SQQS1013-CHP03 r1

SQQS1013 Elementary Statistics
Chapter 3: Introduction to Probability

3.1 INTRODUCTION
 The principles of probability help bridge the worlds of descriptive statistics
and inferential statistics.
 Probability can be defined as the chance of an event occurring or to be
specific the numerical value representing the chance, likelihood, or
possibility that a particular event will occur.
 Situations that involve probability:
 Observing or playing a game of chance such as card games and slot machines
 Insurance
 Investments
 Weather Forecasting etc.
 It is the basis of inferential statistics such as predictions and testing the
hypotheses.
3.2 SAMPLE SPACE & PROBABILTY CONCEPTS

Some basic concepts of probability:
1. A Probability Experiment
- A chance process that leads to well-defined results called outcomes.
2. An Outcome
- The result of a single trial of a probability experiment.
3. A Sample Space
- The set of all possible outcomes of a probability experiment.
- Some sample spaces for various probability experiments are shown below
EXPERIMENT SAMPLE SPACES
Toss one coin Head, Tail

Roll a die 1, 2, 3, 4, 5, 6
Answer a true/false questions True, False
Toss two coins Head-Head, Head-Tail, Tail-Tail, Tail-Head
Chapter 3: Introduction to Probability 1

Example 1
Find the sample space for rolling two dice.
Die 2
Die1
1 2 3 4 5 6
1 (1,1) (1,2) (1,3) (1,4) (1,5) (1,6)

2 (2,1) (2,2) (2,3) (2,4) (2,5) (2,6)
3 (3,1) (3,2) (3,3) (3,4) (3,5) (3,6)
4 (4,1) (4,2) (4,3) (4,4) (4,5) (4,6)
5 (5,1) (5,2) (5,3) (5,4) (5,5) (5,6)
6 (6,1) (6,2) (6,3) (6,4) (6,5) (6,6)
Example 2
Find the sample space for the gender of the children if a family has three
children. Use B for boy and G for girl.
Solution:
There are two genders, male and female, and each child could be of
either gender. Hence, there are eight possibilities.
BBB BBG BGB GBB GGG GGB GBG BGG
4. A Tree Diagram
- Another way to determine all possible outcomes (sample space) of a

probability experiment.
- It is a device consisting of line segments originating from a starting point and
also from the outcome point.

Example 3
Use a tree diagram to find the sample space for the gender of three children in a
family.
3rd child Outcome

2nd child s
B BBB
1st child B
G BBG
B
B BGB
G
G BGG
B B GBB
G
G GBG
G B GGB
G GGG
Example 4
You are at a carnival. One of the carnival games involves asking you to pick a
door and then pick a curtain behind the door. There are 3 doors and 4
curtains behind each door. Use a tree diagram to find the sample spaces for
all the possible choices.
Door Curtain Outcomes
A 1, A
B 1, B
1 C 1, C
D 1, D
A 2, A
2 B 2, B
C 2, C
D 2, D
A 3, A
3 B 3, B
C 3, C
D 3, D

5. Venn Diagrams:
- developed by John Venn and are used in set theory and symbolic logic.
- have been adapted to probability theory.
- A picture (a closed geometric shape such as a rectangle, a square, or a circle)
that depicts all the possible outcomes for an experiment.
- The symbol  represents the union of two events and P(A  B) corresponds
to A OR B.
- The symbol  represents the intersection of two events and P(A  B)
corresponds to A AND B.
Venn diagram representing two events; Venn diagram representing three events;
A and B A, B and C
6. An Event
- Consists of a set of outcomes of a probability experiment.
- An event can be :
a) Simple event – the outcome that is observed in a single
experiment
- an event with one outcome
e.g: If a die is rolled once and a 6 shows. There is
only 1 outcome as a result of a single trial.
b) Compound event – an even with more than one outcome.
e.g : The event of getting an odd number when a die
is rolled since it consists of three outcomes or three
simple events.
Probabilities can be expressed as fractions, decimals or percentages (where

appropriate).

3.2.1 Basic Probability Rules

There are four basic probability rules:
1. The probability of any event E is a number between and including 0 and 1.

0≤P( E )≤1
2. If an event E cannot occur, its probability is 0 (impossible event).
3. If an event is certain, then the probability of E is 1 (certain event).
4. The sum of the probabilities of all the outcomes in the sample space
is 1.
3.2.2 Basic Interpretation of Probability

Three basic interpretations of probability that are used to solve a variety of
problems in business, engineering and other fields:
1. Classical Probability
- Uses sample spaces to determine the probability that an event will happen.
- Assumes that all outcomes in the sample space are equally likely to occur
which means that all the events have the same probability of occurrence.
- The probability of any event E is:
Number of outcomes in E
Total number of outcomes in the sample space
Or denoted as,
n( E )
P ( E )= n ( S )
e.g.: When a single die is rolled, each outcome has the same probability of
occurring. Since there are six outcomes, each outcome has a probability of
1
6.
2. Empirical Probability
- Relies on actual experience to determine the likelihood of outcomes.
- Is based on observation.
- Given a frequency distribution, the probability of an event being in a given
class is:
Frequency for the class or denoted as,

P( E )= fn
Total frequencies in the distribution

Example 5
Hospital records indicate that maternity patients stayed in the hospital for the
number of days shown in the following distribution:
Number of days stayed Frequency

3 15
4 32
5 56
6 19
7 5
Total 127
Find these probabilities,

a) A patient staying exactly 5 days
b) A patient staying less than 6 days
P(less than 6 days) =
c) A patient staying at most 4 days
P(at most 4 days) =
3. Subjective Probability
- Uses a probability value based on an educated guess or estimate, employing
opinions and inexact information.
- This guess is based on the person’s experience and evaluation of a solution.
e.g.: A physician might say that, on the basis of her diagnosis, there is a 30%
chance the patient will need an operation.
Any other examples?

3.3 FIELD OF EVENTS & TYPE OF PROBABILITIES

3.3.1 Field of Events
 Intersection vs. Union events
 Intersection event
 Let A and B be two events defined in a sample space.
 The intersection of events A and B is the event that occurs when both A and B
occur.
 It is denoted by either A  B or AB.
Example 6
A = event that a family owns a DVD player

B = event that a family owns a digital camera
A B
A
and
B
Intersection of A and B
 Union event
 Let A and B be two events defined in a sample space.
 The union of events A and B is the event that occurs when either A or B or both
occur.
 It is denoted as A  B.
Example 7
A = event that a family owns a DVD player
B = event that a family owns a digital camera
A B Shaded area
gives the union of
events A and B.

Example 8
A senior citizens centre has 300 members. Of them, 140 are male, 210 take at least
one medicine on a permanent basis and 95 are male and take at least one medicine
on a permanent basis. Draw a Venn diagram to describe,
a) the intersection of the events “male” and “take at least one medicine on
a permanent basis”.
b) the union of the events “male” and “take at least one medicine on a
permanent basis”.
c) the intersection of the events “female” and “take at least one medicine
on a permanent basis”.
d) the union of the events “female” and “take at least one medicine on a
permanent basis”.
Solution:
Take at least one medicine

Male Female
45 95 115 45
 Independent vs. Dependent Events

 Independent events
 Two events A and B are independent events if the fact that A occurs does not
affect the probability of B occurring.
Example 9
Rolling a die and getting a 6, and then rolling a second die and getting a 3.
Note:
The outcome of the rolling the first die does not affect the probability
outcome of rolling the second die.

 Dependent events
 When the outcome or occurrence of the first event affects the outcome or
occurrence of the second event in such a way that the probability is changed, the
events are said to be dependent events.
 Some examples of dependent events:
o Drawing a card from a deck, not replacing it, and then drawing a second
card.
o Selecting a ball from an urn, not replacing it, and then selecting a second
ball.
o Having high grades and getting a scholarship.
o Parking in a no-parking zone and getting a parking ticket.
3.3.2 Type of Probabilities

NOTE: the examples of joint, marginal and conditional probabilities will be based on the
following contingency table
Table 1: Two-way classification of all employees of a company by gender and

college degree
Category College Not a college

graduate, G graduate, Total
Male, M 7 20 27
Female, F 4 9 13
Total 11 29 40
1. Joint Probability
 The probability of the intersection of events.
 Written by either P(A  B) or P(AB).
Example
(Refer to Table 1)
If one of those employees is selected at random for membership on the employee
management committee, there are 4 joint probabilities that can be defined. That
is,
a) the probability that this employee is a male and a college graduate
b) the probability that this employee is a female and a college graduate

c) the probability that this employee is a male and not a college graduate
d) the probability that this employee is a female and not a college graduate
2. Marginal Probability
 The probability of a single event without consideration of any event.
 Also called as simple probability.
 Named so as they calculated in the margins of the table (divide the
corresponding totals for the row or column by the grand total).
Example
(Refer Table 1)
If one of those employees is selected at random for membership on the
employee management committee, find the probabilities for each of the
followings:
a) the chosen employee is a male
b) the chosen employee is a female
c) the chosen employee a college graduate
d) the chosen employee is not a college graduate

3. Conditional Probability
 Often used to gauge the relationship between two events.
 Conditional probability is the probability that an event will occur given
that another event has already occurred.
 Written as:
P(event 2 will occur | event 1 has already occurred)
 The probability of event A given event B is
 The probability of event B given event A is
Example
(Refer to Table 1)
If one of those employees is selected at random for membership on the
employee management committee, find the probabilities for each of the
followings:
a) the chosen employee is a male given that he is graduated from college
P(M | G) =
b) the chosen employee is not a college graduate given that this employee is
female
P( | F) =

Example
A person owns a collection of 30 CDs, of which 5 are country music.
a) 2 CDs are selected at random and with replacement. Find the probability
that the second CD is country music given that the first CD is country
music.
P(CM |CM) =
b) This time the selection made is without replacement. Find the probability
that the second CD is country music given that the first CD is country
music.
P(CM |CM) =
Summarize what you have learned so far.

3.4 EVENTS & PROBABILITIES RULES
3.4.1 Mutually Exclusive Events & Non-Mutually Exclusive Events

 Two events are mutually exclusive if they cannot occur at the same time
(they have no outcomes in common).
 The probability of two or more events can be determined by the addition
rules.
 There are two addition rules to determine either the two events are mutually
exclusive or not mutually exclusive.
Addition Rule 1
When two events A and B are mutually exclusive, the probability that A or
B will occur is
P(A or B) = P(A) + P(B) or

P(A and B) = 0
P(A) P(B)
Addition Rule 2
When two events A and B are not mutually exclusive, then
P(A or B)= P(A) + P(B) – P(A and B)
P(A and B)
P(A) P(B)

Example
Consider the following events when rolling a die:
A = an even number is obtained = 2,4,6

B = an odd number is obtained = 1,3,5
Are events A and B mutually exclusive?
Solution:
Yes, the two events are mutually exclusive since event A and event B have no
common element.
A B
2 1
4 3
6 5
Example
Determine which events are mutually exclusive and which are not when a
single die is rolled.
a) Getting a 3 and getting an odd number.

Answer: Not Mutually Exclusive
b) Getting a number greater than 4 and getting a number less than 4.
Answer: Mutually Exclusive
c) Getting an odd number and getting a number less than 4.
Answer: Not Mutually Exclusive

Example
There are 8 nurses and 5 physicians in a hospital unit; 7 nurses and 3 physicians
are females. If a staff person is selected, find the probability that the subject is a
nurse or a male.
Solution:
Staff Female, F Male, M Total

Nurses, N 7 1 8
Physicians, PY 3 2 5
Total 10 3 13
P(N or M) = P(N  M)
=
Example
At a convention there are 7 mathematics instructors, 5 computer sciences
instructors, 3 statistics instructors, and 4 science instructors. If an instructor is
selected, find the probability of getting a science instructor or a math instructor .
Solution:
P(science instructor or math instructor)
=

Example
A grocery store employs cashiers, stock clerks and deli personnel. The
distribution of employees according to marital status is shown here.
Marital Status Cashiers Clerks Deli Personnel

Married 8 12 3
Not Married 5 15 2
If an employee is selected at random, find these probabilities:
a. the employee is a stock clerk or married
P(clerk  married) =
b. the employee is not married
P(not married) =
c. the employee is a cashier or is unmarried

P(cashier  not married) =
3.4.1 Independent & Dependent Events

 For two independent events, A and B, the occurrence of event A does not
change the probability of B occurring.
 The probability of independent events can be determined as:
P( A | B ) = P(A) Or P( B | A ) = P(B)
Multiplication Rule 1
When two events are independent, the probability of both occurring
P(A  B) = P(A)  P(B)
ChapterExample
3: Introduction to Probability 16
A box contains 3 red balls, 2 blue balls, and 5 white balls. A ball is selected and
its colour noted. Then it is replaced. A second ball is selected and its colour
noted. Find the probability of each of these:
a) selecting two blue balls.
P (blueblue) = P(blue)  P(blue)
b) selecting 1 blue ball and then 1 white ball.
P (bluewhite) = P(blue)  P(white)
c) selecting 1 red ball and then 1 blue ball.
P(redblue) = P(red)  P(blue)
Example
A survey found that 68% of book buyers are 40 years or older. If two book
buyers are selected at random, find the probability that both are 40 years or
older.
P (buyer) =
 On the other hand, two events, A and B are dependent when the
occurrence of the event A changes the probability of the occurrence of
event B.
 When two events are dependent, another multiplication rule can be used to
find the probability.

Multiplication Rule 2
When two events are dependent, the probability of both occurring
P (A  B) = P(A)  P( B | A )
Example
In a scientific study there are 8 tigresses, 5 of which are pregnant. If 3 are
selected at random without replacement, find the probability that:
a) all tigresses are pregnant.
1st tigress 2nd tigress 3rd tigress Outcomes

3 PG PG, PG,PG
6
4
PG
7 3
4PG PG
5 PG 6
6 PG
8 3
PG PG
7 2
5 6 4 PG PG
7 PG
6 PG PG
3
PG 2
8
2
6 5 PG PG
7 PG 6 PG PG
1
6 PG PG
P(PGPGPG) =

b) two tigresses are pregnant.

Let A be an event of two tigresses being pregnant
P(A) =
3.4.3 Complementary Events
 The set of outcomes in the sample space that is not included in the outcomes
of event E.
 Denoted as E’ (read “E prime”)
Example
Find the complement of each event.
a) Rolling a die and getting a 4
Answer:
b) Selecting a letter of the alphabet and getting a vowel

Answer:
c) Selecting a day of the week and getting a weekday

Answer:
 The outcomes of an event and the outcomes of the complement make up

the entire sample space.
 The rule of complementary events can be stated algebraically in three
ways:
Or
P( E )=1−P (E ) Or
P( E )+ P( E)=1
 The concept can be represented pictorially by the following Venn

Diagrams.
P(E) P(E)
P(S)=1 P (E )
Example
In a group of 2000 tax payers, 400 have been audited by the IRS at least once.
If one tax payer is randomly selected from this group, what is the probability of
that tax payer never been audited by the IRS?
Solution:
Let, A = the selected tax payer has been audited by the IRS at least once
A = the selected tax payer has never been audited by the IRS
 The multiplication rules can be used with the complementary event rule to
simplify solving probability problems involving “at least”.
Example
In a department store there are 120 customers, 90 of whom will buy at least one
item. If 4 customers are selected at random, one by one, find the probability that
at least one of the customers will buy at least one item. Would you consider this
event likely to occur? Explain.
Solution:
Let C = at least one customer will buy at least one item
C = none of the customers will buy at least one item
P(will buy at least one item) = 90 / 120 = ¾
So, P(won’t buy any items) = 1 - 3/4 = ¼
By using the complementary event rule,
P(C )=1−P(C )
=

Yes, this event is most likely to occur (certain event) since the probability of it
occurring is almost 1.
NOTE: The following examples are based on the overall understanding of the entire
probability concepts.
Example
A random sample of 400 college students was asked if college athletes should be
paid. The following table gives a two-way classification of the responses.
Should not be
Should be paid,
paid, PAID
Total
PAID
Student athlete, SA 90 10 100
Student non-athlete, SNA 210 90 300
400
Total 300 100
a) If one student is randomly selected from these 400 students, find the probability
that this student
i. Is in favour of paying college athletes
P(PAID) =
ii. Favours paying college athletes given that the student selected is a non-
athlete
P(PAID | SNA) =
iii. Is an athlete and favours paying student athletes
P(SA PAID) =
iv. Is a non-athlete or is against paying student athletes
P(SNA  PAID ) =

b) Are the events “student athlete” and “should be paid” independent? Are they
mutually exclusive? Explain why or why not.
P(SAPAID) =
Since, P(SAPAID)  P(SA)  P(PAID), those two events are not independent
(dependent).
Also, since P(SAPAID)  0, those two events are not mutually exclusive.
Example
A screening test for a certain disease is prone to giving false positives or false
negatives. If a patient being tested has the disease, the probability that the test
indicates a false negative is 0.13. If the patient does not have the disease, the
probability that the test indicates a false positive is 0.10. Assume that 3% of the
patients being tested actually have the disease.
Let D = the patient has the disease
D = the patient does not have the disease
PO = the patient tests positive
NE = the patient tests negative
Suppose that one patient is chosen at random and tested. Find the probability
that:
Joint Probability
0.87 PO P(DPO)
0.03 D
0.13 NE P(DNE)
0.10 PO D
0.97
D
0.90 NE D

This patient has the disease and tests positive

P(DPO) =
a) This patient does not have the disease and tests positive
P( D PO) =
b) This patient tests positive

P(PO) =
c) This patient does not have the disease and tests negative
P( D NE) =
d) This patient has the disease given that he/she tests positive
P(D | PO) =

EXERCISE 1
1. For each of the following, indicate whether the type of probability involved is an
example of classical probability, empirical probability or subjective probability:
a)The next toss of a fair coin will land on head.
b)Italy will win soccer’s World Cup the next time the competition is held.
c) The sum of the faces of two dice will be 7.
d)The train taking a commuter to work will be more than 10 minutes late.
2. A test contains two multiple-choice questions. If a student makes a random guess to

answer each question, how many outcomes are possible? Draw a tree diagram for
this experiment. (Hint: Consider two outcomes for each question – either the answer
is correct or wrong).
3. Refer to question 2. List all the outcomes included in each of the following events
and mention which are simple and which are compound events.
a) Both answers are correct.
b) At most one answer is wrong.
c) The first answer is correct and the second is wrong.
d) Exactly one answer is wrong.
4. State whether the following events are independent or dependent.

a) Getting a raise in salary and purchasing a new car.
b) Having a large shoe size and having a high IQ.
c) A father being left-handed and a daughter being left-handed.
d) Eating an excessive amount of ice cream and smoking an excessive amount
of cigarettes.
5. 88% of American children are covered by some type of health insurance. If four
children are selected at random, what is the probability that none are covered?
6. A box of nine golf gloves contains two left-handed gloves and seven right-handed
gloves.
a) If two gloves are randomly selected from the box without replacement, what is
the probability that both gloves selected will be right-handed?
b) If three gloves are randomly selected from the box without replacement, what
is the probability that all three will be left-handed?
c) If three gloves are randomly selected from the box without replacement, what
is the probability that at least one glove will be right-handed?

7. A financial analyst estimates that the probability that the economy will experience a
recession in the next 12 months is 25%. She also believes that if the economy
encounters recession, the probability that her mutual fund will increase in value is
20%. If there is no recession, the probability that the mutual fund will increase in
value is 75%. Find the probability that the mutual fund’s value will increase.
8. A car rental agency currently has 44 cars available. 18 of which have a GPS
navigation system. One of the 44 cars is selected at random, find the probability that
this car,
a) has a GPS navigation system.
b) does not have a GPS navigation system.
Now, two cars are selected at random from these 44 cars. Find the probability that
at least one of these cars have GPS navigation system.
9. A recent study of 300 patients found that of 100 alcoholic patients, 87 had elevated
cholesterol levels, and 200 non-alcoholic patients, 43 had elevated cholesterol
levels.
a) If a patient is selected at random, find the probability that the patient is the
following,
i. an alcoholic with elevated cholesterol level.
ii. a non-alcoholic.
iii. a non-alcoholic with non-elevated cholesterol level.
b) Are the events “alcoholic” and “non-elevated cholesterol levels” independent?
Are they mutually exclusive? Explain why or why not.
10. The probability that a randomly selected student from college is a female, is 0.55
and that a student works more than 10 hours per week, is 0.62. If these two events
are independent, find the probability that a randomly selected student is a
a) male and works for more than 10 hours per week.
b) female or works for more than 10 hours per week.
11. A housing survey studied how City Sun homeowners get to work. Suppose that the
survey consisted of a sample of 1,000 homeowners and 1,000 renters.
Drives to Work Homeowner Renter
Yes 824 681
No 176 319
a) If a respondent is selected at random, what if the probability that he or she

i. drives to work?
ii. drives to work and is a homeowner?
iii. does not drive to work or is a renter?
b) Given that the respondent drives to work, what then is the probability that he or
she is a homeowner?

c) Given that the respondent drives to work, what then is the probability that he or
she is a renter?
d) Are the two events, driving to work and being a homeowner, independent?
12. Due to the devaluation which occurred in country PQR, the consumers of that
country were buying fewer products than before the devaluation. Based on a study
conducted, the results were reported as the following:
Brands Number of Products Purchased

Purchased
Fewer Same More
Same 10 14 24
Changed 262 82 8
What is the probability that a consumer selected at random:

b) purchased fewer products than before?
c) purchased the same number or same brands?
d) purchased more products and changed brands?
e) given that a consumer changed the brands they purchased, what then is the
probability that the consumer purchased fewer products than before?
13. A soft-drink bottling company maintains records concerning the number of

unacceptable bottles of soft drink from the filling and capping machines. Based on
past data, the probability that a bottle came from machine I and was non-conforming
is 0.01 and the probability that a bottle came from machine II and was non-
confirming is 0.0025. Half of the bottles are filled on machine I and the other half are
filled on machine II. If a filled bottle of soft drink is selected at random, what is the
probability that:
a) it is a non-confirming bottle?
b) it was filled on machine I and is a conforming bottle?
c) it was filled on machine II or is a conforming bottle?
d) suppose you know that the bottle was produced on machine I, what is the
probability that it is non-conforming?
14. Each year, ratings are compiled concerning the performance of new cars during the
first 90 days of use. Based on a study, the probability that the new car needs a
warranty repair is 0.04, the probability that the car manufactured by Country ABC is
0.60, and the probability that the new car needs a warranty repair and was
manufactured by Country ABC is 0.025.
a) What is the probability that the car needs a warranty repair given that Country
ABC manufactured it?
b) What is the probability that the car needs a warranty repair given that Country
ABC did not manufacture it?

c) Are need for a warranty repair and country manufacturing the car statistically
independent?
15. CASTWAY is a direct selling company which has 350 authorized sales agents from
all over the country. It is known that 168 of them are male. 40% of male sales
agents have permanent jobs while half of female sales agents do not have
permanent jobs.
a) Draw a tree diagram to illustrate the above events.
b) What is the probability that a randomly selected sales agent,
i. has a permanent job?
ii. is a male given that he does not have a permanent job?
EXERCISE 2
1. Given P(M) = 0.53, P(N) = 0.58 and P(MN) = 0.33.
S M N
a) Complete the Van Diagram above with the probability values.

b) Are event M and event N mutually exclusive? Prove it.
c) Are event M and event N independent? Prove it
2. The organizer has organized three games during Lam’s family day. There are run
with one leg (G), fill water in the bottle (B) and tug & war (T). 40 participants had
participated in these games. Below is the Vann Diagram showing the number of
participants for every game during the family day.
S
B G
5
9 2
2a
2a 7
5
T
a) Based on the Diagram above, find:

i. The value of a.
ii. The number of participants who participated in tug & war only.
iii. The number of participants who participated in one game only.
iv. The number of participants who participated in more than one
game.

b) If one participant has been selected at random, find the probability the
participant;
i. Participated in fill water in the bottle game and run with one leg
game only.
ii. Participated in all games.
iii. Participated in tug & war game given he/she has participated in
run with one leg game.
3. Harmony Cultural Club has organized three competitions; singing, dancing and
acting contests. The competition has been organized at different times and each
contestant can participate in more than one contest. Below is the Venn Diagram for
100 contestants during these competitions.
singing act
5
20 12
18
2a a
15
dance
Based on the Venn diagram;

a) Find the number of contestants who participated in dance and act contests.
b) If one contestant has been selected at random, what is the probability the
contestant participated in;
i. one contest only
ii. more than one contest
iii. singing contest given he/she had joined in the act contest
iv. any contest except the dance contest.
4. Xpress Link is a courier company with 300 employees with qualification levels as
shown in the Venn diagram below. Some of the employees hold more than one
qualification.
bachelor degree diploma
36 2k k 50
4k
master degree 102
Based on the Venn diagram above,

a) Find the number of employees who holds a diploma and a bachelor degree
only.
b) What is the probability one employee who has been selected at random
holds;
i. qualifications other than a master degree
ii. three qualifications.
c) Are the event that an employee holds a diploma and the event that an
employee holds a master degree independent? Prove it.
5. Given P(A) = 0.3, P(B) = 0.6 and P (A  B) = 0.2. Draw a Venn diagram to
represent this statement. Then, find:
a) P(B’)
b) P(A  B)
c) P(B|A)
d) P(A’  B)
e) Are A and B mutually exclusive? Prove it.
6. 5% from the total radio sales at Nora’s electric shop will be returned back for
repair by the buyer due to malfunction in the first six month. Given two radios have
been sold last week,
a) Draw a tree diagram to represent the above event.
b) Find the probability that:
i. both radios will be returned back for repair
ii. none of the radios has been returned back for repair
iii. one of the radios will be returned back for repair
iv. the second radio will be returned back for repair given the first
radio had been returned for repair.
c) Are the events of returning back both the radios for repair independent?
Prove it.
7. There are three shipping companies in Baltravia country; company R, S and T.

These three companies have cargo ships and passenger ships. Table below shows
the information about the companies.
Ship Type
Company Total
Cargo Passenger
R 20 20 40
S 40 20 60
T 30 40 70
Total 90 80
a) Find the probability of choosing a cargo ship from company S

b) Find the probability of choosing a ship belonging to company T given that the
ship is a passenger ship.
c) Build a tree diagram to show the selection of a ship from each company.
d) Based on the answer in (c), find the probability:
i. all ships are cargo ships

8. A marketing manager wants to promote a new product of his company named Osom.
He has two marketing plans which are plan A and plan B. The probability that he will
choose plan A is 1/3. The probabilities that he does not succeed to promote the
product when using plan A and plan B is 1/5 and 1/6 respectively.
a) Draw a tree diagram to represent the situation

b) What is the probability that he does not succeed to promote the product?
c) If he fails to promote the product, what is the probability he has used plan
B?
9. Two shooters have been selected to represent Malaysia in USIA game. The
probability the first shooter hits the target is ½ and the probability second shooter
misses the target is 1/3. The game will start with the first shooter and followed by the
second shooter. Draw a tree diagram to represent the events. Then, find the
probability:
a) first shooter and second shooter hit the target

b) only one shooter hits the target
c) none of the shooter hits the target

Matric No: ________________ Group: _________
TUTORIAL CHAPTER 3
QUESTION 1
Nora Kindergarten would like to conduct a Sports Day. TABLE 1 shows the number of
children based on their sport’s group.
TABLE 1
Group Boy (B) Girl (G) Total
Tuah (T) 60 70 130
Jebat (J) 30 10 40
Lekiu (L) 50 20 70
Total 140 100 240
a. If a child is selected at random, what is the probability that the child is:
i. in Tuah or Jebat group
ii. a boy and in Lekiu group

Tutorial
iii. in group Jebat given that the child is a girl.
b. Are the events being “female” and being in“Tuah” group dependent? Prove it?

Tutorial
QUESTION 2
There are 100 students enrolled at the Faculty of Sciences. Courses offered are
Mathematics (M), Physics (F) and Chemistry (K).
10 students enrolled in all courses.

25 students enrolled in Mathematics and Physics courses.
20 students enrolled in Physics and Chemistry courses.
28 students enrolled in Mathematics and Chemistry courses.
60 students enrolled in Mathematics course.
50 students enrolled in Physics course.
53 students enrolled in Chemistry course.
a. By using the given information,
i. plot a Venn diagram.
ii. how many students do not enrolled in either Mathematics course or Physics
course?

Tutorial
b. Based on a (i), if the students were randomly selected, what is the probability that a
student:
i. enrolled in only one course?
ii. enrolled in Physics and Chemistry courses but do not enrolled in Mathematics
course.
Revision History
Rev. Initiated by Description of change Last revised
no.
1 Tan Wen Dee (A132 D) Amended Q. 13 of Ex. 1 to 14 Apr 2014
include the proportion of bottles
on each machine.

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SQQS1013 Elementary Statistics

Chapter 3: Introduction to Probability

3.2 SAMPLE SPACE & PROBABILTY CONCEPTS

EXPERIMENT SAMPLE SPACES

Toss one coin Head, Tail

Chapter 3: Introduction to Probability 1

1 (1,1) (1,2) (1,3) (1,4) (1,5) (1,6)

BBB BBG BGB GBB GGG GGB GBG BGG

- Another way to determine all possible outcomes (sample space) of a

Chapter 3: Introduction to Probability 2

3rd child Outcome

Door Curtain Outcomes

Chapter 3: Introduction to Probability 3

Probabilities can be expressed as fractions, decimals or percentages (where

Chapter 3: Introduction to Probability 4

3.2.1 Basic Probability Rules

1. The probability of any event E is a number between and including 0 and 1.

3.2.2 Basic Interpretation of Probability

Frequency for the class or denoted as,

Chapter 3: Introduction to Probability 5

Number of days stayed Frequency

Find these probabilities,

b) A patient staying less than 6 days

P(less than 6 days) =

c) A patient staying at most 4 days

P(at most 4 days) =

Any other examples?

Chapter 3: Introduction to Probability 6

3.3 FIELD OF EVENTS & TYPE OF PROBABILITIES

A = event that a family owns a DVD player

Chapter 3: Introduction to Probability 7

Take at least one medicine

 Independent vs. Dependent Events

Chapter 3: Introduction to Probability 8

3.3.2 Type of Probabilities

Table 1: Two-way classification of all employees of a company by gender and

Category College Not a college

a) the probability that this employee is a male and a college graduate

b) the probability that this employee is a female and a college graduate

Chapter 3: Introduction to Probability 9

b) the chosen employee is a female

c) the chosen employee a college graduate

d) the chosen employee is not a college graduate

Chapter 3: Introduction to Probability 10

 The probability of event B given event A is

Chapter 3: Introduction to Probability 11

Summarize what you have learned so far.

Chapter 3: Introduction to Probability 12

3.4 EVENTS & PROBABILITIES RULES

3.4.1 Mutually Exclusive Events & Non-Mutually Exclusive Events

P(A or B) = P(A) + P(B) or

When two events A and B are not mutually exclusive, then

P(A or B)= P(A) + P(B) – P(A and B)

Chapter 3: Introduction to Probability 13

A = an even number is obtained = 2,4,6

a) Getting a 3 and getting an odd number.

Chapter 3: Introduction to Probability 14

Staff Female, F Male, M Total

Matric No: ________ Group: _