BUSSINESS Statistics
CHAPTER FIVE :
PROBABILITY
Objectives
At the end of the chapter you will be able to
 Introduction
 Sample Spaces and Probability
 Classical Probability
 Empirical Probability
 Probability rules and trees
The Addition Rules for Probability
 The Multiplication Rules and Conditional Probability
 The Multiplication Rules
 Joint, Marginal and Conditional Probability
 Factorials, permutations, and combination
Introduction
• We often make statements about probability. For
example,
• a weather forecaster may predict that there is an
80% chance of rain tomorrow.
• A health news reporter may state that a smoker has a
much greater chance of getting cancer than does a
nonsmoker
• A college student may ask an instructor about the
chances of passing a course or getting an A if he or
she did not do well on the midterm examination.
What is the probability
• What is meant by the word probability?
• A probability is the numeric value representing the
chance, likelihood, or possibility that a particular
event will occur, such as the price of a stock
increasing, a rainy day, a defective product, or the
outcome five dots in a single toss of a die.
• the probability involved is a proportion or fraction
whose value ranges between 0 and 1, inclusive
Cont.….
• Probability measures the likelihood that an event will occur and its an
important part of statistics.
• It is the basis of inferential statistics, which will be introduced in later
chapters.
• In inferential statistics we make decisions under conditions of
uncertainty
• Probability theory is used to evaluate the uncertainty involved in
those decisions.
• For example, estimating next year’s sales for a company is based on
many assumptions, some of which may happen to be true and others
may not.
•
Cont.…
• Probability theory will help us make decisions
under such conditions of imperfect information and
uncertainty.
• Combining
probability
and
probability
distributions descriptive statistics will help us make
decisions about populations based on information
obtained from samples.
• This chapter presents the basic concepts of
probability and the rules for computing
probability
Conti……
• the probability involved is a proportion or fraction whose value
ranges between 0 and 1, inclusive.
• An event that has no chance of occurring (the impossible event) has
a probability of 0.
• An event that is sure to occur (the certain event) has a probability
of 1.
• Probability as a general concept can be defined as the chance of
an event occurring
• Probability is the basis of inferential statistics. For example,
predictions are based on probability, and hypotheses are tested
by using probability
Sample Spaces and Probability
• The theory of probability grew out of the study
of various games of chance using coins, dice,
and cards. Since these devices themselves well
to the application of concepts of probability,
they will be used in this chapter as examples.
• This section begins by explaining some basic
concepts of probability. Then the types of
probability and probability rules are discussed
Basic Concepts
• Experiment, Outcomes, and Sample Space
• Experiment: In statistics anything that results in a count or
a measurement is called an experiment
• Probability experiment is a chance process that leads to
well-defined results called outcomes.
• Outcome: A particular result of an experiment.
• A sample space is the set of all possible outcomes of a
probability experiment.
• Some sample spaces for various probability experiments
are shown here.
Find the sample space for rolling two dice.
Find the sample space for drawing one
card from an ordinary of cards
EXAMPLE 4–1 One Toss of a Coin
• Draw the tree diagram for the experiment of
tossing a coin once
EXAMPLE 4–2 Two Tosses of a Coin
• Solution
• Sample space = {HH, HT, TH, TT}
Simple and Compound Events
• Event is a collection of one or more of the outcomes of an
experiment
• An event may be a simple event or a compound event.
• A simple event is also called an elementary event, and a
compound event is also called a composite event.
• Simple Event
• Each of the final outcomes for an experiment is called a simple
event. In other words, a simple event includes one and only one
outcome. Usually, simple events are denoted by E1, E2, E3, and so
forth.
•
Simple Event
An event that includes one and only one of
the (final) outcomes for an experi- ment is called a simple event
and is usually denoted by Ei.
EXAMPLE 4–4 Selecting Two Workers
• Reconsider Example 4–3 on selecting two workers from a
company and observing whether the worker selected each
time is a man or a woman. Each of the final four outcomes
(MM, MW, WM, and WW) for this experiment is a simple
event. These four events can be denoted by E1, E2, E3, and
E4, respectively. Thus,
• E1 = {MM}, E2 = {MW}, E3 = {WM}, and E4 =
{WW}
• A compound event consists of more than one outcome
• Compound Event A compound event is a collection of more
than one outcome for an experiment.
Types of the Probability
• There are three basic interpretations of probability:
A. Classical probability
B. Empirical or relative frequency probability
C. Subjective probability
• Classical Probability
Classical probability assumes that all outcomes in the sample space are
equally likely to occur. For example, 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
occurring
Two Properties of Probability
• There are two important properties of probability that
we should always remember. These properties are
mentioned below.
Rule 1.
• The probability of an event always lies in the range 0
to 1.
Whether it is a simple or a compound event, the
probability of an event is never less than 0 or greater
than 1. We can write this property as follows.
• First Property of Probability
• 0 ≤ P(Ei) ≤ 1
• 0 ≤ P(A) ≤ 1
Cont.….
• An event that cannot occur has zero probability
and is called an impossible (or null) event.
• An event that is certain to occur has a probability
equal to 1 and is called a sure (or certain) event.
In the following examples, the first event is an
impossible event and the second one is a sure
event.
• P(a tossed coin will stand on its edge) = 0
• P(a child born today will eventually die) = 1.0
Cont…
Rule 2.
The sum of the probabilities of all simple events (or
final outcomes) for an experiment, denoted by
∑ P(Ei), is always 1.
• Second Property of Probability For an experiment
with outcomes E1, E2, E3, . . . ,
• ∑ P(Ei) = P(E1) + P(E2) + P(E3) + . . . = 1.0
Examples
• if you buy a lottery ticket, you may either win or
lose. The probabilities of these two events must add
to 1.0, that is:
• P(you will win) + P(you will lose) = 1.0 Similarly, for
the experiment of one toss of a coin,
• P(Head) + P(Tail) = 1.0 For the experiment of two
tosses of a coin,
• P(HH) + P(HT) + P(TH) + P(TT) = 1.0
• For one game of football by a professional team,
• P(win) + P(loss) + P(tie) = 1.0
Complementary Events
• Another important concept in probability theory is that of
complementary events. When a die is rolled, for instance, the
sample space consists of the outcomes 1, 2, 3, 4, 5, and 6. The
event E of getting odd numbers consists of the outcomes 1, 3, and
5. The event of not getting an odd number is called the
complement of event E, and it consists of the outcomes 2, 4, and 6.
• The complement of an event E is the set of outcomes in the
sample space that are not included in the outcomes of event E.
The complement of E is denoted by
Empirical Probability
• The difference between classical and empirical probability is that
classical probability assumes that outcomes are equally chances
•
while empirical probability relies on actual experience to determine
the likelihood of outcomes.
• Formula for Empirical Probability
• P(E) =
𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 𝑓𝑜𝑟 𝑡ℎ𝑒 𝑐𝑙𝑎𝑠𝑠
𝑡𝑜𝑡𝑎𝑙 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑖𝑒𝑠 𝑖 𝑡ℎ𝑒 𝑑𝑖𝑠𝑡𝑟𝑖𝑏𝑢𝑡𝑖𝑜𝑛
=
𝑓
𝑛
• This probability is called empirical probability and is based on
observation
Example 1 .
P(airplane) = 3/50 = 0.06 = 6%
Exercise
Subjective probability
• The third type of probability is called subjective probability.
• Subjective Probability Subjective probability is the probability
assigned to an event based on subjective judgment, experience,
information, and belief
• Subjective probability uses a probability value based on an
educated guess or estimate, employment,opinions and inexact
information.
• Subjective probability differs from the other two approaches
because subjective probability differs from person to person
The Addition Rules for Probability
• Mutually exclusive Events
• Events are mutually exclusive if even A and B can not happen
at the same time the outcomes of a single experiment.
• 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)
Example 2
Somaliland Business Development Center including three local companies has a sample
of the following number of employees:
• Dahabshiil
250
• TELESOM
175
• SOMCABLE
115
• If a research is selected at random one employee , find the probability that the
employee is employed by Dahabshiil or SOMCABLE.
P(Dahabshiil or SOMCABLE) =
P(Dahabshiil) + P(SOMCABLE)
•
= 250/540 + 115/540
•
= 365/540
•
= 73/108
Exercise : A city has 9 coffee shops: 3 Starbuck’s, 2 Caribou Coffees, and 4 Crazy
Mocho Coffees. If a person selects one shop at random to buy a cup of coffee, find the
probability that it is either a Starbuck’s or Crazy Mocho Coffees.
Addition Rule 2
• If A and B are not mutually exclusive, then P(A or B) =P(A)
+P(B) -P(A andB)
The Multiplication Rules and Conditional Probability
The Multiplication Rules
• The multiplication rules can be used to find the probability of two or
more events that occur in sequence.
• For example, if you toss a coin and then roll a die, you can find the
probability of getting a head on the coin and a 4 on the die. These
two events are said to be independent since the outcome of the first
event (tossing a coin) does not affect the probability outcome of the
second event (rolling a die).
Con…
Example 2 .
Box contains 3 red balls, 2 blue balls, and 5 white balls. A ball is
selected and its color noted. Then it is replaced. A second ball is
selected and its color noted. Find the probability of each of these.
a) Selecting 2 blue balls
b) Selecting 1 blue ball and 1 white ball
c) Selecting 1 red ball and 1 blue ball
• Solution
• P(blue and blue) = P(blue) . P(blue) = 2/10 . 2/10 = 4/100 =
1/25
• P(blue and white) = P(blue) . P(white) = 2/10 . 5/10 = 10/100 =
1/10
• P(red and blue) = P(red) . P(blue) = 3/10 . 2/10 = 6/100 = 3/50
Multiplication Rule for Dependent Events
Cont..
• Conditional probability was discussed in Section 4.3.1. It is
obvious from the formula for joint probability that if we know
the probability of an event A and the joint probability of
events A and B, then we can calculate the conditional
probability of B given A.
EXAMPLE 4–24 Major and Status of a Student
• The probability is a a randomly selected student
from a college is a senior is .20, and the joint
probability that the student is a computer science
major and a senior is .03. Find the conditional
probability that a student selected at random is a
computer science major given that the student is a
senior.
Marginal and Conditional Probabilities
• Definition
• Marginal Probability Marginal probability is the probability
of a single event without consideration of any other event.
Marginal probability is also called simple probability.
Conditional Probability Conditional probability is the
probability that an event will occur given that another event
has already occurred.
• If A and B are two events, then the conditional probability of
A given B is written as P(A/B) and read as “the probability
of A given that B has already occurred.”
•
EXAMPLE 4–27 Taking a Course in Ethics
• A university president proposed that all students must take a
course in ethics as a requirement for graduation. Three hundred
including faculty members and students from this university were
asked about their opinions on this issue. Table 4.10 gives a twoway classification of the responses of these faculty members and
students
Factorials
• The symbol ! (read as factorial) is used to denote factorials.
The value of the factorial of a number is obtained by
multiplying all the integers from that number to 1. For example,
7! is read as “seven factorial” and is evaluated by multiplying
all the integers from 7 to 1.
EXAMPLE 4–34 Factorial of a Number
• Evaluate 7!
Solution To evaluate 7!, we multiply all the
integers from 7 to 1.
7! = 7 ⋅ 6 ⋅ 5 ⋅ 4 ⋅ 3 ⋅ 2 ⋅ 1 = 5040
• Thus, the value of 7! is 5040
EXAMPLE 4–35 Factorial of a Number
• Evaluate 10!
Solution The value of 10! is given by the product
of all the integers from 10 to 1. Thus,
• 10! = 10 · 9 · 8 · 7 · 6 · 5 · 4 · 3 · 2 · 1 =
3,628,800
EXAMPLE 4–36 Factorial of the
Difference of Two Numbers
Evaluate (12 − 4)!.
Solution The value of (12 − 4)! Is (12 − 4)! = 8! = 8 ⋅ 7 ⋅ 6 ⋅ 5 ⋅ 4 ⋅ 3
⋅ 2 ⋅ 1 = 40,320.
• EXAMPLE 4–37 Factorial of Zero
Evaluate (5 − 5)!.
• Solution The value of (5 − 5)! is 1.
(5 − 5)! = 0! = 1
Note that 0! is always equal to 1.
Combinations
Combinations give the number of ways x
elements can be selected from n elements. The
notation used to denote the total number of
combinations is nCx
• which is read as “the number of combinations
of n elements selected x at a time.”
• Note that some calculators use r instead of x,
so that the combinations notation then reads
nCr.
• In the combinations formula, n represents the total items and x is the
number of items selected out of n. Note that in combinations, n is
always greater than or equal to x. If n is less than x, then we cannot
select x distinct elements from n.
EXAMPLE 4–38 Choosing Ice Cream
An ice cream parlor has six flavors of ice cream. Kristen wants to buy
two flavors of ice cream. If she randomly selects two flavors out of six,
how many combinations are possible?
Solution For this example,
n = total number of ice cream flavors = 6
x = number of ice cream flavors to be selected = 2
Therefore, the number of ways in which Kristen can select two flavors of
ice cream out of six is
6C2 = 6!
= 6! = 6 · 5 · 4 · 3 · 2 · 1
= 15
2!(6 − 2)!
2! 4! 2 · 1 · 4 · 3 · 2 · 1
• Thus, there are 15 ways for Kristen to select two ice cream flavors
out of six
EXAMPLE 4–39 Selecting a Committee
Three members of a committee will be randomly selected from
five people. How many different combinations are possible?
Solution There are a total of five persons, and we are to select
three of them. Hence,
n = 5 and x = 3 Applying the combinations formula, we get
5C3 =
5!
= 5! = 120 = 10
3!(5 − 3)!
3! 2! 6 · 2
• If we assume that the five persons are A, B, C, D, and E, then
the 10 possible combinations for the selection of three
members of the committee are
• ABC ABD ABE ACD ACE ADE BCD BCE BDE
CDE
EXAMPLE 4–40 Selecting Candidates for an
Interview
• Marv & Sons advertised to hire a financial
analyst. The company has received applications
from 10 candidates who seem to be equally
qualified. The company manager has decided
to call only 3 of these candidates for an
interview. If she randomly selects 3 candidates
from the 10, how many total selections are
possible
Permutations
The concept of permutations is very similar to that of combinations but with one
major difference— here the order of selection is important. Suppose there are
three marbles in a jar—red, green, and purple—and we select two marbles
from these three
EXAMPLE 4–41 Selecting Office Holders of a
Club
A club has 20 members. They are to select three office holders—
president, secretary, and treasurer— for next year. They always select
these office holders by drawing 3 names randomly from the names of
all members. The first person selected becomes the president, the
second is the secretary, and the third one takes over as treasurer. Thus,
the order in which 3 names are selected from the 20 names is
important. Find the total number of arrangements of 3 names from these
20
Solution For this example,
• n = total members of the club = 20
• x = number of names to be selected = 3
•
Soultion
Since the order of selections is important, we find the number
of permutations or arrangements using the following formula:
•
Thus, there are 6840 permutations or arrangements for
selecting 3 names out of 20.