College of Engineering

Mindanao State University

ES 85 – LECTURE NOTES
Probability, also theory of probability, branch of 1.If S = {1, 2, 3, 4, 5, 6, 7, 8, 9} and A = {0, 2, 4, 6, 8},
mathematics that deals with measuring or B = {1, 3, 5, 7, 9}, C = {2, 3, 4, 5} and D = {1, 6, 7},
determining quantitatively the likelihood that an list the elements of the sets corresponding to the
event or experiment will have a particular outcome
following events:
Statistics, branch of mathematics that deals with the a)AC; b) AB; c) C’;
collection, organization, and analysis of numerical
d) (C’D)B; e) (SC)’; f) ACD’
data and with such problems as experiment design
and decision-making.
2.Suppose that a family is leaving on a summer
vacation in their camper and that event M is the
Definition 1.1
The set of all possible outcome of a statistical event that they will experience mechanical
experiment is called sample space and is problem, T is the event that they will receive ticket
represented by the symbol S. for committing a traffic violation, and V is the
event that they will arrive at a campsite with no
-Each outcome in a sample space is called an element
vacancies. Referring to the Venn Diagram shown,
or a member of the sample space or simply a sample
point state in words the events represented by the
following regions:
S = {H, T} sample space for tossing a coin a.Region 5
S = {1, 2, 3, 4, 5, 6} sample space for tossing a die
b.Region 3
Sample Space can be represented by a tree diagram,
c. Regions 1 and 2 together
a statement or a rule
d.Regions 4 and 7 together
Definition 1.2 e.Regions 3, 6, 7, and 8 together
An event is a subset of a sample space

Definition 1.3
The complement of an event A with respect to S is
M T
the subset of all elements of S that are not in A. 4
Complement of A is denoted by the symbol A’ 5 7
8
1
Definition 1.4 2
3
The intersection of two events A and B, denoted by 3.Referring to the Venn Diagram, list the numbers of
the symbol AB, is the event containing all elements 6
the regions that represent the following events:
that are common to A and B. V
a) The family will experience no mechanical
problems and commit no traffic violation but
Definition 1.5
Two events A and B are mutually exclusive or disjoint will find a campsite with no vacancies.
if AB = , that is, if A and B have no elements in b) The family will experience both mechanical
common. problems and trouble in locating a campsite
with a vacancy, but will not receive a ticket for
Definition 1.6 violation.
The union of the two events A and B, denoted by the c) The family will either have mechanical trouble
symbol A B, is the events containing all the
or find a campsite with no vacancies but will
elements that belong to A or B or both.
not receive a ticket for committing for traffic
violation.
d) The family will not arrive at a campsite with no
Sample Problems
vacancies.
COUNTING SAMPLE POINTS
Theorem 1 (Multiplication Rule)
If an operation can be performed in n1 ways, and if
for each of these a second operation can be
Theorem 1.7
performed together in n2 ways, then the two
The number of ways of partitioning a set of n objects
operations can be performed together in n 1n2 ways.
into r cells with n1 elements in the first cell, n2
elements in the second, and so forth, is
Theorem 2 (Generalized Multiplication Rule)
If a operation can be performed in n 1 ways, and if for
each of these a second operation can be performed
where n1 + n2 + ….+ nr = n
in n2 ways, and for each of the first two a third
operation can be performed in n3 ways, and so forth,
Theorem 1.8
then the sequence of k operations can be performed
The number of combinations of n distinct objects
in n1n2,……,nk ways.
taken r at a time is

Definition 1.7
A permutation is an arrangement of all or part of a
set of objects. Problems

Theorem 1.3 1.In how many ways can true-false test consisting of
The number of permutation of n distinct objects is n! 9 questions be answered?
2.(a) How many three-digit numbers can be formed
Theorem 1.4 from the digits 0, 1, 2, 3, 4, 5, and 6, if each digit
The number of permutations of n distinct objects can be used only once?
taken r at a time is (b) How many of these are odd numbers?
(c) How many are greater than 330?
P
n r 3.Four married couples have bought 8 seats in a row
for a concert. In how many different ways can they
Theorem 1.5 be seated
The number of permutations of n distinct objects a.with no restrictions?
arranged in a circle is (n-1)!. b.If each couple is to sit together?
c. If all the men sit together to the right of all the
Theorem 1.6 women?
The number of distinct permutations of n things of 4. A college plays football games during a season.
which n1 are of one kind, n2 of a second kind,…., n k of In how many different ways can the team end
a kth kind is the season with 7 wins, 3 loses, and 2 ties?