Chapter 05
Probability
Dr. Halil İbrahim CEBECİ
Statistics Lecture Notes
Assigning Probability to Events
Key components of the statistical inference process is
probability because it provides the link between sample
and population.
Random Experiment:
An action or process that leads to one of several possible
outcomes
 E.g. Flip a coin (Heads and Tails), Record student
evaluations of a course (poor, fair, good, very good,
excellent)
Statistics Lecture Notes – Chapter 05
Assigning Probability to Events
Sample Space:
A sample space of a random experiment is a list of all
possible outcomes of the experiment. The outcomes must
be exhaustive and Mutually exclusive
 All the possible outcomes must be included (exhaustive)
 No two outcomes can occur at the same time (Mutually
exclusive)
𝑆𝑎𝑚𝑝𝑙𝑒 𝑠𝑝𝑎𝑐𝑒 = 𝑆 = 𝑂1 , 𝑂2 , … , 𝑂𝑘
Statistics Lecture Notes – Chapter 05
Assigning Probability to Events
Requirements of probebilities:
Given a 𝑆𝑎𝑚𝑝𝑙𝑒 𝑠𝑝𝑎𝑐𝑒 = 𝑆 = 𝑂1 , 𝑂2 , … , 𝑂𝑘 , the probabilities
assigned to the outcomes must satisfy two requirements:
1. The Probability of any outcome must lie between
0 and 1. That is ,
0 ≤ 𝑃 𝑂𝑖 ≤ 1
for each 𝑖
2.The sum of the probabilities of all the outcomes in a
sample space must be 1. That is,
𝑘
𝑃 𝑂𝑖 = 1
𝑖=1
Statistics Lecture Notes – Chapter 05
Approaches to Assigning Probabilities
Event:
An event is a collection or set of one or more simple
events is a sample space.
E.g. Achieve grade of A (𝐴 = 80, 81, 82, … , 99,100 )
Probability of events:
Sum of the probabilities of the simple events that
constitute the event.
Ex5.1 – Probabilities of the courses grade are
𝑃 𝐴 = 0.2, 𝑃 𝐵 = 0.3, 𝑃 𝐶 = 0.25, 𝑃 𝐷 = 0.15, 𝑃 𝐹 = 0,1
Probability of the event, pass the course, is
𝑃 𝑃𝑎𝑠𝑠 𝑡ℎ𝑒 𝑐𝑜𝑢𝑟𝑠 = 𝑃 𝐴 + 𝑃 𝐵 + 𝑃 𝐶 + 𝑃 𝐷 = 0.90
Statistics Lecture Notes – Chapter 05
Interpreting Probability
One way to interpret probability is this:
If a random experiment is repeated an infinite number of
times, the relative frequency for any given outcome is the
probability of this outcome.
For example, the probability of heads in flip of a balanced
coin is .5, determined using the classical approach. The
probability is interpreted as being the long-term relative
frequency of heads if the coin is flipped an infinite number
of times.
Statistics Lecture Notes – Chapter 05
Joint Probability (Intersection)
Joint probability is the probability that two events will
occur simultaneously. (Intersection of Events A and B is
the event that occurs when both A and B occur.)
Ex5.2 – Suppose that a potential investor examined the
relationship between how well the mutual fund performs
and where the fun manager earned his or her MBA.
Analyze the probabilities given below and interpret the
results.
Mutual Fund
Outperforms Market
Mutual Fund does not
Outperforms Market
Top 20 MBA Programs
0.11
0.29
Not Top 20 MBA Programs
0.06
0.54
Statistics Lecture Notes – Chapter 05
Joint Probability (Intersection)
A5.2 – Evens notaion presented below.
𝐴1 = 𝐹𝑢𝑛𝑑 𝑚𝑎𝑛𝑎𝑔𝑒𝑟 𝑔𝑟𝑎𝑑𝑢𝑎𝑡𝑒𝑑 𝑓𝑟𝑜𝑚 𝑡𝑜𝑝 20 𝑀𝐵𝐴 𝑝𝑟𝑜𝑔𝑟𝑎𝑚
𝐴2 = 𝐹𝑢𝑛𝑑 𝑚𝑎𝑛𝑎𝑔𝑒𝑟 𝑑𝑖𝑑 𝑛𝑜𝑡 𝑔𝑟𝑎𝑑𝑢𝑎𝑡 𝑓𝑟𝑜𝑚 𝑡𝑜𝑝 20 𝑀𝐵𝐴 𝑝𝑟𝑜𝑔𝑟𝑎𝑚
𝐵1 = 𝐹𝑢𝑛𝑑 𝑜𝑢𝑡𝑝𝑒𝑟𝑓𝑜𝑟𝑚 𝑡ℎ𝑒 𝑚𝑎𝑟𝑘𝑒𝑡
𝐵2 = 𝐹𝑢𝑛𝑑 𝑑𝑜𝑒𝑠 𝑛𝑜𝑡 𝑜𝑢𝑡𝑝𝑒𝑟𝑓𝑜𝑟𝑚 𝑡ℎ𝑒 𝑚𝑎𝑟𝑘𝑒𝑡
Joint probabilities are;
𝑃
𝑃
𝑃
𝑃
𝐴1 𝑎𝑛𝑑 𝐵1
𝐴2 𝑎𝑛𝑑 𝐵1
𝐴1 𝑎𝑛𝑑 𝐵2
𝐴2 𝑎𝑛𝑑 𝐵2
Statistics Lecture Notes – Chapter 05
= 0.11
= 0.06
= 0.29
= 0.54
Marginal Probability
Marginal probability is the probability of the occurrence of
the single event
Mutual Fund
Outperforms Market
Mutual Fund does not
Outperforms Market
Totals
Top 20 MBA Programs
𝑃 𝐴1 𝑎𝑛𝑑 𝐵1 = 0.11
𝑃 𝐴1 𝑎𝑛𝑑 𝐵2 = 0.29
𝑃 𝐴1 = 0.40
Not Top 20 MBA Programs
𝑃 𝐴2 𝑎𝑛𝑑 𝐵1 = 0.06
𝑃 𝐴2 𝑎𝑛𝑑 𝐵2 = 0.54
𝑃 𝐴2 = 0.60
𝑃 𝐵1 = 0.06
𝑃 𝐵2 = 0.06
Totals
Marginal Probablilites
Statistics Lecture Notes – Chapter 05
Conditional Probability
Conditional probability is used to determine how two
events are related; that is, we can determine the
probability of one event given the occurrence of another
related event.
Conditional probabilities are written as 𝑷(𝑨 | 𝑩) and read as
“the probability of event A given event B” and is calculated
as:
𝑃(𝐴 𝑎𝑛𝑑 𝐵)
𝑃 𝐴𝐵 =
𝑃(𝐵)
Statistics Lecture Notes – Chapter 05
Conditional Probability
Ex5.3 - The Dean of the School of Business at
Owens University collected the following information
about undergraduate students in her college:
Male
Female
Totals
Accounting
170
110
280
Finance
120
100
220
Marketing
160
70
230
Management
150
120
270
Totals
600
400
1000
Given that the student is a female, what is the
probability that she is an accounting major?
Statistics Lecture Notes – Chapter 05
Conditional Probability
A5.3 – Random experiment given below
𝐴: 𝑆𝑡𝑢𝑑𝑒𝑛𝑡 ′ 𝑠 𝑟𝑒𝑔𝑖𝑠𝑡𝑒𝑟𝑒𝑑 𝑎𝑡 𝐴𝑐𝑐𝑜𝑢𝑛𝑡𝑖𝑛𝑔 𝑀𝑎𝑗𝑜𝑟
𝐹: 𝑆𝑡𝑢𝑑𝑒𝑛𝑡𝑠 𝑔𝑒𝑛𝑑𝑒𝑟
𝑃(𝐴 𝑣𝑒 𝐹) 110 1000
𝑃 𝐴𝐹 =
=
= 0.275
400
𝑃(𝐹)
1000
Statistics Lecture Notes – Chapter 05
Independence
One of the objectives of calculating conditional probability
is to determine whether two events are related.
In particular, we would like to know whether they are
independent, that is, if the probability of one event is
not affected by the occurrence of the other event.
Two events A and B are said to be independent
𝑃(𝐴|𝐵) = 𝑃(𝐴)
𝑃(𝐵|𝐴) = 𝑃(𝐵)
Statistics Lecture Notes – Chapter 05
Independence
Ex5.4 – Refer to table of Ex5.2, calculate probability of
funds outperforms the market when the manager
graduation probability is given.
𝑃 𝐴1 𝐵1
𝑃(𝐴1 𝑎𝑛𝑑 𝐵1 ) 0.11
=
=
= 0.647
𝑃(𝐵1 )
0.17
The marginal probability that a manager graduated from a
top-20 MBA program is
𝑃 𝐴1 = 0.40
Since the two probabilities are not equal, we conclude that
the two events are dependent.
Statistics Lecture Notes – Chapter 05
Union
Another event that is the combination of other events is
the union
Union of Events 𝐴 and 𝐵 is the event that occurs when
either 𝐴 or 𝐵 or both occur.
Ex5.5 – Refer to table of Ex5.2, Determine the probability
that a ramdomly selected fund outperforms the market or
the manager graduated from a top-20 MBA programs.
𝑃 𝐴1 𝑜𝑟 𝐵1 = 𝑃 𝐴1 𝑎𝑛𝑑 𝐵1 + 𝑃 𝐴1 𝑎𝑛𝑑 𝐵2 + 𝑃 𝐴2 𝑎𝑛𝑑 𝐵1
𝑃 𝐴1 𝑜𝑟 𝐵1 = 0.11 + 0.06 + 0.29 = 0.46
Shortcut:
𝑃 𝐴1 𝑜𝑟 𝐵1 = 1 − 𝑃 𝐴2 𝑜𝑟 𝐵2 = 1 − 0.54 = 0.46
Statistics Lecture Notes – Chapter 05
Probability Rules and Trees
Complemet Rule:
The complement of an event A is the event that occurs
when A does not occur.
The complement rule gives us the probability of an
event NOT occurring. That is:
𝑃(𝐴) = 1 – 𝑃(𝐴)
For example, in the simple roll of a die, the probability of
the number “1” being rolled is 1/6. The probability that
some number other than “1” will be rolled is 1 – 1/6 = 5/6.
Statistics Lecture Notes – Chapter 05
Probability Rules and Trees
Multiplication Rule:
The multiplication rule is used to calculate the joint
probability of two events. It is based on the formula for
conditional probability defined earlier:
𝑃(𝐴 𝑎𝑛𝑑 𝐵)
𝑃 𝐴𝐵 =
𝑃(𝐵)
If we multiply both sides of the equation by 𝑃(𝐵) we have:
𝑃(𝐴 𝑎𝑛𝑑 𝐵) = 𝑃(𝐴 | 𝐵) ∗ 𝑃(𝐵)
Likewise, 𝑃(𝐴 𝑎𝑛𝑑 𝐵) = 𝑃(𝐵 | 𝐴) ∗ 𝑃(𝐴)
If A and B are independent events, then
𝑃 𝐴 𝑎𝑛𝑑 𝐵 = 𝑃 𝐴 ∗ 𝑃(𝐵)
Statistics Lecture Notes – Chapter 05
Probability Rules and Trees
Addition Rule:
The Probability that event A, or event B, or both occur is
𝑃(𝐴 𝑜𝑟 𝐵) = 𝑃(𝐴) + 𝑃(𝐵) – 𝑃(𝐴 𝑎𝑛𝑑 𝐵)
Why do we subtract the joint probability P(A and B) from
the sum of the probabilities of A and B?
𝑃(𝐴 𝑜𝑟 𝐵) = 𝑃(𝐴) + 𝑃(𝐵) – 𝑃(𝐴 𝑎𝑛𝑑 𝐵)
When two events are mutually exclusive
Statistics Lecture Notes – Chapter 05
Probability Rules and Trees
Probability Trees:
An effective and simpler method of applying rules is the
probability tree, wherein the events in an experiment are
represented by lines.
Ex5.6 – Student who graduate from law school must still
pass a bar exam. First time test takers passes the exam with
the ratio of 72%. Candidates who fail the first exam may
take it again. Second time test takers passes with ratio of
88%. Find the probability that a randomly selected student
Joint Probability
becomes a lawyer.
First Exam
Pass 0.72
Second Exam
Pass (0.72)
0.72
Pass 0.88
Fail and Pass (0.28*0.88) 0.2464
Fail 0.28
Fail 0.12
Statistics Lecture Notes – Chapter 05
Fail (0.28*0.12)
0.0336
Bayes’ Theorem

Bayes’ Theorem is a method for revising a probability
given additional information.
P( A1 ) P( B / A1 )
P( A1 | B) 
P( A1 ) P( B / A1 )  P( A2 ) P( B / A2 )
Statistics Lecture Notes – Chapter 05
Bayes’ Theorem
Ex5.7 - Duff Cola Company recently received several
complaints that their bottles are under-filled. A
complaint was received today but the production
manager is unable to identify which of the two
Springfield plants (A or B) filled this bottle. What is
the probability that the under-filled bottle came from
plant A?
% of total production % of underfilled bottle
A
55
3
B
45
4
Statistics Lecture Notes – Chapter 05
Bayes’ Theorem
A5.7 - Duff Cola Company recently received several
P ( A) P (U / A)
P( A / U ) 
P ( A) P (U / A)  P ( B ) P (U / B )
.55(.03)

 .4783
.55(.03)  .45(.04)
The likelihood the bottle was filled in Plant A is .4783.
Statistics Lecture Notes – Chapter 05
Exercises
Q5.1 - A manufacturing plant conducted a survey to
determine its employees’ reactions toward a proposed
change in working hours. A breakdown of the responses is
shown in the following table:
Reaction
Work Area
Agree
Disagree
Production
17
23
Office
8
2
Suppose an employee is chosen at random, with the
relevant events being defined as follows:
A: The employee works in production.
B: The employee agrees with the proposed change.
Express each of the following events in words, and find
the probabilities
a) 𝐴
b) (𝐴 𝑜𝑟 𝐵)
Statistics Lecture Notes – Chapter 05
c) (𝐴 𝑎𝑛𝑑 𝐵)
d) (𝐴 𝑜𝑟 𝐵 )
Exercises
Q5.2 - A manufacturing plant conducted a survey to
determine its employees’ reactions toward a proposed
change in working hours. A breakdown of the responses is
shown in the following table:
Men
Women
Less than 2 years
28
26
2 years or more
82
64
One employee is selected at random, and two events are
defined as follows:
A: The employee is male.
B: The employee has worked for the company for two
years or more.
find the following probabilities
a) 𝐵
b) (𝐴 𝑜𝑟 𝐵)
c) (𝐴 𝑎𝑛𝑑 𝐵)
Statistics Lecture Notes – Chapter 05
d) (𝐴 𝑜𝑟 𝐵 )
Exercises
Q5.3 - An accounting firm has advertised the availability of its
report describing recent changes to the federal income tax act.
The first 200 callers requesting a copy of the report are
classified in the following table according to the medium by
which the caller became aware of the report and the caller’s
primary interest.
Primary Interest
Radio
Newspaper
Word of Mouth
Personel Tax
34
20
26
Coorporate Tax
36
70
14
One caller is selected at random, and two events are:
A: The caller is primarily interested in corporate tax.
B: The caller became aware of the report through the newspaper.
Express each of the following probabilities in words, and find its
numerical value:
a) 𝑃 𝐴 𝐵
b) 𝑃 𝐵 𝐴
c) 𝑃 𝐴 𝐵
d) 𝑃 𝐴 𝐵
Statistics Lecture Notes – Chapter 05
Exercises
Q5.4 - A firm’s employees were surveyed to determine
their feelings toward a new dental plan and a new life
insurance plan. The results showed that 81% favored the
insurance plan, while only 35% favored the dental plan. Of
those who favored the insurance plan, 30% also favored
the dental plan.
a. What percentage of the employees favored both plans?
b. What percentage of the employees favored at least one
of the plans?
Statistics Lecture Notes – Chapter 05
Exercises
Q5.4 - Consider two events, A and B, for which 𝑃(𝐴) = 0.2,
𝑃(𝐵) = 0.6, and 𝑃(𝐴 𝑜𝑟 𝐵) = 0.68
a. Find 𝑃(𝐴 𝑎𝑛𝑑 𝐵)
b. Are A and B independent events?
c. Are A and B mutually exclusive events?
Q5.5 - An electrical contractor has observed that 90% of
his accounts are paid within 30 days. Of those that are not
paid within 30 days, 40% remain unpaid after 60 days. If
one account is selected at random, what is the probability
that it is paid within 60 days?
Q5.6 - A mechanic has removed six spark plugs from an
engine and finds two to be defective. If two spark plugs
are selected at random from among these six, what is the
probability that exactly one of them is defective?
Statistics Lecture Notes – Chapter 05