PROBABILITY

Lesson
3
 TOPICS
1. Fundamental Counting Principle
2. Factorial
3. Permutation
4. Combination
5. Types of Probability
6. Rules of Probability
7. Normal Probability Distribution
 LEARNING OUTCOMES
At the end of the lesson, you should be able to:
1. Define and differentiate permutation from
combination;
2. Simplify and evaluate factorial problems;
3. Solve different problem of events using the
fundamental counting principle and probability;
4. Apply probability in solving problems;
5. Solve different probability problems using rules of
probability.
6. Apply the concept of normal probability distribution
in word problems.
 01
FUNDAMENTAL
COUNTING
PRINCIPLE
FUNDAMENTAL Example:
COUNTING Suppose a mall has three
PRINCIPLE doors, in how many ways can
a shopper enter and leave
If ACTIVITY 1 can be done
the mall.
in n1 ways, ACTIVITY 2 can
be done in n2 ways,
ACTIVITY 3 can be done in
n3 ways, and so forth, then
the number of ways of doing
three activities on a
specified order is the
product of n1, n2, n3, and
so on. In symbols,
1. BY LISTING
The different entrance-exit pairs are:

There are nine different ways of doing the said

activities.
2. BY USING A TABLE
In the table, the ordered pair (D1, D1) means that the
shopper can enter and leave the mall using the same
door. The order pair (D1, D2) means that the shopper
can enter the mall in door D1 and can leave using D2.

Entrance Exit Door

Door D1 D2 D3
D1 (D1, D1) (D1, D2) (D1, D3)
D2 (D2, D1) (D2, D2) (D2, D3)
D3 (D3, D1) (D3, D2) (D3, D3)
3. BY USING A TREE DIAGRAM
The diagram show below shows the nine different ways of
entering and leaving the mall.
4. BY USING THE FUNDAMENTAL COUNTING
PRINCIPLE
There are three doors to enter the mall, n1 = 3, and
using the same for leaving, n2 = 3. Thus, (3)(3) = 9
different ways.
 02
FACTORIAL
DEFINITION OF Example:
CONCEPT
Factorial
 Factorial n is denoted by
n! and is define as n! =
n(n-1)(n-2)(n-3)…
(3)(2)(1).
 The product (3)(2)(1) can
be written in brief
factorial notation as 3!
(read as “three
factorial” or “factorial
3”. Thus, six factorial
is written as:
 03
PERMUTATION
DEFINITION OF
CONCEPT
Permutation
 It refers to the wherein,
arrangement of objects n = number of objects
with reference to order. r = number of objects
 Given a set with n taken from n at a time
objects, then we can take
r objects from the set.
The total of permutations
of n distinct objects “The number of permutations
taken r at a time is of n objects taken r at a
presented by the notation time
nPr and can be evaluated
issuing the formula,
PERMUTATION RULE 1
The number of permutation of n distinct objects taken
all together is n!.

Example:
How many different signals can be made using four flags
if all flags must be used in each signal?
PERMUTATION RULE 2
The arrangement of n objects in a specific order using
r objects at a time is given by the formula,

Example:
How many different ways can a president and vice-
president be selected for a group if there are seven
members available?
PERMUTATION RULE 3
The arrangement of n objects in a circular pattern is
given by the formula,

Example:
In how many ways can five persons be seated around a
circular table?
PERMUTATION RULE 4
The number of permutations of n objects in which r1 are
alike,r2 are alike, r3 are alike,… etc., is

Example:
How many different permutations can be made from the
letters of the word “STATISTICS”?
 04
COMBINATION
DEFINITION OF
CONCEPT
Combination
 If the order is not wherein,
important, the total n = number of objects
number of orders or r = number of objects
arrangement is called taken from n at a time
combination.
 The number of
combinations of n objects
taken r at a time denoted “The number of combinations
by nCr, or C (n,r), and of n objects taken r at a
is given by the formula: time
Example:
Given the letters A, B, C, and D, list the permutations
and combinations for selecting two letters.
Note that in permutation, AB is different from BA but
in combination, AB is the same as BA.

Permutations Combinations

AB BA CA DA AB BC
AC BC CB DB AC BD
AD BD CD DC AD CD
 05
TYPES OF
PROBABILITY
PROBABILITY TYPES OF
CONCEPTS PROBABILITY
A foundation of Science of
Statistics is the concept Classical Probability
of the probability.
Theories in Statistics are
rooted in this concept. It Relative Probability
is associated with the
ideas of chance and
likehood. Subjective Probability
1. CLASSICAL PROBABILITY
The probability of an event is the ratio of the number
of favorable outcomes to the total number of possible
outcomes.
𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝐹𝑎𝑣𝑜𝑟𝑎𝑏𝑙𝑒 𝑂𝑢𝑡𝑐𝑜𝑚𝑒𝑠
𝑃𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑜𝑓 𝑎𝑛 𝐸𝑣𝑒𝑛𝑡 =
𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝐴𝑙𝑙 𝑃𝑜𝑠𝑠𝑖𝑏𝑙𝑒 𝑂𝑢𝑡𝑐𝑜𝑚𝑒𝑠

Example:
What is the probability of getting a head in one toss
of a fair coin?
2. RELATIVE PROBABILITY
Known as Probability as Relative Frequency which is
also referred to as experimental or empirical
probability.
𝐹𝑟𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 𝑜𝑓 𝑂𝑐𝑐𝑢𝑟𝑒𝑛𝑐𝑒𝑠 𝐹𝑎𝑣𝑜𝑟𝑎𝑏𝑙𝑒 𝑡𝑜 𝑡ℎ𝑒 𝐸𝑣𝑒𝑛𝑡
𝑃𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑜𝑓 𝑎𝑛 𝐸𝑣𝑒𝑛𝑡 =
𝑇𝑜𝑡𝑎𝑙 𝐹𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦

Example:
An examination is given to a class of 40 students. 4
got perfect scores. What is the probability that a
student who is picked at random from the group, is a
student: a) who got a perfect score b) who did not get
a perfect score?
3. SUBJECTIVE PROBABILITY
This probability is the chance of occurrence assigned
by a person to an event based on personal experience,
intuition, and even beliefs.

Example:
What is the probability that a business competitor will
launch its marketing program next week?

What is the probability that it will rain tomorrow?

What is the probability that you will celebrate

Valentine’s Day on March?
 06
RULES OF
PROBABILITY
DEFINITION OF Event
CONCEPT It is an outcome of an
experiment which is often
Sample Space symbolized by the letter x
A complete roster or or E.
listing of all elementary
events for an experiment. Example:
It is the set of all S = {1,2,3,4,5,6}
possible outcomes of S = {(H,H),(H,T),(T,H),(T,T)}
statistical experiment and
is denoted by the symbol S.

Experiment
The process that produces
outcomes. It is any
activity that generates set
of data.
DEFINITION OF Example:
CONCEPT
Union of Events
The union of two events, A
and B, denoted by the
symbol A ∪ B, is the event
containing all the elements
that belong to A or to B or
both.
Intersection of Events
The intersection of two
events A and B, denoted by
the symbol A ∩ B, is the
event that consists all
elements that are common to
A and B.
DEFINITION OF Example:
CONCEPT 𝐴∩𝐵 =∅

Mutually Exclusive
Events
Two events A and B are 𝐴 𝐵
mutually exclusive if A ∩ B
= ∅ ; that is A and B have
no elements in common.
Complement of an Event 𝐴′
𝛺
The complement of an event
A with respect to S is the
set of all elements of S 𝐴
that are not in A. We
denote the complement of A
by the symbol A’
1. THE ADDITION RULES
 If A and B are two events, then

 If A and B are two mutually exclusive events, then

 If A and A’ are complementary events, then

Example:
What is the probability of drawing a black card or a
face card?
Example:
What is the probability of getting a total of nine or
ten when a pair of dice is tossed?
Example:
From a bag containing five red, seven blue, and three
green balls, a ball is drawn. Find the probability that
the ball is not red.
2. CONDITIONAL PROBABILITY
The conditional probability of B, given A, denoted by
P (B/A), is given by the equation:
Example:
The probability that Albert will pass Statistics is
P(S)=0.70. The probability that he will pass English is
P(E)=0.80; and the probability that he will pass
Statistics and English is P(S ∩ E)=0.50. Find the
probability that Albert will pass Statistics given that
he will pass English.
3. INDEPENDENT EVENTS
Two events are said to be independent events if
information of one does not change the probability of
other.
Two events are said to be dependent events if
information of one does change the probability of
other.
Two events, A and B, are independent if either

Otherwise, A and B are dependent.

Example:
Suppose a box contains seven red and nine blue balls.
If two balls are drawn in succession, the first ball
drawn is red and is replaced in the box before the
second ball is drawn. What is the probability that the
second ball drawn is blue?
4. MULTIPLICATION RULES
 If in an experiment the events A and B can both
occur, then

 If two events A and B are independent, then

 If in an experiment the events A1,A2,A3,…,AK are

dependent, then
4. MULTIPLICATION RULES
 If the events A1,A2,A3,… Ak are independent, then;
Example:
Two pieces of marble are drawn from a jar containing 5
green, seven yellow, and nine orange marbles. Find the
probability that the first marble is green and the
second is yellow:
a. if the marble is replaced into the jar before the
second marble is drawn.
Example:
Two pieces of marble are drawn from a jar containing 5
green, seven yellow, and nine orange marbles. Find the
probability that the first marble is green and the
second is yellow:
b. if the first is not replaced.
Example:
Three cards are drawn in succession without replacement
form an ordinary deck of playing cards. Find the
probability that the first card is a black king, the
second card is an ace or queen, and the third card is
greater than five but less than 10.
 07
NORMAL
PROBABILITY
DISTRIBUTION
 In this, the value
DEFINITION OF 
cluster around an average
CONCEPT value and fewer values
are found at increasing
Normal Distribution distances from the
 It is represented by a average.
bell-shaped and its  It is associated with the
probability distribution terms:
is called a normal  “Bell Shaped”
distribution.  Symmetric
 It occupies the central  Range of Possible Values
place in the study of the is Infinite on Both
Theory of Statistics, as Directions
it is the basis of
solving different types
of statistical problems.
PROPERTIES OF 4. The total area under the
normal curve and above
NORMAL CURVE the horizontal axis is
1. The curve is symmetric equal to 1 or 100%.
about the vertical axis 5. The normal curve is
through the mean. divided into ±3 standard
2. The mean, median, and deviations.
mode have the same value 6. Along the horizontal
and located at the same line, the distance from
point (center) along the one integral standard
horizontal axis. score to the next
3. The tails of the curve integral standard score
are asymptotic relative is measured by the
to the horizontal line standard deviation.
as we proceed in either 7. For a normal curve, the
direction away from the area within:
mean.
PROPERTIES OF
NORMAL CURVE
 1 SD from the mean is
about 68%
 2 SD from the mean is
about 95%
 3 SD from the mean is
about 99.7%
 STANDARD SCORE
To determine the areas under the normal curve, we shall
convert scores into standard scores or z-scores. This
means that empirical distribution will be standardized to
the theoretical normal curve.

It is the position of a value of x in terms of the number

of standard deviations, which is located from the mean.

Formula: where:
= standard score
= mean
= standard deviation
= empirical value
Example:
Given that 𝑃(2.5 ≤ 𝑐 ≤ 7.5), 𝜇 = 5, and 𝜎 = 10. Determine the
areas under the normal curve.
STANDARD NORMAL DISTRIBUTION TABLE
Example:
In a course audit test in the College of Teacher
Education, with sample of 100 cases, the mean score is
70 and the standard deviation is 4. Assuming normality,
what is the percentage of the cases fall between the
mean and a score of 76?
STANDARD NORMAL DISTRIBUTION TABLE
Example:
In a course audit test in the College of Teacher
Education, with sample of 100 cases, the mean score is
70 and the standard deviation is 4. Assuming normality,
what is the probability that the score picked at random
will lie above the score 76%?
STANDARD NORMAL DISTRIBUTION TABLE
Example:
In a course audit test in the College of Teacher
Education, with sample of 100 cases, the mean score is
70 and the standard deviation is 4. Assuming normality,
what is the probability that a score will lie below
score 76?
STANDARD NORMAL DISTRIBUTION TABLE
Example:
In a course audit test in the College of Teacher
Education, with sample of 100 cases, the mean score is
70 and the standard deviation is 4. Assuming normality,
how many cases fall between scores of 72 and 78?
STANDARD NORMAL DISTRIBUTION TABLE
THANK
YOU!