CHAPTER 3.pdf ELEMENTARY PROBABILITY FOR STUDENTS

Deterministic and non-deterministic models
3.2 Review of set theory: sets, union, intersection,
complementation, De Morgan’s rules
3.3 Random experiments, sample space and events
3.4 Finite sample spaces and equally likely outcomes
3.5 Counting techniques
3.6 Definitions of probability
3.7 Derived theorems of probability

1. CHAPTER THREE
ELEMENTARY PROBABILITY

2. Outline
3.1 Deterministic and non-deterministic models
3.2 Review of set theory: sets, union, intersection,
complementation, De Morgan’s rules
3.3 Random experiments, sample space and events
3.4 Finite sample spaces and equally likely outcomes
3.5 Counting techniques
3.6 Definitions of probability
3.7 Derived theorems of probability

3. 1.1. Deterministic and non-deterministic
models
•Models are integral parts of both social and natural
sciences. In both cases we construct or fit models to
represent the interrelationship between two or more
variables.
•Particularly, in the fields like Statistics and
Economics models are fitted for the sake of
forecasting.
•It is possible to classify models in to different
groups based on varied attributes or criteria.
•Based on the type of experiment for which we fit the
model, we classify model as Deterministic and
Nondeterministic models.

4. Experiment
•Experiment is any activity (process or action) that
we intended to do under certain condition to obtain a
well defined results, usually called the outcome of
an experiment.
•The possible results of an experiment may be one or
more. Based on the number of possible results, we
may classify an experiment as Deterministic and
Non-deterministic experiment.
•Steps involved in an Experiment:
–Input Equipments, material, input data etc.
–Action to be performed
–Output list of all results of the experiment

5. Deterministic Experiments
•A precisely deterministic input yields a precisely
deterministic output. This is an experiment for which the
outcomes can be predicted in advance and is known prior
to its conduct.
•For this type of experiment we have only one possible
result (certain and unique).
•The result of an experiment is assumed to be dependent on
the condition under which an experiment is performed.
•A mathematical form of equations to be defined on this
experiment is called a deterministic model.
•Ex: Distance covered by a car traveling at a constant
speed; An experiment conducted to verify the Newton
Laws of Motion: F = ma; an experiment conducted to
determine the economic law of demand: Q
t =a+bP
t where
Q is a quantity demand, P is the price and t is a time; etc…

6. Non-deterministic Experiments
•Even exact knowledge of input and action does not allow exact
prediction of outcome.
•This is an experiment for which the outcome of a given trial
cannot be predicted in advance prior to its conduct.
•We also call this experiment as unpredictable or probabilistic or
stochastic or random experiment.
•Usually the result of this experiment is subjected to chance and
is possibly more than one.
•Ex:
–Tossing a coin;
–Throwing a dice;
–Life of an electric bulb;
–Number of road accidents in a day at Addis Ababa;
–Queue size at a railway reservation counter; etc.
•In probability theory we are mainly concerned with the random
experiments.

7. Non-deterministic Experiments cont’
•In this experiment, whatever the condition under which an
experiment is performed, one cannot tell with certainty
which outcome occurs at any particular execution of an
experiment though it is possible to list those outcomes.
•This types of experiments are characterized by the
following three properties:
I.The experiment is repeatable under identical conditions.
II.The outcome in any particular trial is variable, i.e., it
depends on some chance or random mechanism.
III.If the experiment is repeated a large number of times,
then some regularity becomes apparent in the outcomes
obtained. This regularity enables us to set some
mathematical form of equations called non-
deterministic model.

8. •Outcome: is the result of a single trial of a random experiment.
Example: in tossing a fair coin {T}, {H}
•Sample Space: is the Set of all possible outcomes of a probability
experiment
•Example 1: Rolling a die:
•Example 2: Tossing a coin once: .
•Example 3: Tossing a coin twice:
Event: is a subset of sample space. It is a statement about one or
more outcomes of a random experiment.
–They are denoted by capital letters.
Example: Getting an odd numbers in rolling a die.
Solution: Let “A” is an event of getting odd numbers. Then
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9. •Equally Likely Events
–Events which have the same chance of occurring.
•Roll a die, let A be observing a number less than 4 and B be observing a
number greater than 3.
–A and B are equally likely events
•Complement of an Event
–the complement of an event A means non-occurrence of A and contains
those points of the sample space which don’t belong to A.
•Example: Toss a fair die, let A be observing an odd number, then
A={1, 3, 5}
A
C
={2, 4, 6}
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10. •Elementary Event: an event having only a single element or
sample point.
•Mutually Exclusive (disjoint) Events: Two events which cannot
happen at the same time (no intersection).
•Independent Events: Two events are independent if the
occurrence of one does not affect the probability of the other
occurring.
•Dependent Events: Two events are dependent if the first event
affects the outcome or occurrence of the second event in a way the
probability is changed

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11. 1.2. Review of set theory
•Set: A set is a well-defined collection or list of objects.
•An object that belongs to a particular set is called an element.
•Sets are usually denoted by capital letters (A, B, C etc). On the other
hand, elements of sets are usually denoted by small letters (a, b, c etc).
Two sets are equal if they have exactly the same elements in them
A set that contains no elements is called a null set or an empty set
If every element in Set A is also in Set B, then Set A is a subset of Set B
•Examples: The set of students in a class; the set of even numbers; the
set of possible outcomes of an experiment; etc.
•Note: If X belongs to set A we write X ∈ A, and if X does not belong to
set A we write X ∉ A

12. Definition and Types of Sets
Universal set (U): Universal set is the collection of all objects
under consideration.
Example: The set of real numbers can be seen to be
universal set of numbers.
Empty set (φ or {} ): A set with no element is called empty set.
Example: If the universal set is the set of positive integers,
then getting a negative integer is impossible
Subset: A ⊆ B iff X ∈ A ⇒ X ∈ B for all X element in the
universal set.

Review of set theory
12

13. Set operations
Complement: For any set A, the complement of A denoted by A
/
, or A
c
or
Ā is given by:
{X ∈ U/ X∉A}
Note: U
′
= φ ; φ
′
= U ; ( A
′
)
′
= A

Union: Given two sets A and B, the union of A and B denoted by A∪ B is
the set of all elements, which belong to set A or B, or both.
A∪B ={ X: X ∈ A ∨ X∈ B}
Review of set theory
13

14. Set operations
Intersection: For any two sets A and B the intersection of A and B is
defined to be the set of all elements that occur in both set A and also set B.
Symbolically, we write
A ∩ B = { x U | x  A and x  B }




Set difference: elements in A but not in B
A - B = { x | x  A and x  B }
A - B = A ∩ B
c
 Important!


Review of set theory
14

15. Properties of set operations
I.Commutative law
A∪ B = B∪ A, and
A ∩ B = B ∩ A
II.Associative law
A∪ (B ∪ C) = (A ∪ B) ∪ C, and
A ∩ (B ∩ C) = (A ∩ B) ∩ C
III.Distributive law
A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C),
A ∩ (B ∪ C) = (A ∩B) ∪ (A ∩ C),
Review of set theory
15

16. Properties of set operations…
V. Demorgan’s law
•(A∪ B)
c
= A
c
∩ B
c
and (A ∩B)
c
= A
c
∪ B
c

Remark
•A ∩ B = B - (A
c
∩ B) or
•A ∩ B = A - (A ∩ B
c
)

Review of set theory
16

17. Definition of terms (Probability experiment)
There are different types of sample space:-
Finite sample space
Tossing a coin. S={Heads, Tails}
Throwing a die. S={1, 2, 3, 4, 5, 6}
Countably infinite sample space
S={0, 1, 2, 3, …}
Uncountable sample space
S={t: t>5}

Deterministic and Non-Deterministic Models
17

18. Definition of terms (Probability experiment)
Event – any subset of basic outcomes from the sample space.
Mutually Exclusive events: cannot occur simultaneously (together)
A ∩ B = φ
Independent Events: Two events are said to be independent if the
occurrence of one does not affect probability of the other occurring.
Dependent Events: Two events are dependent if the first event affects the
outcome or occurrence of the second event in a way the probability is
changed.

Deterministic and Non-Deterministic Models
18

19. Combination of events
Union: The event A∪ B occurs if either A or B or both occur.
Intersection: The event A ∩ B occurs if both A and B occur
Complement of an Event: the complement of an event A means
nonoccurrence of A
Deterministic and Non-Deterministic Models
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20. Exercise 1: A person is selected at random from a population of a given
town
A: be the event that the person is male
B: be the event that the person is under 30 years
C: be the event that the person speaks foreign language
Describe the following events symbolically
a)A male under 30 years who does not speak foreign language.
b)A female who is either under 30 or speaks foreign language.
c)A person who is either under 30 or female but not both.
d)Male who is either under 30 or speaks foreign language but not both.
Deterministic and Non-Deterministic Models
20

21. Solution 1
a)A male under 30 years who does not speak foreign language.
(A ∩ B) ∩ C
′

b)A female who is either under 30 or speaks foreign language.
A
′
∩ (B ∪ C)
c)A person who is either under 30 or female but not both.
(B ∪ A
′
) ∩ (B ∩ A)
′

d)Male who is either under 30 or speaks foreign language but not both.
A ∩ [(B ∪ C) ∩ (B ∩ C)
′
]
Deterministic and Non-Deterministic Models
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22. Counting Techniques
•In order to calculate probabilities, we have to know
–The number of elements of an event
–The number of elements of the sample space.
•That is in order to judge what is probable, we have to know what
is possible.
•In order to determine the number of outcomes, one can use several
rules of counting.
– The Addition rule
–The multiplication rule
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− The Permutation rule
− The Combination rule

23. 1. Addition Principle (Rule)
•If a task can be accomplished by "k distinct" procedures
where the i
th
procedures has “n
i” alternatives , then the
total number of ways of accomplishing the task is:


Example: There are two transportation means from city A to
city B, either using bus transportation or train transportation.
There are 3 buses and 2 trains. How many ways of
transportation is there from city A to city B?
Solution: A person can take any of 5 means of transportation
from city A to B.

24. –If an operation consists of k steps and
•the 1
st
step can be performed in n
1 ways,
•the 2
nd
step can be performed in n
2 ways (regardless of how the
1
st
step was performed) ,


•The k
th
step can be performed in n
k ways (regardless of how the
preceding steps were performed) ,
then the entire operation can be performed in
n
1 ∙ n
2 ∙… ∙ n
k ways.
Example: If we have 6 different shirts, 4 different under wears, 5
different pairs of socks and 3 different pairs of shoes, how many
different outfits could we wear? (Ans: 360)
Exercise: How many 7-character license plates are possible if the
first three characters must be letters, the last four must be digits 0-9,
and repeated characters are allowed?
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Multiplication Rule
…

25. –A permutation is an arrangement of "n distinct" objects in a
specific order.
–The number of ways of selecting r distinct objects from n distinct
objects and rearranging those r objects is given by the formula


–The number of permutations of n distinct objects taken all together
is n! or

–The number of permutations of n objects in which k1
are alike (the
same), k2
are alike, ... etc, then the total number of arrangements is



25 )!(
!
rn
n
P
rn


Permutation Rule

26. Example 1: In how many ways can the letters A,B and C be arranged
taken two at a time.
Example 2: a) In how many ways can 3 students be arranged in
rows of 3 chairs?
b)In how many ways can a student arrange his/her 4 different
books on a shelf?
Example 3: If 2 different mathematics books, 3 different statistics
books and 2 different Chemistry books are to be arranged in a shelf,
then how many different arrangements are possible if:
a)The books in each particular subject must "stand all together".
b) Only the Mathematics books must stand all together.
c)There is no restriction.
Example 4: How many different permutations of n objects can be
made from the letters in the word MISSISSIPPI.





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27. Combination Rule
–Combination is a selection of n distinct objects without regard to
order.
–It is used when the order of arrangement is not important, as in the
selection process.
–The number of combinations of r objects selected from n objects is
denoted by



–Example: A committee of two people must be chosen from a group
of five people. How many different committees can be formed?
(Ans: 10 ways)
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28. Example 1: Given the letters A,B,C & D. List the number of
permutations & combinations for selecting two letters.



Example 2: Out of 30 male students and 20 female students in
Statistics department, a committee consists of 3 male students and 2
female students is to be formed. In how many ways can this be done
if:
a)any male students and any female students can be included (all
students are eligible). (Ans= )
b)One particular female must be a member. (Ans= )
c)Two particular male students cannot be member for some reasons.
(Ans= )

29. Some remarks
1.When we select r objects from n distinct objects,
we have (n - r) objects unselected and hence
there are as many ways of selecting r from n as
there are not selected n - r objects.
•Thus

2.Binomial Coefficients
•The quantity is also known as a binomial
coefficient because it is the coefficient of the term
a
n-r
b
r
in the expansion of the expression (a+ b)
n
.

30. Binomial Theorem
•For any two real numbers a and b and any
positive integer n we have


Special cases

31. Basic approaches to probability
–There are four different conceptual approaches to the study of
probability theory. These are:
•The Classical Approach.
•The Frequents (Empirical) Approach.
•The Axiomatic Approach.
•The Subjective Approach.
1. The Classical Approach
This approach is used when all outcomes are equally likely. Total
number of outcome is finite, say n.

32. Example 1: When a single die is rolled, then what is the probability
of getting an odd numbers?
Solution: let A- be an event that getting an odd numbers in rolling a
die. Then



Example 2: A box of 80 candles consists of 30 defective and 50 non
defective candles. If 10 of these candles are selected at random
without replacement, what is the probability that:
a)All will be defective?
b) 6 will be non defective?
c)All will be non defective?
Exercise: From a group of 5 men and 7 women, it is required to form
a committee of 5 persons. If the selection is made randomly, then
what is the probability that 2 men and 3 women will be in the
committee?
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33. 2. The Frequents Approach (Empirical Probability)
•This is based on the relative frequencies of outcomes belonging to
an event.
•In a given frequency distribution, the probability of an event A
being in a given class is:



Example 2: In a sample of 50 people, 22 had type "A", 5 had type
"B", 2 had type "AB" and 21 had type "O" blood. Find the
probability that a person has blood type "O"?
Solution: Let A- be the event that a person has blood type "O". Then

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34. 3. Axiomatic Approach
–Let E be a random experiment and S be a sample space associated
with E. With each event A a real number called the probability of A
(P(A)) satisfies the following properties called axioms of
probability or postulates of probability


1)P(A)≥0
2)P(S)=1
–P(A
c
)=1- P(A)
–P(Ø) =0
–P(AnB
c
) = P(A)-P(AnB)
3)If A and B are mutually exclusive events, then
P(AUB) = P(A) + P(B)
–If If A1, A2, A3 ... is a finite or infinite sequence of mutually
exclusive events of S, then
P(A
1 u A
2 u A
3 u ...) = P( A
1) + P( A
2) + P( A
3) + ...=
34   )(
iAP

35. Derived Probabilities
P(φ)=0 for any sample space S
P(A′)=1-P(A)
P(A U B)=P(A)+P(B) - P(A ∩B)
For any two events say, A and B, the probability that exactly one of the
events A or B occurs but not both is :
P(A U B)=P(A)+P(B) - 2P(A ∩B)


Approaches to Measuring Probability
35

36. Example: Sixty percent of the families in a certain community own
their own car, thirty percent own their own home, and twenty percent
own both their own car and their own home. If a family is randomly
chosen,
a)what is the probability that this family do not have a car?
b)what is the probability that this family owns a car or a house?
c)what is the probability that this family owns a car or a house
but not both?
Solution: Let A represents that the family owns a car and B
represents that the family owns a house. P(A)=0.6, P(B)=0.3, and
P(A n B)=0.2.
a)P(A
c
)=0.4
b)P(AUB) =0.7
c)P((AnB
c
)U(A
c
nB))=0.5

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37. Subjective Approach
–Subjective probability is a prediction that is based on an
individual's personal judgment, not on mathematical calculations.
–Subjective probabilities, like the name suggests, are probabilities
that come from an individual's personal judgment of an event
happening.
–Subjective probability differ from person to person, and because
they are subjective, they can be based on a person's beliefs or
other factors.
–Used when no historical data.
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