The probability of an event A is a

number between 0 and 1 inclusive

( )0 1P A≤ ≤

EXAMPLE


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Sample space, { }1,2,3,4,5,6,7,8,9,10S = , given A = { } { }1,3,5,7,9oddnumbers =

and B = { } { }2,4,6,8,10evennumbers = . Find ( ) ( )P A andP B

A card is drawn at random from an ordinary pack of 52 playing cards. Find

the probability that the card is a seven.

Two fair coins are tossed. Find the probability that the two heads are

obtained.

A ball is drawn from a box containing 10 red, 15 black, 5 green and 10 yellow

balls. Find the probability that the ball is

a. Black

b. Not green or yellow

c. Not yellow

d. Red or black or green

e. Not blue

EXAMPLE:

EXAMPLE:

EXAMPLE

EXAMPLE:


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Venn diagram : A simple way of illustrating the relationship

between sets. The sets are represented by a

simple plane area, usually bounded by a

circle or a closed space.

The data below shows a survey of 305 college students.

125 take Mathematics

115 take Accounting

110 take Science

35 take Mathematics and Accounting

30 take Mathematics and Science

34 take Accountings and Science

10 take Mathematics, Science and Accounting

a. Illustrates the information using a Venn Diagram

b. How many take Mathematics only

c. How many takes Accounting but not Science

d. How many take Science or Accounting

e. How many students who are not taking any subject

EXAMPLE:


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Tree diagram : Illustrates experiments that can be

generated in stages. Each level of

branching on the tree corresponds to a

step required to generate the final

outcome.

A medical technician records a person’s blood type and Rh factor. List the

sample events in the experiment.

Two balls are drawn from a box containing 10 red, 15 black, 5 green and 10

yellow balls. If the balls are selected

i) First with replacement and again without replacement.

ii) First without replacement and again without replacement.

Find the probability if the balls selected are:

a. Of the same color

b. Not the same colors

c. Green and yellow

d. Both black

EXAMPLE:

EXAMPLE:


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Probability table : A tabulated form to illustrate experiments

that can be generated in stages.

A medical technician records a person’s blood type and Rh factor. List the

sample events in the experiment.

Probability Table:

Blood Type

Rh Factor A B AB O

Negative

Positive

Refer to the table illustrated below.

Length of Service

Loyalty < 1 year 1-5 years 6-10 years >10 years Total

Remain 10 30 5 75 120

Not Remain 25 15 10 30 80

Total 35 45 15 105 200

a. What is the probability of selecting at random an executive who has

been working for more than 10 years?

b. What is the probability of selecting at random an executive who would

not remain with the company and has less than one year of service?

EXAMPLE:

EXAMPLE:


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Operations of set theory:

i. Unions A B∪ : The union of events A and B, is the event

that A or B or both occur.

ii. Intersection A B∩ : The intersection of events A and B, is the

event that both A and B occur.

iii. Complements A′ : Also denoted by CA and A represents the

area (values) not in A

iv. Disjoint 0A B∩ = : Also called mutually exclusive events are

two events that does not contain have

similarities.


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Two fair coins are tossed, and the outcome is recorded. These are the events

of interest:

A: Observe at least one head

B: Observe at least one tail

Define the events A, B, A ∩ B, and A ∪ B as collections of simple events, and

find their probabilities.

EXAMPLE:


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An experiment consists of tossing a single dice and observing the number of

dots that shows on the upper face. Events A, B and C are defined as follows:

A : Observe a number less than 4

B : Observe a number less than or equal to 2

C : Observe a number greater than 3

Draw a Venn diagram that illustrates the situation:

Find the following probabilities:

1. P( S )

3. P( B )

4. P(A ∩ B ∩ C)

5. P(A ∩ B)

6. P(A ∩ C)

7. P(B ∩ C)

8. P(A ∪ C)

9. P(B ∪ C)

EXAMPLE:


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Addition Rule:

The addition rule helps solve probability problems that involve two events.

When asked to find the probability of A or B, mean that A can happen, or B

can happen, or both can happen together. This is what is stated in the

addition rule.

Notation for Addition Rule:

P (A or B) = P (in a single trial, event A occurs or event B occurs or

they both occur)

Additive Rule of

Probability : Deals with unions of events

Given two events, A and B, the probability of their union, A ∪ B, is equal to

* Mutually exclusive : Two events are mutually exclusive when

one event occurs the other cannot.

When events A and B are mutually

exclusive, P( A ∩ B) = 0

Suppose we roll two dice and want to find the probability of rolling a sum of 6

or 8. This can be written in words as P(6 or 8) or more mathematically is

P(6 8). So what is the probability of getting a 6 or an 8 or both?

P (6) = 5/36

P (8) = 5/36

P (6 and 8 together) is impossible so the probability is 0.

So P(6 8) = 5/36 + 5/36 - 0 = 10/36 = 5/18

For the additive rule: P (A ∪ B) = P (A) + P(B) – P (A ∩ B)

If A and B are mutually exclusive, then P (A ∩ B) = 0

thus P (A ∪ B) = P (A) + P (B)

P (A ∪ B) = P (A) + P(B) – P (A ∩ B)


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1. You are going to pull one card out of a deck. Find P(Ace King).

2. You are going to roll two dice. Find P(sum that is even or sum that is a

multiple of 3).

3. Drawing a card from an ordinary deck of cards. Find P(three or jack),

P(three or jack), and P(club or four).

Remember that:

- OR (the union symbol ) means that one

or the other or both events can happen. - AND (the intersection symbol ∩ ) means

that two events happen.

EXAMPLE


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Multiplication Rule:

The multiplication rule also deals with two events, but in these problems the

events occur as a result of more than one task (rolling one die then another,

drawing two cards, spinning a spinner twice, pulling two marbles out of a

bag, etc).

When asked to find the probability of A and B, we want to find out the

probability of events A and B happening.

Notation for Multiplication Rule:

P (A and B) = P (event A occurs in a first trial and event B occurs in a

second trial / event A and event B occur together)

Multiplicative Rule of

Probability : Deals with intersections of events

Given two events, A and B, the probability

that both of two events occur is

* Independent events : Two events A and B are said to be

Independent if and only if either

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

Otherwise, the events are said to be

dependent.

For multiplicative rule: P (A ∩ B) = P (A) P (B|A)

@ P (A ∩ B) = P (B) P (A|B)

If A and B are independent, then P (A ∩ B) = P (A) P (B)

Similarly, if A, B and C are independent events, then the probability

that A, B, and C occur is P (A ∩ B ∩ C) = P (A) P (B) P (C)

P (A ∩ B) = P (A) P (B|A)

@

P (A ∩ B) = P (B) P (A|B)


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1. Suppose we roll one die followed by another and want to find the

probability of rolling a 4 on the first die and rolling an even number on

the second die.

P (4) = 1/6

P (even) = 3/6

So P (4 even) = (1/6) (3/6) = 3/36 = 1/12

� While the rule can be applied regardless of dependence or

independence of events, we should note here that rolling a 4 on

one die followed by rolling an even number on the second die

are independent events.

� Each die is treated as a separate thing and what happens on

the first die does not influence or effect what happens on the

second die. This is our basic definition of independent events:

the outcome of one event does not influence or affect the

outcome of another event.

Notice:

In this problem we are not dealing with the sum of both dice. We are

only dealing with the probability of 4 on one die only and then, as a

separate event, the probability of an even number on one dies only.

2. Suppose you have a box with 3 blue marbles, 2 red marbles, and

4 yellow marbles. We are going to pull out the first marble, leave it out,

and then pull out another marble. What is the probability of pulling out

a red marble followed by a blue marble?

� We can still use the multiplication rule which says we need to

find P(red) P(blue). But be aware that in this case when we go

to pull out the second marble, there will only be 8 marbles left in

the bag.

P (red) = 2/9 P (blue) = 3/8

P (red blue) = (2/9)(3/8) = 6/72 = 1/12

Notice:

The events in this example were dependent. When the first marble was

pulled out and kept out, it affected the probability of the second

event. This is what is meant by dependent events.

EXAMPLE:


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1. There are 11 marbles in a bag. Two are yellow, five are pink and four

are green. Suppose you pull out one marble, record its color, put it

back in the bag and then pull out another marble. Find

a. P (yellow and pink)

b. P (pink and green)

2. Suppose you are going to draw two cards from a standard deck. What

is the probability that the first card is an ace and the second card is a

jack.

Conditional Probability:

This rule is applied when you have two events and you already know the

outcome of one of the events.

Conditional Probability : Probability obtained with the additional

information that some other event has

already occurred.

The conditional probability of B, given that A has occurred, is:

The conditional probability of A, given that B has occurred, is:

( )( | ) ( ) 0

( )

P A BP B A if P A

P A

∩= ≠

( )( | ) ( ) 0

( )

P A BP A B if P B

P B

∩= ≠

EXAMPLE:


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1. A math teacher gave her class two tests. 25% of the class passed both

tests and 42% of the class passed the first test. What percent of those

who passed the first test also passed the second test?

� This problem describes a conditional probability since it asks us

to find the probability that the second test was passed given

that the first test was passed.

( )( )

( )

%606.042.0

25.0===

=FirstP

SecondandFirstPFirst|SecondP

2. A survey of 500 adults asked about college expenses. The survey asked

questions about whether or not the person had a child in college and

about the cost of attending college. Results are shown in the table

below.

Cost Too Much Cost Just Right Cost Too Low

Child in College 0.30 0.13 0.01

Child not in College 0.20 0.25 0.11

Suppose one person is chosen at random. Given that the person has a

child in college, what is the probability that he or she ranks the cost of

attending college as “cost too much”?

This problem reads:

P (cost too much | child in college) or

P (cost too much given that there is a child in college)

According to the conditional probability rule:

P(cost too much child in college) =

P(cost too much child in college) = 0.30

P(child in college) = 0.30 + 0.13 + 0.01 = 0.44

Therefore;

EXAMPLE:


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=

35

Complementation : The complement of an event A, denoted

by CA , consists of all the simple events in

the sample space S that are not in A.

1. For events A and B it is known that

( ) ( ) ( )2 7 5,

3 12 12P A P A B andP A B= ∪ = ∩ = . Find

a. ( )P B

b. P( B|A )

2. For events A and B it is known that ( ) ( ) ( ) 0.1P A P B andP A B= ∩ = and

( ) 0.7P A B∪ = . Find ( )P A .

3. Given ( ) 4.0=AP and ( ) 5.0=BP

a. If A and B are mutually exclusive events, find P (both A and B)

b. If A and B are independent events, find P (either A and B)

( ) ( )' 1P A P A= −

EXAMPLE:

36

4. In a color preference experiment, 8 toys are placed in a container. The

toys are identical except for color where 2 are red, and six are green.

A child is asked to choose two toys at random, once with replacement

and once without replacement.

What is the probability that the child chooses the two red toys for both

ways? Draw tree diagram for each method to illustrate your working.

(Relate the tree diagram using the multiplicative rule of probability)

5. An experiment consists of tossing a single dice and observing the

number of dots that shows on the upper face. Events A, B and C are

defined as follows:

A : Observe a number less than 4

B : Observe a number less than or equal to 2

C : Observe a number greater than 3

Are events A and B independent? Are the events A and B mutually

exclusive?

37

6. Toss two coins and observe the outcome. Define these events:

A: Head on the first coin.

B: Tail on the second coin.

Are events A and B independent?

TAKE NOTE:

For this type of question,

ensure you prove the steps

accordingly.

38

Additional Rules using complements of events:

P (A ∪ B )’ = P ( A’ ∩ B’ )

P (A ∩ B )’ = P ( A’ ∪ B’)

P ( A ∩ B’ )

P ( A’ ∩ B )

P ( A’ )

39

Given ( ) 0.59P A = , ( ) 0.30P B = and ( ) 0.21P A B∩ = , find

a. ( )P A B∪

b. ( )' 'P A B∪

c. ( )'P A B∩

d. ( )' 'P A B∩

EXAMPLE:

40

Given ( ) 0.40P A = , ( ) 0.70P B = and ( ) 0.2P A B∩ = , find

a. ( )/P A B

b. ( )'/P A B

c. ( )/ 'P A B

d. ( )'/ 'P A B

EXAMPLE:

41

Bayes’ Theorem : The probability of event A, given that event B has subsequently occurred, is

( )( ) ( )

( ) ( )[ ] ( ) ( )[ ]A|BPAPA|BPAP

A|BPAPB|AP

+=

Suppose 5% of the population of Umen (a fictional country) have a disease

that is peculiar to that country. Consider there is technique to detect the

disease although it is not very accurate. The probability that the test indicates

disease is present is 0.90. The probability that a person actually does not have

the disease but test indicates the disease is present is written as 0.15.

Illustration using tree diagram:

Define the situation

Let D : Have the disease N : Does not have the disease

X : Test show the disease is present (positive)

X : Test shows the disease is not present (negative)

Draw tree diagram using the terms defined above

EXAMPLE:

42

Possible questions:

a. What is the probability of the patient is tested positive?

b. What is the probability of the patient is tested negative?

c. What is the probability that the patient has the disease and tested positive?

d. What is the probability that a person has the disease, given that he or she is

tested positive?

e. What is the probability that a person has the disease, given that he or she is

tested negative?

43

When a person needs a taxi, it is hired from one of the three firms : X, Y and Z.

Of the hiring, 40% are from X, 50% are from Y and 10% are from Z. For taxi hired

from X, 9% arrived late. The corresponding percentages for taxis hired that

arrived late from firm Y and Z are 6% and 20% respectively.

a. Construct a tree diagram for the above information.

b. Calculate the probability that the next taxi hired

i. will be from X and will not arrive late

ii. will arrive late

c. Given that a call is made for a taxi and that it arrives late, find the

probability that it came from Y.

44

Permutations : The number of ways we can arrange n

distinct objects, taking them r at a time is

!

( )!

n

r

n

n rP =−

Permutation is an ordered arrangement of all or part of a set o items.

The 3 letters, P, Q and R can be arranged in the following ways

PQR PRQ QPR RPQ RQP QRP

Each of the arrangement is called permutation of the letters P, Q and R.

Using the formula for permutation, the number of permutations of the 3 letters

taken 3 at a time is

( )3

3

3!

3 3 !

3!

0!

3 2 1

6

P =−

=

= × ×

=

Similarly, if the question requires the number of permutations of 3 letters taken

2 at a time is

( )3

2

3!

3 2 !

3!

1!

3 2 1

6

P =−

=

= × ×

=

The permutations are PQ, PR, QP, QR, RP, RQ.

.

TAKE NOTE:

PR and RP are considered as

2 different permutations

45

The numbers of permutations of n items with p are alike of a first kind, q is alike

of a second kind, r is alike of a third kind, and so on is

!

! ! !.....

n

p q r

The number of permutations if the n items are arrange in a circle

( )1 !

! ! !.....

n

p q r

−

Calculate the number of permutations that can be formed using letters from

the word STATISTICS.

3 lottery tickets are drawn from a total of 50. If the tickets are distributed to

each of 3 employees in the order in which they are drawn, the order will be

important. How many simple events are associated with the experiment?

46

In how many ways can the letters A, B, C, D and E can be arranged without

repetition when

a. all the 5 letters are taken at a time

b. 4 of the letters are taken at a time

c. 3 of the letters are taken at a time

47

Find the number of arrangements of the word MATHEMATICS and THEMSELVES

How many 3-letter words can be formed from the letters in the word

ABSOLUTE?

How many of these 3-letters words

a. Contain the letter S,

b. Do not contain any vowel

48

Combinations : The number of distinct combinations of n

distinct objects that can be formed, taking

them r at a time is

)!(!

!

rnr

n

Cn

r −=

Combination of a set of items is a selection of one or more of the items with

no consideration given to the order or arrangement of the items

The number of combinations of 3 letters, P, Q and R, taken 2 at a time is

( )3

2

3!

2! 3 2 !

3 2 1

2 1 1

3

C =−

× ×=

× ×

=

The combinations are: PQ or QP, QR or RQ and PR or RP.

The relation between permutation and combination can be written as

!r

PC

n

rn

r=

TAKE NOTE:

PR and RP are considered

the same combination

49

A printed circuit board may be purchased from 5 suppliers. In how many

ways can 3 suppliers be chosen from the 5.

A committee is to be formed from 8 men and 4 women. Find the number of

ways this committee can be formed consisting of

a. 7 members

b. 7 members, 5 men and 2 women

c. 7 members, men more than the women

EXAMPLE:

EXAMPLE:

TAKE NOTE:

Permutations and

Combinations techniques

are easier to perform by

using the box method

50

1. One card will be randomly selected from a standard 52-card deck.

What is the probability the card will be queen?

2. The National Center for Health Statistics reports that of 883 deaths, 24

resulted from an automobile accident, 182 from cancer and 333 from

heart disease. What is the probability that a particular death is due to

an automobile accident?

3. An ordinary dice is thrown. Find the probability that the number

obtained

a. Is a multiple of 3

b. Is less than 7

c. Is a factor of 6

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

{ }1,6,7D = . List the elements of the sets and the probabilities

corresponding to the following events:

a. CAc ∪

b. c

CB ∩

c. ( )cc

BS ∩

d. ( ) BDCc ∪∩

e. ( )c

BA ∩

5. If 3 coins are to be thrown simultaneously together, list down all the

elements.

6. Referring to question 5, what is the probability of obtaining

a. One head

b. At least two heads

c. All tails

EXERCISE:

51

7. A trainee has conducted a survey on the hand phone market among

students in private higher educational institutes (IPTS) around Kuala

Lumpur. From the survey among 130 students, the following data were

gathered:

53 students use MOTOROLA phones

58 students use NOKIA phones

54 students use SONY ERRICSON phones

30 students use NOKIA phones only

25 students use SONY ERRICSON phones only

28 students use MOTOROLA phones only

8 students use all three phones

a. Present the information gathered using a Venn Diagram

b. How many students use only MOTOROLA and NOKIA

phones?

c. How many students use at least 2 hand phones?

d. How many students did not use any off the phones

above?

8. The probability of surviving a certain transplant operation is 0.55. If a

patient survives the operation, the probability that his or her body will

reject the transplant within a month is 0.20. What is the probability of

surviving both of these critical stages?

9. In a college graduating class of 100 students, 54 studied mathematics,

69 studied history and 35 studied both mathematics and history.

a. Draw a Venn diagram that illustrates the above situation

b. Using the diagram drawn in (a), find the probability of

i. The student takes mathematics and history

ii. The student does not take either of these subjects

iii. The student takes history but not mathematics

52

10. A selected group of employees of Samsung Manufacturing is to be

surveyed about a new pension plan. In-depth interviews are to be

conducted with each employee selected in the sample. The

employees are classified as follows:

Classification Event No of

employees

Supervisors A 120

Maintenance B 50

Production C 1460

Management D 302

Secretarial E 68

What is the probability that the first person selected is

a. Either in maintenance or secretarial

b. Not in management

11. There are 100 students in a first year college intake, 45 are male

students of which 36 are studying programming. From the total female

students, 42 are studying programming. Draw a probability table to

illustrate the situation. What is the probability that a student selected

randomly is studying programming knowing that the student is a male?

12. A coin is loaded so that the probabilities of heads and tails are 0.52

and 0.48 respectively. If the coin is tossed three times, what is the

probabilities of getting

a. All heads

b. Two tails and a head in that order

13. Given that A and B are two events with probabilities P(A) = 0.4, P(A/B)

= 0.2 and P(B) = 0.15. Find

a. P(B/A)

b. P(A ∪ B)

14. Two event R and T are defined in a sample space with probabilities

P(R) = 0.35, P(T) = 0.28 and P(RT) = 0.06. Find

53

a. P(R ∪ T)

b. P(RT’)

c. P(R’T’)

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

scores obtained

a. is a multiple of 5

b. is greater than 9

c. is a multiple of 5 or is greater than 9

d. is a multiple of 5 and is greater than 9

16. Two events A and B are such that P(A) = 0.3 and P(B) = 0.4 and P(A/B)

= 0.1. Calculate the probabilities

a. that both the events occur

b. at least one of the event occur

c. B occur, given that A has occurred.

17. Three people in an office decide to enter a marathon race. The

respective probabilities that they will complete the marathon are 0.9,

0.7, and 0.6. Find the probability that at least one will not complete the

marathon. Assume that of each is independent of the performances of

the others.

18. Suppose A and B are two event with P(B) = 0.5, P(A / B) = 0.4 and P(A’ /

B’) = 0.3. Find

a. P(A ∩ B)

b. P(A ∪ B)

19. Let S and T are two events such that P(S) = 0.6, P(T) = 0.5 and P(S’ ∩ T) =

0.3. Obtain

a. P(S ∩ T)

b. P(T / S’)

54

20. Suppose that A and B are two events with P(A) = 1/5, P(B) = 1/4 and P(A ∪ B) = 1/3. Find

a. P(A ∩ B)

b. P(A / B)

c. P(B’ / A’)

21. A and B are two events where P(B) = 1/6 , P(A ∩ B) = 1/12, and P(B / A)

= 1/3. Find

a. P(A)

b. P(A / B’)

22. Given that P(A) = 0.8, P(A / B) = 0.8 and P(A ∩ B) = 0.5. Find

a. P(B)

b. P(B / A)

c. P(A ∪ B)

23. A sharpshooter hits a target with probability 0.75. Assuming

independence, find the probability of getting

a. A hit followed by two misses

b. Two hits and a miss in any order

55

24. A box contains 5 red bulbs, 4 blue bulbs and 3 yellow bulbs. Three balls

are selected at random from the box.

a. Find the probability that all three bulbs are of the same color.

b. Find the probability that the three bulbs are of different colors if

it is known that one of them is yellow.

25. A box contains 15 mathematics books and 10 music books. Two books

are selected one at a time without replacement, find

a. The probability that the first book selected is a mathematics

book

b. The probability that the second book selected is a music book if

the first book selected is a mathematics book

26. Given that P(C ∩ D) = ¼, P(C/D) = 1/3 and P(D/C) = 3/5.Find

a. P(C)

b. P(D)

c. P(C / D’)

27. If A and B are mutually exclusive events with ( ) 0.5P A = and ( ) 0.3P B = ,

find

a. ( )'P A

b. ( )'P B

c. ( )P A B∪

d. ( )' 'P A B∪

e. ( )'P A B∩

f. ( )' 'P A B∩

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28. Given the following probabilities ( ) ( ) 95.0,3.0 =∪= BAPAP

a. Find ( )BP if A and B are mutually exclusive

b. Find ( )BP if A and B are independent events

29. A box contains 4 red balls, 5 green balls and 8 blue balls

a. A ball is drawn at random from a box. Find the probability that it

is

i. Red

ii. Green or blue

b. Three balls are drawn one by one from the box. Find the

probability that the balls are drawn in the order of red, green

and blue if the ball is

i. Replaced

ii. Not replaced

30. From a box containing 6 black beads and 4 green beads. 3 beads are

drawn in succession, each beads being replaced in the box before the

next draw is made. Draw a tree diagram that represents the above

experiment. Find the probability that

a. All three are of the same color

b. Each color is represented

31. If the probability that student A will fail a certain statistics examination is

0.5, the probability that student B will fail the examination is 0.2 and the

probability that both student A and student B will fail the examination is

0.1, what is the probability that:

a. At least one of the two students will fail the examination?

b. Neither student A nor student B will fail the examination?

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32. Routine physical examinations are conducted annually as part of a

health service program for a company for its employees. It was

discovered that 8% of the employees need corrective shoes, 15% need

dental work and 3% need both health service. What is the probability

that an employee selected at random will need either corrective

shoes or major dental work?

33. An automatic Shaw machine inserts mixed vegetables into a plastic

bag. Experience revealed some packages were underweight and

some overweight, but mostly had satisfactory weight.

Weight Probability No of packages

Underweight, A 0.025 100

Satisfactory, B 0.900 3600

Overweight, C 0.075 300

1.000 4000

a. What is the probability of selecting three packages from the

production line and finding all three are underweight?

b. What does this probability mean?

34. The probability that a regularly schedule flight departs on time is 83.0)( =DP , the probability that it arrives on time is 92.0)( =AP , and

the probability that it departs and arrive on time is 78.0)( =∩ ADP .

Find the probability that a plane

a. Arrives on time given that it departed on time

b. Departed on time given that it has arrived on time

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35. A firm uses 3 local hotels to provide accommodations for its clients.

From past experience, it is known that 15% of its clients are assigned

rooms at Hotel A, 55% at Hotel B and 30% at Hotel C. It is known that

the probabilities Hotel A, Hotel B and Hotel C will be facing water

problems are 0.03, 0.02 and 0.07 respectively.

a. Draw a tree diagram for the above data.

b. What the probability that a client will be assigned a room

without water problems?

36. The Credit Department of a supermarket reported that 75% of their

sales are in cash while the remaining 25% are using credit cards. 80% of

the cash sales and 30% of the sales using credit cards are for the sales

amount of less than RM100.

a. Draw a tree diagram for the above problem.

b. What is the probability of a sale less than RM100?

c. Puan Rahimah has just bought a calculator that cost less than

RM100, what is the probability that she paid cash?

37. The number of employees of store A, B and C are distributed in the

percentage 20%, 30% and 50%. 50%, 60% and 70% of these are women

respectively. One employee is selected at random.

a. Draw a tree diagram to summarize the above problem.

b. What is the probability that the employee is a male?

c. Given that the employee is a male, what is the probability that

he works in store C?

38. In an annual sport election, there are three possible candidates. The

probability of Mr. Ahmad, Mr. Yaman and Mr. Zain being nominated

are 0.25, 0.45 and 0.3 respectively. If the nominated candidates are

taking part in the election, the probability that the election is won by

Mr. Ahmad, Mr. Yaman and Mr. Zain are 0.6, 0.35 and 0.60 respectively

a. Draw a probability tree to represent the above situation.

b. Given that someone won the election, what is the probability

that he is Mr. Zain?

39. The Snapquick Store gets its supply of camera from three suppliers A, B

and C in the ratio of 5 : 3 : 2. However, some of the cameras supplied

59

are faulty. 10% of the cameras obtained from supplier A are faulty, so

are 5% from supplier B and 3% from supplier C.

a. Construct a tree diagram.

b. One camera is chosen at random from the store. What is the

probability that the camera

i. is faulty

ii. was obtained from supplier A if it is found to be faulty.

40. Of a group of students studying at a School of Management, 48% are

male and 52% are female. 20% of the males and 30% of the females

from this group, major in Marketing. Find the probability that:

a. A student selected at random from this group is a female

majoring in Marketing.

b. A student selected at random from this group is not majoring in

Marketing

c. A marketing student selected at random from this group is a

male.

41. In how many ways can the letters A, B, C, D, E and F can be arranged

without repetition when

a. All the 6 letters are taken at a time

b. 5 of the letters are taken at a time

c. 3 of the letters are taken at a time

42. Find the number of arrangements of the following words

a. CALENDAR

b. MALAYSIA

c. CALCULATOR

d. PROBABILITIES

43. Suppose repetitions are not permitted

a. How many three digit numbers can be formed from the six digits

2, 3, 4, 5, 7 and 9?

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b. How many of these numbers are less than 400?

c. How many are even?

d. How many are odd?

e. How many are multiples of 5?

44. Solve the problem in question 43, if repetitions are permitted.

45. { }: 2,4,6,7A . How many 4 digits numbers can possibly be formed from

set A if

a. each digit can be used only once

b. each digit can be used more than once

c. 4th digit is odd number

46. In how many ways can the first, second and third prizes are awarded in

a class of 30?

47. If there are 8 vacant seats in a bus, in how many ways can 5 persons

seat themselves?

48. In how many ways can 7 persons sit at a round table?

49. How many 4-digit numbers can be formed by using the digits 0 to 9 if

a. Repetition is allowed

b. Repetition is not allowed

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50. Using the word MISSISSIPPI, find the number of permutations that can

be formed

a. From all the letters of the word

b. If the words are to begin with the letter I

c. If the words are to begin and end with the letter S

d. If the two P’s are to be next to each other

e. If the four S’s are to be next to each other

51. A class consists of seven men and five women

a. Find the number of committees of five that can be formed from

the class

b. Find the number of committees that consist of three men and

two women

c. Find the number of committees that consist of at least one man

and at least one woman

52. A bag contains five red marbles and six white marbles.

a. Find the number of ways that four marbles can be drawn from

the bag

b. Find the number of ways that four marbles can be drawn from

the bag, if two of the marbles must be red and two of the

marbles must be white

c. Find the number of ways that four marbles can be drawn from

the bag if the four marbles must be of the same color

53. How many ways can the letters of the word ‘DIGIT’ can be arranged?

How many of these arrangement

a. The I’s are together?

b. The I’s are separated?

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54. A committee consist of 4 members will be selected from 5 polices, 2

lawyers and 3 doctors. How many ways the committee can be formed

if

a. The committee will consist of 3 polices

b. The committee will consist of no more than a lawyer

55. A delegation of 4 members is to be formed from 7 chemistry lecturers

and 5 biology lecturers. In how many ways can the group be formed if

at least 2 chemistry lecturers should be in the group?

56. There are 7 women and 8 men in a committee. In how many ways can

a group of 3 people be selected from the committee if the 3 should be

of all men or all women?

57. In how many ways can a president, an assistant president, a treasurer

and a secretary be chosen from 8 selected members of an

organization?

58. Students of Diploma in Computer Science consist of 10 males and 15

females. What is the probability of selecting a committee of 5 males

and four females if Siti must be one of them?

59. The eleven letters of the word BOOKSHELVES are arranged in a line.

a. How many distinct arrangements can be done?

b. If an arrangement is chosen at random, what is the probability

that the two O’s are together?

60. In an examination a student has to answer 6 out of 10 questions.

a. How many choices does the student have?

b. How many choices does the student have if he/she must answer

the first two questions?

c. How many choices does the student have if he/she must answer

at least 3 of the first four questions?

topic 5 probability permutation and combination

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