3 Prob

The objective interpretation of probability
Some basic concepts

The set theory interpretation of probability
The subjective interpretation of probability
Lesson 6: Elements of probability theory
Le Thi Xuan Mai
The university of natural sciences
January 5, 2022
T.X.M. Le Descriptive statistics

Some basic concepts The classical interpretation of probability
The set theory interpretation of probability The relative frequency interpretation of probability
The objective interpretation of

probability

The probability theory allows to quantify the chance. We begin

with the example of flipping a coin. If a coin is fair, we expect that
the head and the tail are the same chance to occur.
The probability of an event is the likelihood or chance that the
event will occur. It is stated as a number from 0 to 1.
A probability of 0 means that the event cannot possibly occur
A probability of 1 means that the event is certain to occur.

Example 1. The probability of the event that a man can give birth
to a baby is equal to 0. The probability of the event that the sun
rises in the east is equal to 1.
There are four different methods for interpreting and calculating
probabilities:
the classical interpretation
the relative frequency interpretation
the set theory interpretation
the subjective interpretation.

I The classical interpretation was developed from studies of the

games of chance used in gambling: throwing dice, flipping coins,
picking cards, etc.
I It deals with idealized games in which every trial of an
experiment is done under uniform and perfect conditions.
I Such perfect games are always ”fair” in that all possible
outcomes are equally likely occur.
One such idealized game would be the rolling of a uniformly dense,
perfectly symmetrical die onto a flawless surface, using identical
hand motions with each trial. Under these conditions all six faces
of the die are equally likely to be upward at the end of the roll.
I In this interpretation, the probability is determined before the
experiment is attempted, classical probabilities are also called
a prior probabilities.
In the classical interpretation of probability, it is restricted to

idealized experiments where all possible outcomes are known in
advance and all are equally likely. That is probabilities are
determined before any experiments are done.
I There are many instances, these outcomes cannot be assumed
to be equally likely. In the relative frequency interpretation,
probabilities are determined from the results of previous
experiments.
I The relative frequency of the event, taken from previous data, is
considered to be an estimate of the probability of future
occurrences of the event.

Example 2. We have a coin that we think may be ”unfair”. The

two possible outcomes of a flip (head or tail) may be not equally
likely. To investigate this possibility, we flip the coin 100 times and
get 70 heads and 30 tails.
I The relative frequency of heads is 70/100 = 0.7. Then our
estimate of the probability of getting a head in future flips of this
coin is 0.7.
I It is only an estimate, an approximation because if we went on
to repeat the experiment 1000 times or 1000000 times, we may get
somewhat different.
I 100 trials is sufficient to confirm your suspicion that this coin is
unfair (this coin gives more heads than tails).

The law of large number
For n trial of an experiment, if nA is the number of times A

occurred in those trials, then the relative frequency nA /n will get
closer and closer to P(A) as n increases
nA
P(A) = lim . (1)
n→∞ n

Some basic concepts
In discussing the interpretations of probability, we use concepts of

experiments, trials, outcomes, and events.
In statistics, an experiment is an any process that yields a
measurement.
Example 3. Roll a six-side die and observe, when the die stops
rolling, the number of dots on the upward face.
Each identical repetition of an experiment is called a trial of

the experiment. Each result of a trial is called an outcome.
The set whose elements are all the possible outcomes of the
experiment is called a sample space (Ω).

Some basic concepts
If P(A) and P(A0 ) are the probability that event A will occur and
the probability that event A0 will not occur, then the odds in
favor of A (the event occurring) are defined as the ratio of P(A)
to P(A0 )
P(A) c
0
= . (2)
P(A ) d
This ratio have no factors in common.
Example 4. Two football teams play each other 10 times a year.
Over the past eight years, team A has won 48 times and team B
has won 32 times. Then the odds in favor of A are
P(A) 48/80 3
0
= = . (3)
P(A ) 32/80 2

Some basic concepts
Any specific outcome of an experiment can be classified in

different categories called events. In mathematics view, an
event is a subset of elements of sample space Ω. We say that
an event A occurs if the result of the experience belongs to A.
The sample space is the certain event.
In the die-rolling example, each roll of the six-sided die is a trial of
the experiment, and each trial yields one of six possible outcomes:
1, 2, 3, 4, 5,or 6 dots.
The sample space is S = {1, 2, 3, 4, 5, 6}. There are six
elementary events in the sample space:
e1 = {1}, e2 = {2}, e3 = {3}, e4 = {4}, e5 = {5}, e6 = {6}.
Let A be an event: Rolling an even number (2, 4, 6). A is an event.
Rolling a die, if you get 4, we say that the event A occurs.

Some basic concepts
The probability of event A (denoted by P(A)) is the ratio of the

number of possible outcomes favorable to A to the total number of
possible outcomes
number of outcomes favorable to A
P(A) = . (4)
the total number of possible outcomes
For the die-rolling experiment, if A is rolling an odd number, then
there is three favorable outcome out of a total of six, and
NA 3
P(A) = = = 0.5
N 6

Some basic concepts Axioms of probability
The set theory interpretation of probability Some properties of probability
The set theory interpretation of

probability

A probability function is defined as any mathematical function that

both assigns real numbers called probabilities to events in a sample
space and also satisfies the three axioms
Axiom I: For event A in Ω
P(A) ≥ 0 (5)
Axiom II: For sample space Ω
P(Ω) = 1 (6)
Axiom III: If events A and B in Ω are mutually exclusive

(A ∩ B = ∅), then
P(A ∪ B) = P(A) + P(B) (7)

Axiom I states that the probability of event A is always a

nonnegative real number, that P(A) is always greater of equal
to zero.
Axiom II states that one of the events in Ω must occur in
100%. It is because Ω is an certain event.
Axiom III states that the probability of the union of two
mutually exclusive events A and B is equal to the sum of their
separate probabilities.

For the empty event ∅ in Ω
P(∅) = 0. (8)
For event A in Ω
0 ≤ P(A) ≤ 1. (9)
For event A and its complement A0
P(A) + P(A0 ) = 1. (10)
For event A and B
P(A ∪ B) = P(A) + P(B) − P(A ∩ B). (11)
If events A1 , A2 , . . . , Ak in S are all mutually exclusive, then
P(A1 ∪ A2 ∪ . . . ∪ Ak ) = P(A1 ) + P(A2 .) + . . . + P(Ak ). (12)

If Ω contains n elementary events ei that each have a

probability P(ei ), then
n
X
P(ei ) = 1. (13)
i=1
If event A in Ω contains k elementary events ei , then

n
X
P(A) = P(ei ). (14)
i=1
If Ω contains N equally likely elementary events ei , then

1 1
P(ei ) = and P(A) = NA . (15)
N N

Some basic concepts
The subjective interpretation of

probability

Some basic concepts
I Probabilities determined with classical or relative frequency

probability functions are called objective probabilities.
I Objective probabilities are determined from purely objective
information: clear factual information about the likelihood of an
event, that has not been distorted by personal feelings or
prejudices.
I In many instances, it is impossible to know in advance all
possible, equally likely outcomes of an experiment or the
proportion of times an event has occurred in a long series of trials.
Sometimes, an experiment has never been done before will be
attempted only once.

Some basic concepts
I These instances require ”personal judgments” or ”educated

guesses”.
I In such instances, a numerical value is assigned to a personal
degree of belief (or degree of certainty) in the likelihood of the
event.
I Such measures of degree of belief are subjective, and thus this
version of probability is called the subjective interpretation of
probability.

Some basic concepts
Example 5 A business manager is about to introduce a new

product into a market. To determine the probability of success for
the product, they evaluate available information (e.g successes or
failures of similar products, your previous experiences when
introducing new products,etc), consult there feeling and intuitions,
and then put it all together into this subjective probability value.
By this mental integration, they have decided that the probability
of success is 0.8, that is the product is four times more likely to
success than to fail.

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The objective interpretation of probability

Some basic concepts

Lesson 6: Elements of probability theory

Le Thi Xuan Mai

The university of natural sciences

T.X.M. Le Descriptive statistics

The objective interpretation of

T.X.M. Le Descriptive statistics

The probability theory allows to quantify the chance. We begin

T.X.M. Le Descriptive statistics

T.X.M. Le Descriptive statistics

I The classical interpretation was developed from studies of the

In the classical interpretation of probability, it is restricted to

T.X.M. Le Descriptive statistics

Example 2. We have a coin that we think may be ”unfair”. The

T.X.M. Le Descriptive statistics

The law of large number

For n trial of an experiment, if nA is the number of times A

T.X.M. Le Descriptive statistics

In discussing the interpretations of probability, we use concepts of

Each identical repetition of an experiment is called a trial of

T.X.M. Le Descriptive statistics

T.X.M. Le Descriptive statistics

Any specific outcome of an experiment can be classified in

T.X.M. Le Descriptive statistics

The probability of event A (denoted by P(A)) is the ratio of the

T.X.M. Le Descriptive statistics

The set theory interpretation of

T.X.M. Le Descriptive statistics

A probability function is defined as any mathematical function that

Axiom II: For sample space Ω

Axiom III: If events A and B in Ω are mutually exclusive

P(A ∪ B) = P(A) + P(B) (7)

T.X.M. Le Descriptive statistics

Axiom I states that the probability of event A is always a

T.X.M. Le Descriptive statistics

For the empty event ∅ in Ω

P(A) + P(A0 ) = 1. (10)

For event A and B

P(A ∪ B) = P(A) + P(B) − P(A ∩ B). (11)

If events A1 , A2 , . . . , Ak in S are all mutually exclusive, then

P(A1 ∪ A2 ∪ . . . ∪ Ak ) = P(A1 ) + P(A2 .) + . . . + P(Ak ). (12)

T.X.M. Le Descriptive statistics

If Ω contains n elementary events ei that each have a

If event A in Ω contains k elementary events ei , then

If Ω contains N equally likely elementary events ei , then

T.X.M. Le Descriptive statistics

The subjective interpretation of

T.X.M. Le Descriptive statistics

I Probabilities determined with classical or relative frequency

T.X.M. Le Descriptive statistics

I These instances require ”personal judgments” or ”educated

T.X.M. Le Descriptive statistics

Example 5 A business manager is about to introduce a new

T.X.M. Le Descriptive statistics

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