CHAPTER 2 : PROBABILITY

LEARNING OBJECTIVES

At the end of this chapter, the student would be able to

 apply the concepts of set theory to probability problems

 learn the concept of the sample space associated with a random experiment
 learn the concept of an event associated with random experiment
 learn the concept of probability of an event
 define and calculate the probability of two events, namely, independent events, mutually
exclusive events and conditional events
 to construct tree diagrams as a means of solving probability problems

INTRODUCTION
In the study of statistics, we are concerned basically with the presentation and interpretation of chance
outcomes that occur in a planned study or scientific investigation. Probability is the likelihood or chance
that a particular event will occur. Probability theory is used in the fields of investments and weather
forecasting and in various other fields. For example, we may record the number of accident occur monthly
in Pagoh Jaya road, justify the installation of a traffic light, we might classify items coming off an assembly
lines as “defective” or “nondefective”; or we maybe interested with the velocity of a missile at specified
times. Statisticians use the word experiment to describe any process that produces a definite outcome
that cannot be predicted with certainty. A simple example is tossing a coin which the possible outcomes
are either head or tail. Even when a coin is tossed repeatedly, we cannot be certain that a given toss will
result in a head. Probability is also the basis of inferential statistics.

FUNDAMENTAL OF PROBABILITY

SET – A group of objects. Capital letters will

denote sets and small letters will represent
their elements

The following notations can be useful

 𝑎 ∈ 𝐴 means “𝑎 is an element of set 𝐴

 𝑎 ∉ 𝐴 means “𝑎 is not an element of set 𝐴
 𝑛(𝐴) means “the number of elements in set 𝐴

Set notation is used as a symbolic tool for subsets, unions and intersection of sets. Figure 1 shows two
arbitrary set 𝐴 and 𝐵. Every element of set 𝐴 is in set 𝐵, thus 𝐴 is subset of 𝐵. We denote this by 𝐴 ⊂ 𝐵
 𝐴⊂𝐵
B
A

Figure 1

In studying probability, developing a language of terms and symbols is very helpful. Four basic term of
probability as given

An experiment is a process that produces outcomes. Processes such as

flipping a coin, rolling a die, interviewing 15 randomly selected customer
and asking them which brand of shampoo they prefer, testing new
pharmaceutical drugs on samples of cancer patients and measuring their
improvements are examples of experiments.

An outcome is the result obtained from an experiment carried out.

A sample space is the set of all possible outcomes of an experiment and

its represented by symbol 𝑺. Each outcome in a sample space is called an
element or a sample points

An event is a subset of sample space. If the experiment is to sample 100

items from a production line, the event would be to count the number of
“good” or “defective” items. When a coin is flipped, the event might be
number of” head” or number of “tail”.

Example 1.1
Construct a sample space for the experiment that consist of tossing a single coin.

Solution :
 Example 1.2
Construct a sample space for the experiment that consists of rolling a single dice. Find the events that
correspond to the phrases “ an even number is rolled” and “ a number greater than two is rolled”.

Solution :

For example 1.1 and 1.2, the outcome can be presented by using graphical representation that is Venn
Diagram (refer Figure 2)

1 3 5
H T
2 4 6

Figure 2 : Venn Diagrams for Sample Space

Example 1.3
A random experiment consists of tossing two coins. Construct a sample space

Solution :

A device that can be helpful in identifying all the possible outcomes of a random experiment, it can
present by using tree diagram . It is described in the following example
 Example 1.4
Construct a sample space that describes all three-child families according to the genders of the
children with respect to birth order

Solution :

Definition :
The union of two events 𝐴 and 𝐵, denoted by 𝐴 ∪ 𝐵 is the event containing all the elements of 𝐴 or 𝐵
or both. The keywords to express the union of two events is “or”

S
A B

Let 𝐴 be the event that an employee selected at random is male. Let 𝐵 be the event that the
employee selected is single. Then the event 𝐴 ∪ 𝐵 is the set of all employees who either male or
single or both.
Definition :
The intersection of two events 𝐴 and 𝐵, denoted by 𝐴 ∩ 𝐵 is the event containing all elements in
both 𝐴 and 𝐵 The keywords to express the intersection of two events is “and”

S
A B
𝐴∩𝐵

Let 𝐶 be the event that a person selected at random in library is a college student and let 𝐷 be the
event that the person is a female. Then 𝐶 ∩ 𝐷 is the event of all female college students in the library.

Definition :
If 𝐴 is the subset of sample space 𝑆, then the complement of 𝐴, denoted by 𝐴′ is the set that contains
all elements not in 𝐴. We write it as 𝑆 = 𝐴 ∪ 𝐴′

A
A’
Consider the sample space 𝑆 = {𝐶ℎ𝑒𝑚𝑖𝑠𝑡𝑟𝑦, 𝑀𝑎𝑡ℎ𝑒𝑚𝑎𝑡𝑖𝑐, 𝑆𝑡𝑎𝑡𝑖𝑠𝑡𝑖𝑐𝑠, 𝑃ℎ𝑦𝑠𝑖𝑐𝑠, 𝐵𝑖𝑜𝑙𝑜𝑔𝑦}
Let 𝑀 = {𝐶ℎ𝑒𝑚𝑖𝑠𝑡𝑟𝑦, 𝑃ℎ𝑦𝑠𝑖𝑐𝑠, 𝐵𝑖𝑜𝑙𝑜𝑔𝑦}. Then complement of 𝑀 is 𝑀′ = {𝑀𝑎𝑡ℎ𝑒𝑚𝑎𝑡𝑖𝑐, 𝑆𝑡𝑎𝑡𝑖𝑠𝑡𝑖𝑐𝑠}
 Definition :
Two events 𝐴 and 𝐵 are mutually exclusive if they cannot occur at the same time. Simply says that no
intersection occur (𝐴 ∩ 𝐵 = ∅)

Let 𝐴 = {𝑎, 𝑒, 𝑖, 𝑜, 𝑢} and 𝐵 = {𝑝, 𝑞, 𝑟}, then 𝐴 ∩ 𝐵 = ∅ because 𝐴 and 𝐵 have no elements in
common. Then we can say that 𝐴 and 𝐵 are mutually exclusive events.

PROBABILITY OF AN EVENT
Classical probability approach

Classical probability uses sample spaces to determine the numerical probability that an event will
happen. It assumes that all outcomes of the experiments are equally likely to occur.

Consider the following examples. What is the probability of getting a head when a fair coin is tossed?
One would probably give the answer “ ½ ” without bothering to toss the coin 1000 times. The same
situation would occur if one were to be asked, “What is the probability of getting a “1” when a fair die is
thrown?” The answer given would be “ 1/6 ”.

Classical Definition :

𝑛(𝐴)
Probability of occurrence, 𝑃 (𝐴) = 𝑛(𝑆)

Definition :
The probability of an event 𝐴 is the sum of the probabilities of the all sample points or individual
outcomes in 𝐴. It is denoted , 𝑃(𝐴)

If an event 𝐴 is 𝐴 = {𝑎1 , 𝑎2 , 𝑎3 , ⋯ , 𝑎𝑘 } then 𝑃 (𝐴) = 𝑃 (𝑎1 ) + 𝑃(𝑎2 ) + 𝑃(𝑎3 ) + ⋯ + 𝑃(𝑎𝑘 )

Properties of probability

 0 ≤ 𝑃(𝐴) ≤ 1. Simply says that probability cannot be negative and greater than 1
 𝑃 (𝑆) = 1. Sum of all probabilities of the outcome in the sample space is 1
 𝑃 (∅) = 0. When an event can never happen we say that the probability is 0
 If 𝑃(𝐴) = 𝑝, then the probability of the complement of 𝐴, 𝐴′ ; 𝑃 (𝐴′ ) = 1 − 𝑝

Example 1.5
Flash card with numbers 1 to 20 are put into a bag. A card is chosen at random from the bag. Find the
probability of getting
a) an odd number
b) multiples of 3

Solution :

Example 1.6
A coin is tossed twice. Find the probability that the coins match either both head or both tail.

Solution :
The previous example illustrate how probabilities can be computed simply by counting when the sample
space consists of a finite number of equally likely outcomes. In some situations the individual outcomes
of any sample space unequally likely.

Example 1.7
A box contains 10 white and 6 black marbles. Construct a sample space for the experiment randomly
drawing out, with replacement, two marbles in succession and noting the color each time. Find
a) Probability at least one marble of each color is drawn
b) Probability no white marble is drawn

Solution :

Example 1.8
The breakdown of the student body in a local high school according to race and ethnicity is 51%
white, 27% black, 11% Hispanic, 6% Asian and 5% for all others. A student is randomly selected from
this high school. Find the probabilities of the following events

a) B = the student is black

b) M = the student is minority (that is not white)
c) N = the student is not black

Solution :
EXERCISE

1. A box contains 16 white and 16 black marbles. Three marbles were taken from the box without
replacement. Construct a sample space and find

a) P (at least one marble of each color is drawn)

b) P (no white marble is drawn)

c) P (more black than white marble are drawn)

2. A pencil case contains 8 red, 6 black and 1 blue marker pens. Two marker pens were taken without
replacement. Draw a tree diagram and find

a) P (both different colors)

b) P (at least one red color)

c) P (second draw is black color)

3. A sample space is 𝑆 = {𝑢, 𝑣, 𝑤, 𝑥}. Identify two events as 𝐴 = {𝑣, 𝑤} 𝑎𝑛𝑑 𝐵 = {𝑢, 𝑤, 𝑥}. Suppose
𝑃 (𝑢) = 0.22, 𝑃 (𝑤) = 0.36 𝑎𝑛𝑑 𝑃 (𝑥) = 0.27.
a) Determine 𝑃(𝑣).

b) Find 𝑃(𝐴)

c) Find 𝑃(𝐵)

3 2
4. In a family, probability to get a boy is 5 and to get a girl is 5. If three child are selected, find

a) P (at least one child is a girl)

b) P (at most one child is a girl)

c) P (all of the children are boys)

d) P (the first selection is a girl)

5. Suppose one has in a bag 8 red marbles and 2 green marbles. Find the probability of drawing 1 red
and 1 green , if

a) the first marble is replaced

b) the first marble drawn is not replaced

6. A box contains 500 envelops of which 75 contain $100 in cash, 150 contains $25, and 275 contain $10.
An envelope may be purchased for $25. What is the sample space for the different amount of money?
Assign probabilities to the sample points and then find the probability that the first envelope purchased
contains less than $1000.

7. The following two-way contingency table gives the breakdown of the population in a particular locale
according to age and tobacco usage :

TOBACCO USAGE
AGE
Smoker Non-smoker
Under 30 0.05 0.2
Over 30 0.2 0.55

A person is selected at random. Find the probability of each following events.

a) The person is a smoker

b) The person is under 30

c) The person is a smoker who is under 30

8. The following two way contingency tables give the breakdown of the population of adults in a
particular locale according to highest level of education and whether or not the individual regularly
takes dietary supplements :

Use of supplements
Education
Takes Does not take
No High School Diploma 0.04 0.06
High School Diploma 0.06 0.44
Undergraduate Degree 0.09 0.28
Graduate Degree 0.01 0.02

An adult is selected at random. Find the probability of each events.

a) The person has a high school diploma and takes dietary supplements regularly.

b) The person has an undergraduate degree and takes dietary supplements regularly.

c) The person does not take dietary supplements regularly.

9. A pair of fair dice is tossed. Find the probability of getting

a) a total of 8.

b) at most a total of 5.

10. If 3 books are picked at random from a shelf containing 5 novels, 3 books of poems and a dictionary.
What is the probability that

a) the dictionary is selected?

b) 2 novels and 1 book of poems are selected?

RULES OF PROBABILITY

1. Probability Rules for Complement

If 𝐴 and 𝐴′ are complementary events, then 𝑃 (𝐴) + 𝑃(𝐴′ ) = 1

So, 𝑃 (𝐴) = 1 − 𝑃(𝐴′)

Example 1.9
If the probabilities that an automobile mechanic will service 3, 4, 5, 6, 7, 8 or more cars on any given
workday are respectively, 0.12, 0.19, 0.28, 0.24, 0.10 and 0.07, find the probability that he will service
at least 5 cars on his next day at work.

Solution :
2. Probability Rules of Addition

The rules of addition applies to the situation when we have two or more events and we want to know
the probability that either event occurs. There are have three rules.

Addition Rule 1 :

If 𝐴 and 𝐵 are two events, then 𝑃(𝐴 𝑜𝑟 𝐵) = 𝑃 (𝐴 ∪ 𝐵) = 𝑃(𝐴) + 𝑃 (𝐵) − 𝑃(𝐴 ∩ 𝐵)

Addition Rule 2 :

For three events 𝐴, 𝐵, and 𝐶. then 𝑃(𝐴 ∪ 𝐵 ∪ 𝐶 ) = 𝑃 (𝐴) + 𝑃 (𝐵) + 𝑃(𝐶 ) − 𝑃 (𝐴 ∩ 𝐵) − 𝑃(𝐴 ∩ 𝐶 ) −
𝑃 (𝐵 ∩ 𝐶 ) + 𝑃(𝐴 ∩ 𝐵 ∩ 𝐶)

Addition Rule 3 :

If 𝐴 and 𝐵 are mutually exclusive events, then 𝑃 (𝐴 ∪ 𝐵) = 𝑃(𝐴) + 𝑃 (𝐵)

Note :

𝐴 and 𝐵 are mutually exclusive if there is no intersection between both

events. Then 𝑃 (𝐴 ∩ 𝐵) = 0

Example 1.10
1 2 3
If 𝑃(𝐴) = 5 , 𝑃 (𝐵) = 3 , 𝑃(𝐴 ∩ 𝐵) = 8 , find 𝑃(𝐴 ∪ 𝐵)

Solution :

Since there is intersection between 𝐴 and 𝐵, then use Addition Rule 1

1 2 3 59
𝑃(𝐴 ∪ 𝐵) = 𝑃 (𝐴) + 𝑃(𝐵) − 𝑃 (𝐴 ∩ 𝐵) = + − =
5 3 8 120
 Example 1.11
John is going to graduate from an industrial engineering department in a university by the end of the
semester. After being interviewed at two companies he likes, he assesses that his probability of
getting an offer from company A is 0.8 and the probability that he gets an offer from company B is
0.6. If on the other hand he believes that the probability that he will get offers from both companies
is 0.5, what is the probability that he will get at least one offer from these two companies?

Solution:

Example 1.12
A student goes to the library. The probability that she checks out a work of fiction is 0.40, a work of
non-fiction is 0.30 and both fiction and non fiction is 0.20. Find the probability that either the student
check out a work of fiction or non fiction or both.

Solution :
 Example 1.13
What is the probability of getting a total of 7 or 11 when a pair of fair dice are tossed?

Solution :

3. Probability Rules of Multiplication

The rule of multiplication applies to the situation when we want to know the probability of two or more
events that occur in sequence. There are two rules

Multiplication Rule 1 :

When two events are independent, the probability of both occurring is 𝑃 (𝐴 ∩ 𝐵) = 𝑃(𝐴) ∙ 𝑃(𝐵)

The events are said to be independent when the outcome of first event does not affect the
occurrence of second event. In other word, when the experiment are with replacement.

Multiplication Rule 2 :

When two events are dependent, the probability of both occurring is 𝑃(𝐴 ∩ 𝐵) = 𝑃 (𝐴) ∙ 𝑃 (𝐵|𝐴)
or 𝑃(𝐴 ∩ 𝐵) = 𝑃 (𝐵) ∙ 𝑃(𝐴|𝐵 )

The events are said to be dependent when the outcome of first event affects the outcome or
occurrence of second event in such a way the probability is changed. In other word, when the
experiment are without replacement.
 Example 1.14
An urn contains 6 red marbles and 4 black marbles. Two marbles are drawn without replacement
from the urn. Find the probability that both of the marbles are black.

Solution :

Draw a tree diagram

Example 1.15
If 2 fair coins are tossed, find the probability of getting a tail on the first and the second toss.

Solution :
EXERCISE

1. A tourist who speak English and German but no other language visits a region of Slovenia. If 35% of
the residents speak English, 15% speak German and 3% speak both language English and German, find
the probability that the tourist will be able to talk with a randomly encountered resident of the region.

2. In a certain country 43% of all automobiles have airbags, 27% have anti-lock brakes and 13% have
both. Find the probability that a randomly selected vehicle will have either airbags or anti-lock brakes or
both.

3. A manufacturer examines its records over the last year on a component part received from outside
suppliers. The breakdown on source (supplier A, supplier B) and quality (H : High, U : Usable, D :
Defective) is shown in the two way contingency table.

H U D
A 0.6937 0.0049 0.0014
B 0.2982 0.0009 0.0009

The record of a part is selected at random. Find the probability of each of the following events.

a) The part was defective

b) The part was either of high quality or was at least usable.

c) The part was defective and came from supplier B

d) The part was defective or came from supplier B

4. The breakdown of the students enrolled in a university course by class ( F : Freshman; S : Sophomore;
J : Junior; Se : Senior ) and academic major ( S : Science, Mathematic or Engineering ; L : Liberal Arts ; O :
Other) is show in the two way contingency table.

Class
Major
F S J Se
S 92 42 20 13
L 368 167 80 53
O 460 209 100 67

A student enrolled in the course is selected in random. Find the probability of each following events

a) The student is a freshman.

b) The student is a liberal arts major.

c) The student is a freshman liberal arts major.

 d) The student is either a freshman or a liberal arts major.

e) The student is not a liberal arts major.

5. A manufacturer of a flu vaccine is concerned about the quality of its flu serum. Batches of serum are
processed by three different departments having rejection rates of 0.10, 0.08, and 0.12, respectively.
The inspection by three departments are sequential and independent.

a) Find the probability that a batch of serum survives the first departmental inspection but is rejected
by the second department.

b) Find the probability that a batch of serum is rejected by the third department.

6. After one senior staff retires, three other staffs, Qash, Adam and Putri apply for the vacant post in
that department but only one person will get the post. Find the probability that

a) Qash will get the post.

b) Adam or Putri will get the post.

7. The probability that McDonalds will locate in Muar is 0.7, the probability that it will locate in Pagoh is
0.4 and the probability that it will locate in either Muar or Pagoh is 0.8. Find the probability that the
industry will locate

a) in both cities.

b) in neither city.

8. A town has 2 fire engines operating independently. The probability that a specific engine is available
when needed is 0.96.

a) Find the probability that neither is available when needed.

b) Find the probability that a fire engine is available when needed.

9. The probability that a vehicle entering the Luray Caverns has Canadian license plates is 0.12 ; the
probability that it is a camper is 0.28 and the probability that it is a camper with Canadian license plates
is 0.09. What is the probability that

a) a camper entering the Luray Caverns has Canadian license plates?

b) a vehicle with Canadian license plates entering the Luray Caverns is a camper?

c) a vehicle entering the Luray Caverns does not have Canadian plates or is not a camper?
10. Determine which events are mutually exclusive and which are not when a single die is rolled.

a) Getting an odd number and getting an even number

b) Getting a 3 and getting an odd number

c) Getting an even number and getting a number less than 5

d) Getting a number greater than 3 and getting a number less than 3

CONDITIONAL PROBABILITY

Suppose a fair dice has been rolled and we are asked to give the probability that it was a three. There
are six equally likely outcomes, so the answer is 1/6. But suppose that it is known that the number rolled
was odd. Since there are only have three odd number including three, then we will revised the answer
from 1/6 to 1/3. In general the revised probability that an event 𝐴 has occurred, taking into account the
additional information that another event 𝐵 has definitely occurred on this trial of the experiment is
called the conditional probability of 𝐴 given 𝐵 and it is denoted by 𝑃 (𝐴|𝐵). It is usually read “the
probability of 𝐴 given that 𝐵 occurs”.

Definition :
The conditional probability of 𝐴 given 𝐵, denoted 𝑃 (𝐴|𝐵) is the probability that event 𝐴 has occurred
in a trial of a random experiment for which it is known that event 𝐵 has definitely occurred. It can be
computed by following formula :

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

It also can be written as :

𝑃 (𝐴 ∩ 𝐵) = 𝑃(𝐴|𝐵 )𝑃(𝐵) = 𝑃 (𝐵|𝐴)𝑃 (𝐴) − 𝑀𝑢𝑙𝑡𝑖𝑝𝑙𝑖𝑐𝑎𝑡𝑖𝑣𝑒 𝑅𝑢𝑙𝑒

since 𝑃 (𝐴 ∩ 𝐵) = 𝑃(𝐵 ∩ 𝐴)
 Example 1.16
A fair die is rolled.
a) Find the probability that the number rolled is a five given that it is odd
b) Find the probability that the number rolled is odd given that it is a five

Solution :

Example 1.17
Suppose the sample space 𝑆 is the population of adults in a small town who have completed the
requirements for a college degree. We shall categorized them according to the gender and
employment status. The data are given in the table below.

Employed Unemployed
Male 460 40
Female 140 260

Suppose a person is selected at random

a) Find the probability that the person selected is an employed
b) Find the probability that the person selected is a man given that the one chosen is employed

Solution :
INDEPENDENT EVENTS

Definition :
Events 𝐴 and 𝐵 are independent if

𝑃(𝐴 ∩ 𝐵) = 𝑃(𝐴) ∙ 𝑃(𝐵)

By using algebra, it can be shown that the equality 𝑃 (𝐴|𝐵) = 𝑃(𝐴) and 𝑃 (𝐵|𝐴) = 𝑃(𝐵) hold if and only
if 𝑃(𝐴 ∩ 𝐵) = 𝑃(𝐴) ∙ 𝑃(𝐵) is hold.

The formula in the definition has two practical but exactly opposite uses :

1. In a situation in which we want to check whether or not events 𝐴 and 𝐵 are independent.

2. In a situation, we know two events 𝐴 and 𝐵 are independent and we know (𝐴)𝑎𝑛𝑑 𝑃(𝐵) , then we
can compute 𝑃(𝐴 ∩ 𝐵).

Example 1.18
A single fair die is rolled. Let 𝐴 = {3} and 𝐵 = {1,3,5}. Are 𝐴 and 𝐵 independent?

Solution :
 Example 1.19
Many diagnostic tests for detecting diseases do not test for the disease directly but for a chemical or
biological product of the disease, hence are not perfectly reliable. The sensitivity of a test is the
probability that the test will be positive when administered to a person who has the disease. The
higher the sensitivity, the greater the detection rate and the lower the false negative rate.

Suppose the sensitivity of a diagnostic procedure to test whether a person has a particular disease is
92%. A person who actually has the disease is tested for it using this procedure by two independent
laboratories.

a) Find the probability that both test results will be positive

b) Find the probability that at least one of the two results will be positive

Solution :

EXERCISE
1. For two events 𝐴 and 𝐵, 𝑃(𝐴) = 0.73, 𝑃 (𝐵) = 0.48 𝑎𝑛𝑑 𝑃 (𝐴 ∩ 𝐵) = 0.29. Find

a) 𝑃 (𝐴|𝐵)

b) 𝑃 (𝐵|𝐴)

c) Determine whether or not 𝐴 and 𝐵 are independent.

2. For two events 𝐴 and 𝐵, 𝑃(𝐴) = 0.26, 𝑃 (𝐵) = 0.37 𝑎𝑛𝑑 𝑃 (𝐴 ∩ 𝐵) = 0.11. Find

a) 𝑃 (𝐴|𝐵)

b) 𝑃 (𝐵|𝐴)

c) Determine whether or not 𝐴 and 𝐵 are independent.

3. For independent events 𝐴 and 𝐵, 𝑃 (𝐴) = 0.81 𝑎𝑛𝑑 𝑃(𝐵) = 0.27. Find

a) 𝑃(𝐴 ∩ 𝐵)

b) 𝑃 (𝐵|𝐴)

c) 𝑃 (𝐴|𝐵)

4. Compute the following probabilities in connection with two tosses of a fair coin.

a) The probability that the second toss is head.

b) The probability that the second toss is head given that the first toss is head.

c) The probability that the second toss is head given that at least one of the tosses is head.

5. A jar contains 10 marbles, 7 black and 3 white. Two marbles are drawn without replacement.

a) Find the probability that both marbles are black

b) Find the probability that exactly one marble is black

c) Find the probability that the second marble is white given that the first marble must be white

6. A random experiment gave rise to the two-way contingency table shown. Use it to compute the
probabilities indicated.

R S
A 0.13 0.07
B 0.61 0.09

a) Find 𝑃 (𝐴), 𝑃 (𝑅)𝑎𝑛𝑑 𝑃(𝐴 ∩ 𝑅)

b) Determine either 𝐴 and 𝑅 are independent

7. From Majalah Kini, Azizi reported that due to male entering nursing school, the number of Malaysian
male and female working in Hospital Datuk Abdullah is shown in table below.

Dental Nurse, D Staff Nurse, S Assistant Nurse, A

Male, M 19 45 0
Female, F 36 54 16

a) Complete the contingency table. Find the marginal probabilities of Dental Nurse, Staff nurse,
Assistant Nurse, Male and Female.
 b) Determine the probability that the nurse selected is Dental Nurse and a Male.

c) Calculate the probability that the adult selected is a female, given that she is a Staff Nurse.

8. In a certain city, 40% of the students who like Red football team, 35% of the students who like Blue
football team and 40% of the students who like Green football team. During a particular football season,
45% of the Red fan came to the tournament, 40% of the Blue fan came to the tournament and 60% of the
Green fan came to the tournament. If a randomly selected student came to the tournament, find the
probability that the fan is from

a) Red fan

b) Blue fan

c) Green fan

9. 30% of all computer used by government agencies are supplied by company A and the rest by company
B. 5% of all computer supplied by company A are defective while 2% of all computer supplied by company
B are defective.

a) How many percent of all computer used by the government agencies are defective?

b) A computer is found to be defective. What is the probability it was supplied by company B?

10. A teacher asks each student to write his name on a piece of paper and put it inside the box for a lucky
draw. There are 10 local girls, 20 foreign girls, 15 local boys and 30 foreign boys in the class.

a) Please fill up the contingency table.

Student Boy (B) Girl (G) Total

Local (L)
Foreign (F)
Total

b) Find 𝑃(𝐿 ∩ 𝐵) and 𝑃(𝐺 ∩ 𝐹)

c) After making the draw, the teacher announces that the winner is a girl. What is the probability that
the winner is a local student?

d) After making the draw, the teacher announces that the winner is a foreign student. What is the
probability that winner is a boy?