NUEVA VIZCAYA STATE UNIVERSITY
Bayombong, Nueva Vizcaya
College of Engineering
Chapter 2: Probability
PROBABILITY is an area of study which involves predicting the
relative likelihood of various outcomes. It is a mathematical area
which has developed over the past three or four centuries. One
of the early uses was to calculate the odds of various gambling
games. Its usefulness for describing errors of scientific and
engineering measurements was soon realized. Engineers study
probability for its many practical uses, ranging from quality
control and quality assurance to communication theory in
electrical engineering. Engineering measurements are often
analyzed using statistics, as we shall see later in this book, and
a good knowledge of probability is needed in order to
understand statistics.
SAMPLE SPACE
The set of all possible outcomes of a statistical experiment is
called the sample space and is represented by the symbol S.
ELEMENT
Example #3.
An experiment that tosses a coin 3 times.
S = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}
A = event that has at least 1 head
B = event that has at most 1 head
EA = {HHH, HHT, HTH, HTT, THH, THT, TTH}
EB = {HTT, THT, TTH, TTT}
PROBABILITY | Basic Rules
1. The complement of an event A with respect to S is the
subset of all elements of S that are not in A. We denote
the complement of A by the symbol A’.
2. The intersection of two events A and B, denoted by
the symbol
A ∩ B, is the event containing all
elements that are common to A and B.
Each outcome in a sample space is called an element or a
member of the sample space.
3. Two event A and B are mutually exclusive, or
disjoint, if A ∩ B = Ø that is, A and B have no
elements in common.
Example #1:
4. The union of events A and B, denoted by A∪B, is the
event containing all the elements that belong to A or B
or both.
Consider the experiment of tossing a die. If we are interested in
the number that shows on the top face, the sample space would
be
S = {1,2,3,4,5,6,7,8,9,10}
A = {1,3,5,8,9}
S = {1,2,3,4,5,6}
PROBABILITY | Sample Spaces & Events
Example #2:
An experiment consists of flipping a coin and then flipping it a
second time if a head occurs. If a tail occurs on the first, flip,
then a die is tossed once. To list the elements of the sample
space providing the most information, we construct the tree
diagram
B = {1,4,6,8,10}
Example #4.
If M = {x | 3 < x < 9} and N= {y | 5 < y < 12}, then
M U N = {z | 3 < z < 12}
VENN DIAGRAMS
A∩B=
B∩C =
S = {HH, HT, T1, T2, T3, T4, T5, T6}
A∪C=
B’ ∩ A =
A∩B∩C=
(A ∪ B) ∪ C=
Counting Sample Points
1st Rule: If operations can be performed in n ways, and if for
each of these ways a second operation can be performed in n2
ways, then two operations can be performed in n1n2 ways
PROBABILITY | Sample Spaces & Events
Example#1:
EVENT
How many 4-digit even number can be formed from 0, 1, 2, 5,
6, and 9 if each digit can be used only once?
-
Is any collection of sample points called subset of a
sample space
2nd Rule: The number of permutations of n objects is n!
1|MODULE 2
Prepared by:
MARY ABEGAIL P. CADANG
Emergency Instructor
NUEVA VIZCAYA STATE UNIVERSITY
Bayombong, Nueva Vizcaya
College of Engineering
Chapter 2: Probability
**A permutation is an arrangement of all or part of a set of
objects.
Example#2:
The number of permutations of letters a,b,c,d.
3rd Rule: The number of permutation of n distinct object taken r
at a time is
𝒏!
𝒏𝑷𝒓 =
(𝒏 − 𝒓)!
Example #3:
In one year, three awards (research, teaching, and service) will
be given for a class of 25 graduate students in a statistics
department. If each student can receive at most one award, how
many possible selections are there?
Example #4:
A president and a treasurer are to be chosen from a student club
consisting of 50 people. How many different choices of officers
are possible if
(a) there are no restrictions;
(b) A will serve only if he is president;
(c) B and C will serve together or not at all:
(d) D and E will not serve together?
4th Rule: The number of distinct permutations of n things of
which n1 are one of a kind, n2 of a second kind, …, nk of nth kind
is
𝒏!
𝒏𝟏! 𝒏𝟐! … 𝒏𝒌!
Example#5:
In a college football training session, the defensive coordinator
needs to have 10 players standing in a row. Among these 10
players, there are 1 freshman, 2 sophomores, 4 juniors, and 3
seniors, respectively. How many different ways can they be
arranged in a row if only their class level will be distinguished?
5th Rule: The number of combinations of n distinct objects taken
r at a time is
𝒏!
𝒓! (𝒏 − 𝒓)!
Example#6: A young boy asks his mother to get five GameBoyTM 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?
BASIC RULES OF COMBINING PROBABILITIES
1. Addition Rule
- This can be divided into two parts, depending upon
whether there is overlap between the events being
combined.
a. - If the events are mutually exclusive, there is no
overlap: if one event occurs, other events cannot
occur. In that case the probability of occurrence of
one or another of more than one event is the sum
of the probabilities of the separate events. This
corresponds to a logical or and gives a sum of
separate probabilities.
- Often we can divide all possible outcomes into
two groups without overlap. If one group of
outcomes is event A, the other group is called the
complement of A and is written A or A′ . Since A
and A together include all possible results, the
sum of Pr [A] and Pr [ A ] must be 1. If Pr [ A ] is
more easily calculated than Pr [A], the best
approach to calculating Pr [A] may be by first
calculating Pr [ A ].
EXAMPLE #1.
A sample of four electronic components is taken from
the output of a production line. The probabilities of the
various outcomes are calculated to be: Pr [0
defectives] = 0.6561, Pr [1 defective] = 0.2916, Pr [2
defectives] = 0.0486, Pr [3 defectives] = 0.0036, Pr [4
defectives] = 0.0001. What is the probability of at least
one defective?
Answer: It would be perfectly correct to calculate as
follows:
Pr [at least one defective]
= Pr [1 defective] + Pr [2 defectives] +
Pr [3 defectives] + Pr [4 defectives]
= 0.2916 + 0.0486 + 0.0036 + 0.0001
= 0.3439. But it is easier to calculate instead:
Pr [at least one defective]
= 1 – Pr [0 defectives]
= 1 – 0.6561
= 0.3439 or 0.344.
b. - If the events are not mutually exclusive, there can
be overlap between them. This can be visualized
using a Venn diagram. The probability of overlap
must be subtracted from the sum of probabilities
of the separate events (i.e., we must not count the
same area on the Venn diagram twice).
The circle marked “A” represents the probability
(or frequency) of event “A”, the circle marked “B”
represents the probability (or frequency) of event
“B”, and the whole rectangle represents all
possibilities, so a probability of one or the total
frequency. The set consisting of all possible
outcomes of a particular experiment is called the
sample space of that experiment. Thus, the
rectangle on the Venn diagram corresponds to the
sample space. An event, such as A or B, is any
subset of a sample space. In solving a problem we
must be very clear just what total group of events
2|MODULE 2
Prepared by:
MARY ABEGAIL P. CADANG
Emergency Instructor
NUEVA VIZCAYA STATE UNIVERSITY
Bayombong, Nueva Vizcaya
College of Engineering
Chapter 2: Probability
we are concerned with—that is, just what is the
relevant sample space.
Set notation is useful:
Pr [A ∪ B) = Pr [occurrence of A or B or both], the
union of the two events
A and B.
Pr [A ∩ B) = Pr [occurrence of both A and B], the
intersection of events
A and B.
Then in Figure 2.2, the intersection A ∩ B
represents the overlap between events
A and B.
Figure 2.3 Shows Venn diagrams representing
intersection, union, and complement. The crosshatched area of Figure 2.3(a) represents event A.
The cross-hatched area on Figure 2.3(b) shows
the intersection of events A and B. The union of
events A and B is shown on part (c) of the
diagram. The cross-hatched area of part (d)
represents the complement of event A.
Answer:
Pr [queen] = 4/52.
Pr [heart] = 13/52.
Pr [queen ∩ heart] = 1/52.
These quantities are shown on the Venn
diagram of Figure 2.4
Then Pr [queen ∪ heart]
= Pr [queen] + Pr [heart] – Pr [queen ∩ heart]
=4/52 + 13/52 – 1/52 = 16/52
The simple addition law, sometimes equation 2.1,
and the definitions of intersections and unions can
be used with Venn diagrams to solve problems
involving three events with both single and double
overlaps. This usually requires us to apply some
form of the addition law several times. Often an
appropriate approach is to find the frequency or
probability corresponding to a series of simple
areas on the diagram, each one representing
either a part of only one event without overlap
(such as A ∩ B ∩ C ) or only a clearly defined
overlap (such as A ∩ B ∩ C ).
Multiplication Rule
a. The basic idea for calculating the number of
choices can be described as follows: Say
there are n1 possible results from one
operation. For each one of these, there are n2
possible results from a second operation.
Then there are (n1 × n2) possible outcomes
of the two operations together. In general, the
numbers of possible results are given by
products of the number of choices at each
step. Probabilities can be found by taking
ratios of possible results.
If the events being considered are not mutually
exclusive, and so there may be overlap between
them, the Addition Rule becomes
Pr [A ∪ B) = Pr [A] + Pr [B] – Pr [A ∩ B] (2.1)
In words, the probability of A or B or both is the
sum of the probabilities of A and of
B, less the probability of the overlap between A
and B. The overlap is the intersection between A
and B.
Example
If one card is drawn from a well-shuffled bridge
deck of 52 playing cards (13 of each suit), what is
the probability that the card is a queen or a heart?
Notice that a card can be both a queen and a
heart. Then a queen of hearts (or queen ∩ heart)
overlaps the two categories.
Example 2.7
In one case a byte is defined as a sequence
of 8 bits. Each bit can be either zero or one.
How many different bytes are possible?
Answer:
We have 2 choices for each bit and a
sequence of 8 bits. Then the number of
possible results is (2)8
= 256.
b. The simplest form of the Multiplication Rule
for probabilities is as follows: If the events are
independent, then the occurrence of one
event does not affect the probability of
occurrence of another event. In that case the
probability of occurrence of more than one
event together is the product of the
probabilities of the separate events. (This is
3|MODULE 2
Prepared by:
MARY ABEGAIL P. CADANG
Emergency Instructor
NUEVA VIZCAYA STATE UNIVERSITY
Bayombong, Nueva Vizcaya
College of Engineering
Chapter 2: Probability
consistent with the basic idea of counting
stated above.)
If A and B are two separate events that are
independent of one another, the probability of
occurrence of both A and B together is given
by: Pr [A ∩ B] = Pr [A] × Pr [B]
Example 2.8
If a player throws two fair dice, the probability
of a double one (one on the first die and one
on the second die) is (1/6) (1/6) = 1/36. These
events are independent because the result
from one die has no effect at all on the result
from the other die. (Note that “die” is the
singular word, and “dice” is plural.)
c. If the events are not independent, one event
affects the probability for the other event. In
this case conditional probability must be
used. The conditional probability of B given
that A occurs, or on condition that A occurs,
is written Pr [B | A].
This is read as the probability of B given A, or
the probability of B on condition that A occurs.
Conditional probability can be found by
considering only those events which meet the
condition, which in this case is that A occurs.
Among these events, the probability that B
occurs is given by the conditional probability,
Pr [B | A]. In the reduced sample space
consisting of outcomes for which A occurs,
the probability of event B is Pr [B | A]. The
probabilities calculated in parts (d) and (e) of
Example 2.6 were conditional probabilities.
The multiplication rule for the occurrence of
both A and B together when they are not
independent is the product of the probability
of one event and the conditional probability of
the other:
Pr [A ∩ B] = Pr [A] × Pr [B | A] = Pr [B] × Pr
[A | B]
This implies that conditional probability can
be obtained by
These relations are often very useful.
Example
Four of the light bulbs in a box of ten bulbs
are burnt out or otherwise defective. If two
bulbs are selected at random without
replacement and tested, (i) what is the
probability that exactly one defective bulb is
found? (ii) What is the probability that exactly
two defective bulbs are found?
Answer: A tree diagram is very useful in
problems involving the multiplication rule.
Let us use the symbols D1 for a defective first
bulb, D2 for a defective second bulb,
G1 for a good first bulb, and G2 for a good
second bulb.
At the beginning the box contains
four bulbs which are defective and six which
are good. Then the probability that the first
bulb will be defective is 4/10 and the
probability that it will be good is 6/10. This is
shown in the partial tree diagram at left.
Probabilities for the second bulb vary,
depend-ng on what was the result for the first
bulb, and so are given by conditional
probabilities. These relations for the second
bulb are shown at right in Figure 2.7.
If the first bulb was defective, the box will then
contain three defective bulbs and six good
ones, so the conditional probability of
obtaining a defective bulb on the second draw
is 3/9, and the conditional probability of
obtaining a good bulb is 6/9.
If the first bulb was good the box will contain
four Figure 2.7: Second Bulb defective bulbs
and five good ones, so the conditional
probability of obtaining a defective bulb on the
second draw is 4/9, and the conditional
probability of obtaining a good bulb is 5/9.
Notice that these arguments hold only when
the bulbs are selected “without replacement”;
if the chosen bulbs had been replaced in the
box and mixed well before another bulb was
chosen, the relevant probabilities would be
different.
Now let us combine the separate
probabilities.
The probability of getting two defective bulbs
must be (4/10) (3/9) = 12/90, the probability of
getting a defective bulb on the first draw and
a good bulb on the second draw is
4|MODULE 2
Prepared by:
MARY ABEGAIL P. CADANG
Emergency Instructor
NUEVA VIZCAYA STATE UNIVERSITY
Bayombong, Nueva Vizcaya
College of Engineering
Chapter 2: Probability
(4/10)(6/9)=24/90, the probability of getting a
good bulb on the first draw and a defective
Bulb on the second draw is (6/10)(4/9)=24/90,
and the probability of getting two
good(6/10)(5/9)=30/90. In symbols we have:
Notice that both D1 ∩ G2 and G1 ∩ D2
correspond to obtaining 1 good bulb and 1
defective bulb.
The complete tree diagram is shown in Figure
2.8.
Notice that all the probabilities of events add up to one, as
they must:
Now we have to answer the specific questions which were
asked:
1. Pr [exactly one defective bulb is found]
= Pr [D1 ∩ G2] + Pr [G1 ∩ D2]
The first term corresponds to getting first a defective
bulb and then a good bulb, and the second term
corresponds to getting first a good bulb and then a
defective bulb.
2. Pr [exactly two defective bulbs are found]
= Pr [D1 ∩ D2] = 12/90= 0.133. There is only one
path which will give this result.
Notice that testing could continue until either all 4
defective bulbs or all 6 good bulbs are found.
Example
A fair six-sided die is tossed twice. What is the
probability that a five will occur at least once?
Answer: Note that this problem includes the
possibility of obtaining two fives. On any one toss, the
probability of a five is 1/6, and the probability of no
fives is 5/6. This problem will be solved in several
ways.
5|MODULE 2
Prepared by:
MARY ABEGAIL P. CADANG
Emergency Instructor
NUEVA VIZCAYA STATE UNIVERSITY
Bayombong, Nueva Vizcaya
College of Engineering
Chapter 2: Probability
6|MODULE 2
Prepared by:
MARY ABEGAIL P. CADANG
Emergency Instructor
NUEVA VIZCAYA STATE UNIVERSITY
Bayombong, Nueva Vizcaya
College of Engineering
Chapter 2: Probability
Permutations and Combinations
Permutations and combinations give us
quick, algebraic methods of counting. They are used in
probability problems for two purposes: to count the
number of equally likely possible results for the
classical approach to probability, and to count the
number of different arrangements of the same items to
give a multiplying factor.
a. Each separate arrangement of all or part of a set
of items is called a permutation. The number of
permutations is the number of different
arrangements in which items can be placed.
Notice that if the order of the items is changed, the
arrangement is different, so we have a different
permutation. Say we have a total of n items to be
arranged, and we can choose r of those items at a
time, where r ≤ n. The number of permutations of
n items chosen r at a time is written nPr. For
permutations we consider both the identity of the
items and their order.
and the remaining (n–n1) items are the same of a
different class, the number of permutations can be
𝒏!
shown to be
. The numerator, n!
𝒏𝟏!(𝒏 − 𝒏𝟏)!
would be the number of permutations of n
distinguishable items taken n at a time. But n1 of
these items are indistinguishable, so reducing the
1
number of permutations by a factor
, and
𝑛1!
another (n – n1) items are not distinguishable from
one another, so reducing the number of
𝟏
permutations by another factor
. If we
(𝒏 − 𝒏𝟏)!
have a total of n items, of which n1 are the same
of one class, n2 are the same of a second class,
and n3 are the same of a third class, such that n1
+ n2 + n3 = 1, the number of permutations is
𝒏!
. This could be extended to further
𝒏𝟏! 𝒏𝟐! 𝒏𝟑!
classes.
Example
An engineer in technical sales must visit plants in Vancouver,
Toronto, and Winnipeg. How many different sequences or
orders of visiting these three plants are possible?
Answer: The number of different sequences is equal to 3P3 =
3! = 6 different permutations. This can be verified by the
following tree diagram:
b. The calculation of permutations is modified if
some of the items cannot be distinguished from
one another. We speak of this as calculation of the
number of permutations into classes. We have
already seen that if n items are all different, the
number of permutations taken n at a time is n!
However, if some of them are indistinguishable
from one another, the number of possible
permutations is reduced. If n1 items are the same,
7|MODULE 2
Prepared by:
MARY ABEGAIL P. CADANG
Emergency Instructor
NUEVA VIZCAYA STATE UNIVERSITY
Bayombong, Nueva Vizcaya
College of Engineering
Chapter 2: Probability
8|MODULE 2
Prepared by:
MARY ABEGAIL P. CADANG
Emergency Instructor
NUEVA VIZCAYA STATE UNIVERSITY
Bayombong, Nueva Vizcaya
College of Engineering
Chapter 2: Probability
“…Have I not commanded you? Be strong and
courageous. Do not be frightened, and do not be
dismayed, for the Lord your God is with you wherever
you go.”
Joshua 1:9
The Lord our God be with us, as he was with our
fathers. May he not leave us or forsake us
1 Kings 8:57
O you, O Lord, I lift up my soul. 2 O my God, in you I
trust; let me not be put to shame; let not my enemies
exult over me. 3 Indeed, none who wait for you shall
be put to shame; they shall be ashamed who are
wantonly treacherous. 4 Make me to know your ways,
O Lord; teach me your paths. 5 Lead me in your truth
and teach me, for you are the God of my salvation; for
you I wait all the day long.
Psalms 25:1-5
9|MODULE 2
Prepared by:
MARY ABEGAIL P. CADANG
Emergency Instructor