Preliminary Concepts on
Probability
MATH30-6
Probability and Statistics
Objectives
At the end of the lesson, the students are expected to
• Understand and describe sample spaces and events for
random experiments with graphs, tables, lists, or tree
diagrams;
• Use permutation and combinations to count the
number of outcomes in both an event and the sample
space;
• Define probability; and
• Relate counting techniques to real life situations.
Probability
• A tool to relate the descriptive statistics to inferential
statistics
• Ratio of number of samples derived from the total
population
• Deals with counting elements
Random Experiment
• An experiment that can result in different outcomes,
even though it is repeated in the same manner every
time
Examples:
- Measuring a current in a copper wire with the presence
of uncontrollable inputs resulting the variations in
measurements
- Designing a communication system (computer or voice
communication network) where the information
capacity available to serve individuals using the
network is an important design consideration
Sample Space
• The set of all possible outcomes of a random
experiment
• Denoted as S
Examples:
Consider the experiment of tossing a die.
• Sample space for the number appearing on the top
face:
S1 = {1, 2, 3, 4, 5, 6}
• Sample space for the number appearing on the top
face whether it is even or odd:
S2 = {even, odd}
Discrete Sample Space
• A sample space is discrete if it consists of a finite or
countable infinite set of outcomes.
Examples:
- Sample space for the number appearing on the top
face:
S = {1, 2, 3, 4, 5, 6}
- Sample space for a thrown die until a five occurs:
S = {F, NF, NNF, NNNF, …}
where F = occurrence of 5 and N = nonoccurrence of 5.
Continuous Sample Space
• A sample space is continuous if it contains an interval
(either finite or infinite) of real numbers.
Example:
- Sample space of the life in years (t) of a certain
electronic component:
S = {t|t ≥ 0}
Sample Space
Provide a reasonable description of the sample space for
each of the random experiments in Exercises 2-1 to 2-17.
There can be more than one acceptable interpretation of
each experiment. Describe any assumptions you make.
2-2/28 Each of the four transmitted bits is classified as
either in error or not in error.
2-4/28 The number of hits (views) is recorded at a highvolume Web site in a day.
Sample Space
Provide a reasonable description of the sample space for
each of the random experiments in Exercises 2-1 to 2-17.
There can be more than one acceptable interpretation of
each experiment. Describe any assumptions you make.
2-16/28 An order for a computer system can specify
memory of 4, 8, or 12 gigabytes, and disk storage of 200,
300, or 400 gigabytes. Describe the set of possible orders.
2-17/28 Calls are repeatedly placed to a busy phone line
until a connection is achieved.
Sample Space
2-18/28 In a magnetic storage device, three attempts are
made to read data before an error recovery procedure
that repositions the magnetic head is used. The error
recovery procedure attempts three repositionings before
an “abort” message is sent to the operator. Let
s denote the success of a read operation
f denote the failure of a read operation
F denote the failure of an error recovery procedure
S denote the success of an error recovery procedure
A denote an abort message sent to the operator.
Describe the sample space of this experiment with a tree
diagram.
Event
• Subset of the sample space of a random experiment
Consider the events E1 and E2.
• Union of two events
- Consists of all outcomes that are contained in either of
the two events
- Denoted by E1 ∪ E2
• Intersection of two events
- Consists of all outcomes that are contained in both of
the two events
- Denoted by E1 ∩ E2
Event
• Complement of an event
- Set of outcomes in the sample space that are not in the
event
- The complement of the event E is E′ or EC.
- (E′)′ = E
Mutually Exclusive Events
Two events, denoted as E1 and E2 , such that
𝐸1 ∩ 𝐸2 = ∅
are said to be mutually exclusive or disjoint.
Example:
Let M = {a, e, i, o, u} and N = {r, s, t}; then it follows that
𝑀 ∩ 𝑁 = ∅. That is, 𝑀 and 𝑁 have no elements in
common and, therefore, cannot occur simultaneously.
Event
2-19/28 Three events are shown on the Venn diagram in
the following figure:
Reproduce the figure and shade the region that
corresponds to each of the following events:
(a) A′ (b) A ∩ B
(c) (A ∩ B) ∪ C
(d) (B ∪ C)′
(e) (A ∩ B)′ ∪ C
Venn Diagram
Venn Diagram
Several results that follow from the foregoing definitions,
which may easily verified by means of Venn diagrams, are
as follows:
1. A ∩ ∅ = ∅
2. A ∪ ∅ = ∅
3. A ∩ A′ = ∅
4. A ∪ A′ = S
5. S′ = ∅
6. ∅′ = S
7. A′ ′ = A
Venn Diagram
8. De Morgan’s laws:
𝐴 ∩ 𝐵 ′ = 𝐴′ ∪ 𝐵′
𝐴 ∪ 𝐵 ′ = 𝐴′ ∩ 𝐵′
9. Distributive laws:
𝐴 𝐵 ∪ 𝐶 = 𝐴𝐵 ∪ 𝐴𝐶
𝐴 ∪ 𝐵𝐶 = 𝐴 ∪ 𝐵 𝐴 ∪ 𝐶
Event
2-21/28 A digital scale is used that provides weights to the
nearest gram.
(a) What is the sample space for this experiment?
Let A denote the event that a weight exceeds 11 grams,
let B denote the event that weight is less than or equal to
15 grams, and let C denote the event that a weight is
greater than or equal to 8 grams and less than 12 grams.
Describe the following events.
(b) A ∪ B
(c) A ∩ B
(d) A′
(e) A ∪ B ∪ C
(f) (A ∪ C)′
(g) A ∩ B ∩ C
(h) B′ ∩ C
(i) A ∪ (B ∩ C)
Event
2-23/29 Five bits are transmitted are transmitted over a
digital communications channel. Each bit is neither
distorted or received without distortion. Let Ai denote the
event that the ith bit is distorted, i = 1, … , 5.
(a) Describe the sample space for this experiment.
(b) Are the Ai’s mutually exclusive?
Describe the outcomes in each of the following events:
(c) A1
(d) A1′
(e) A1 ∩ A2 ∩ A3 ∩ A4
(f) (A1 ∩ A2) ∪ (A3 ∩ A4)
Counting Techniques
• An important part of combinatorics (study of
arrangement of objects which is part of discrete
mathematics)
• Methods used for counts of the numbers of outcomes
in the sample space and various events for analyzing
random experiments
• Used for more complicated problems and more difficult
sample space or an event
Multiplication Rule
• If an operation can be performed in 𝑛1 ways, and if for
each of these a second operation can be performed in
𝑛2 ways, then the two operations can be performed in
𝑛1 𝑛2 ways.
Example:
How many sample points are in the sample space when a
pair of dice is thrown once?
The first die can land in any one of 𝑛1 = 6 ways. For each
of these 6 ways the second die can also land in 𝑛2 = 6
ways. Therefore, the pair of dice can land in
𝑛1 𝑛2 = 6 6 = 36 possible ways.
Generalized Multiplication Rule
• If an operation can be performed in 𝑛1 ways, and if for
each of these a second operation can be performed in
𝑛2 ways, and for each of the first and two a third
operation can be performed in 𝑛3 ways, and so forth,
then the sequence of 𝑘 operations can be performed in
𝑛1 𝑛2 … 𝑛𝑘 ways.
Generalized Multiplication Rule
Example:
Sam is going to assemble a computer by himself. He has
the choice of ordering chips from two brands, a hard drive
from four, memory from three and an accessory bundle
from five local stores. How many different ways can Sam
order the parts?
Since 𝑛1 = 2, 𝑛2 = 4, 𝑛3 = 3, and 𝑛4 = 5, there are
𝑛1 × 𝑛2 × 𝑛3 × 𝑛4 = 2 × 4 × 3 × 5 = 120.
Generalized Multiplication Rule
Examples:
2-16/46 How many even four-digit numbers can be
formed from the digits 0, 1, 2, 5, 6, and 9 if each digit can
be used only once?
2-30/51 In how many different ways can a true-false test
consisting of 9 questions can be answered?
Generalized Multiplication Rule
2-34/34 A wireless garage door opener has a code
determined by the up or down setting of 10 switches.
How many outcomes are in the sample space of possible
codes?
2-35/35 An order for a computer can specify any one of
five memory sizes, any one of three types of displays, and
any one of five sizes of hard disks, and can either include
or not include a pen tablet? How many different systems
can be ordered?
Generalized Multiplication Rule
14/340 In a version of the computer language BASIC, the
name of a variable is a string of one or two alphanumeric
characters, where uppercase and lowercase letters are
not distinguished. (An alphanumeric character is either
one of the 26 English letters or one of the 10 digits.)
Moreover, a variable name must begin with a letter and
must be different from the five strings of two characters
that are reserved for programming use. How many
different variable names are there in this version of
BASIC?
Generalized Multiplication Rule
15/340 Each user on a computer system has a password,
which is six to eight characters long, where each character
is an uppercase letter or a digit. Each password must
contain at least one digit. How many possible passwords
are there?
Permutation
• An ordered sequence of the elements
Example:
Consider the three letters a, b, and c. The possible
permutations are abc, acb, bac, bca, cab, and cba.
• The number of permutations of n different elements is
n! (read as n factorial) where
n! = n × (n ─ 1) × (n ─ 2) × … × 2 × 1
Note: 0! = 1
Permutation of Subsets
The number of permutations of subsets of r elements
selected from a set of n different elements is
𝑃𝑟𝑛 = 𝑛 × 𝑛 − 1 × 𝑛 − 2 × ⋯ × 𝑛 − 𝑟 + 1
𝑛!
𝑛
𝑃𝑟 =
𝑛−𝑟 !
Equation 2-2
Note: 𝑃𝑟𝑛 = 𝑛𝑃𝑟 = 𝑃 𝑛, 𝑟
Permutation
Examples:
2-42/30 In the layout of a printed circuit board (PCB) for
an electronic product, there are 15 different locations that
can accommodate chips.
(a) If five different types of chips are to be placed on the
board, how many different layouts are possible?
(b) If the five chips that are placed on the board are of the
same type, how many different layouts are possible?
Permutation
2-45/31 Consider the design of a communication system.
(a) How many three-digit phone prefixes that are used to
represent a particular geographic area (such as an area
code) can be created from the digits 0 through 9?
(b) As in part (a), how many three-digit phone prefixes are
possible that do not start with 0 or 1, but contain 0 or 1 as
the middle digit?
(c) How many three-digit phone prefixes are possible in
which no digit appears more than once in each prefix?
Permutation
2.31/51 A witness to a hit-and-run accident told the police
that the license number contained the letters RLH
followed by 3 digits, the first of which was a 5. If the
witness cannot recall the last 2 digits, but is certain that
all 3 digits are different, find the maximum number of
automobile registrations that the police may have to
check.
2.32/52 (a) In how many ways can 6 people be lined up to
get on a bus?
(b) If 3 specific persons, among 6, insist on following each
other, how many ways are possible?
(c) If 2 specific persons, among 6, refuse to follow each
other, how many ways are possible?
Permutation
2.33/52 If a multiple-choice test consists of 5 questions
each with 4 possible answers of which only 1 is correct,
(a) In how many different ways can a student check off
one answer to each question?
(b) In how many ways can a student check off one answer
to each question and get all the answers wrong?
2.36/52 (a) How many three-digit numbers can be formed
from the digits 0, 1, 2, 3, 4, 5, and 6 if each digit can be
used only once?
(b) How many of these are odd numbers?
(c) How many are greater than 330?
Circular Permutation
The number of permutations of 𝑛 distinct objects
arranged in a circle is 𝑛 − 1 !.
Example:
2-43/52 In how many ways can 5 different trees be
planted in a circle?
Permutation of Similar Objects
• Permutation with repetition
The number of permutations of 𝑛 = 𝑛1 + 𝑛2 + ⋯ + 𝑛𝑟
objects of which 𝑛1 are of one type, 𝑛2 are of second
type, … , and 𝑛𝑟 are of an rth type is
𝑛!
𝑛1 ! 𝑛2 ! 𝑛3 ! … 𝑛𝑟 !
Permutation of Similar Objects
Examples:
2-45/52 How many distinct permutations can be made
from the letters of the word INFINITY?
2-46/52 In how many ways can 3 oaks, 4 pines, and 2
maples be arranged along a property line if one does not
distinguish among trees of the same kind?
2-12/26 A part is labeled by printing with four thick lines,
three medium lines, and two thin lines. If each ordering of
the nine lines represent a different label, how many
different labels can be generated by using this scheme?
Ordered Partition
The number of ways of partitioning a set of 𝑛 objects into
𝑟 cells with 𝑛1 elements in the first cell, 𝑛2 elements in
the second, and so forth, is
𝑛!
𝑛
𝑛1 , 𝑛2 , … , 𝑛𝑟 = 𝑛1 ! 𝑛2 ! … 𝑛𝑟 !
where 𝑛1 + 𝑛2 + ⋯ + 𝑛𝑟 = 𝑛.
Example:
2-20/49 In how many ways can 7 scientists be assigned to
one triple and two double hotel rooms?
Combination
A combination is actually a partition of two cells, the one
cell containing the 𝑟 objects selected and the other cell
containing the 𝑛 − 𝑟 objects that are left.
The number of such combinations, denoted by
𝑛
𝑛
𝑟, 𝑛 − 𝑟 , is usually shortened to 𝑟 ,
since the number of elements in the second cell must be
𝑛−𝑟 .
Combination
The number of combinations of 𝑛 distinct objects taken 𝑟
at a time is
𝑛!
𝑛
= 𝑛𝐶𝑟 = 𝐶 𝑛, 𝑟 =
.
𝑟
𝑟! 𝑛 − 𝑟 !
Combination
Examples:
2-14/27 Sampling without Replacement
A bin of 50 manufactured parts contains three defective
parts and 47 nondefective parts. A sample of six parts is
selected from the 50 parts without replacement. That is,
each part can only be selected once and the sample is a
subset of the 50 parts. How many different samples are
there of size six that contain exactly two defective parts?
Combination
2-22/50 A young boy asks his mother to get five game-boy
cartridges from his collection of 10 arcade and 5 sports
games. How many ways are there that his mother will get
3 arcade and 2 sports games, respectively?
11/358 How many poker hands of five cards can be dealt
from a standard deck of 52 cards? Also, how many ways
are there to select 47 cards from a standard deck of 52
cards?
Combination
12/360 How many ways are there to select five players
from a 10-member tennis team to make a trip to a match
at another school?
13/360 A group of 30 people have been trained as
astronauts to go on the first mission to Mars. How many
ways are there to select a crew of six people to go on this
mission (assuming that all crew members have the same
job)?
Combination
15/360 Suppose that there are 9 faculty members in the
mathematics department and 11 in the computer science
department. How many ways are there to select a
committee to develop a discrete mathematics course at a
school if the committee is to consist of three faculty
members from the mathematics department and four
from the computer science department?
Summary
• A random experiment is an experiment that can result
in different outcomes, even though it is repeated in the
same manner each time.
• The sample space is the set of all possible outcomes of
a random experiment.
• The event is a subset of a sample space.
• Multiplication rule is a formula used to determine the
number of ways to complete an operation from the
number of ways to complete successive steps.
• A permutation is an arrangement of all (n!) or part
(permutation of subsets) of a set of objects.
Summary
• Two circular permutations are not considered different
unless corresponding objects in the two arrangements
are preceded or followed by a different object as we
proceed in a clockwise direction. For example, if 4
people are playing bridge, we do not have a new
permutation if they all move one position in a
clockwise direction. By considering one person in a
fixed position and arranging the other in 3! ways, we
find that there are 6 distinct arrangements for the
bridge game.
• A combination is actually a partition with two cells, the
on cell containing the r objects selected and the other
cell containing the (n ─ r) objects that are left.
References
• Montgomery and Runger. Applied Statistics and
Probability for Engineers, 5th Ed. © 2011
• Rosen, Kenneth H. Discrete Mathematics and Its
Applications, 6th Ed. © 2007
• Walpole, et al. Probability and Statistics for Engineers
and Scientists 9th Ed. © 2012, 2007, 2002