Today, I will introduce the basic concepts
of probability theory. So, probability theory
had its origins in the games of chance in
early sixteenth and seventeenth century when
the owners of gambling houses in Europe - they
became interested to explore that whether
1 can find out the probabilities of various
events which take place during a gambling
game such as tossing of a die, rolling of
a coins, roulettes, etcetera.
So, they contacted some of the prominent mathematicians
of that time say, Pascal and Fermat and through
the correspondence between these mathematicians
the theory of probability started to develop.
One of the fundamental features of this probability
is that the phenomena which we are interested
in are random in nature. So, for example,
if you consider tossing of a die then, we
do not know whether which phase we are in
number 1 2, 3 etcetera will come, but in the
long run if you toss a enough number of times
then the proportion of the number of occurrences
of say one of them say, 6 will be 1 by 6 say
then it means that the probability or the
chance of appearance of 6 is 1 by 6.
Similarly, in the tossing of a coin we do
not know that at each trial whether we will
get a head or a tail; but if we toss a large
number of times then we know that nearly 50
percent of the times there will be head or
50 percent of the times there will be tail.
This long term behavior is known as statistical
regularity and that is what encourages us
to study the subject probability.
A similar kind of observation you can make
in the experiments which are connected with
the real life such as experiments in physics
experiments in genetics or virtually any phenomena
in real life. Consider for example, birth
of a offspring; now, suppose we consider human
beings then for each birth we do not know
whether a the child will be a boy or a girl.
But, in the long run it is very well known
what percentage of children will be boys and
what percentage will be girls.
An insurance company while promoting A new
policy would like to know how many of the
people will survive up to the age of maturity.
For example, if the policy matures at the
age of 60 then it would like to know the percentage
of the people in the target group who will
be surviving beyond the age 60 and therefore,
may get the benefits which are due to them.
Now, for individual person it is not possible
to tell whether you will die at the age of
60 or not, but in the whole population 1 can
tell the percentage of people dying before
60 or dying after the age of 60. A similar
kind of statistical regularity is observed
and it is used for weather predictions; the
prediction of the say, growth of crop, the
economic growth, the financial situation of
the country, etcetera.
Here what happens that in most of these cases
although the things may look that they are
pre-ordained or pre-designed, but pre-determined,
but actually there will be several conditions
which regulate the which regulate the occurrence
of the final phenomena and therefore, 1 can
treat them as random phenomena. Now, we will
introduce some of the basic concepts the first
is experiment the term which I have used repeatedly
just in the discussion. So, an experiment
is observing something happen or conducting
something under certain conditions which result
in some outcome. Let me explain this little
by way of a definition. So, consider say rainfall.
So now, in rainfall is a consequence of several
things finally, we observe that there is a
rainfall. So, there is a cloud formation;
there is some lean occurrence; there is a
humidity; there are various factors which
need to there is a rainfall or there it is
cloudy or it may not rain at all; it may rain
in some other region; now observing of this
weather is a experiment.
Similarly, suppose we consider how much crop
of a particular or how much yield of a particular
crops say wheat, is there in a particular
in a particular field. Now, this is dependent
upon the seeds, the plot of the land where
it is there the irrigation procedure and other
mechanical procedure which are used for farming.
So, the entire process although we are not
conducting, but it is happening and it is
a random experiment; the outcome is recorded
as the final yield - yield of the crop.
You must be very much familiar with lot of
experiments which are done in physical chemical
and biological sciences. For example, we have
various experiments in chemistry where certain
chemicals are mixed and they result in some
compound been made.
So, broadly speaking, we segregate the experiment
into 2 types of experiments: 1 is deterministic
experiment. In the deterministic experiments
under certain conditions, if an experiment
is conducted it results in a known outcome.
So, many of the classroom experiments in physics,
chemistry, biology, etcetera, they are like
this. For example, if I have 2 molecules of
hydrogen and a molecule of oxygen then we
know the outcome is water.
Suppose we consider say a, we put water in
a vessel and heat it then the temperature
reaches 100 degree celsius and the atmospheric
pressure say 700 mg then, the outcome is that
the water will boil. So, these experiments
are called deterministic experiments. However,
we are not concerned with these experiments
in the subject of probability.
We are concerned with the experiments which
are called random experiments. in the random
experiments although we may fix the conditions
under which the trials are conducted or the
experiment is conducted, but the outcome is
still uncertain consider for example, tossing
of a coin. So, although we may fix lot of
conditions such as, what kind of coin we are
having how to hold it when we are tossing,
but even then, when we toss the coin and it
falls the outcome is uncertain. It may be
head or tail or in extreme situation we may
consider that it stands on its side also.
Consider say tossing of a die; so, again the
conditions are similar. We may fix the die
in various ways, but when we toss it and if
it is a, if we are really tossing it then
after falling which face will be upwards is
not known. Suppose, we consider drawing of
a card from a deck of cards; suppose we consider
say birth of a child; suppose we consider
age at death of a person; suppose we consider
say amount of rainfall during a monsoon season
in a geographical area; suppose we want to
consider yield of a crop of a certain food
grains in a state; suppose we consider the
time taken to complete a 100 meter speed by
an athlete. All of these phenomena here - the
conditions of the experiment are fixed. For
example, when we look at say the time taken
to complete a 100 meter speed by an athlete
then the conditions are fixed. For example,
the ground is fixed the starting time is fixed
the athlete is in the perfect condition the
person who will direct the race is prepared
the person who will record the time is prepared.
However, how much actual time the sprinter
will take to complete the 100 meter race,
will always be uncertain it may be 10 seconds,
it may be 10 point 1 second, it may be 9 point
7 seconds etcetera.
Suppose we are considering say, a mechanical
instrument such as say, life of a bulb. So,
when we purchase a bulb from the market and
you light it then, it may work for 1 hour,
it may work for 10 hours it may work for 100
hours. So, although there all these bulbs
may be produced by the same company under
the same conditions, even then the actual
life of the bulb is not fixed; it can vary.
If we look at say working time of a say life
of a mechanical instrument is given; say for
example, a certain turbine or a certain engine.
So, although they are all produced by certain
standard process, but the actual life of that
instrument will not be, cannot be predicted
in advance. We may consider say... So, in
these cases all of these examples they relate
to certain fixed conditions for the conducting
of the experiment. However, the final outcome
is not known in advance.
So, all of these are known as random experiments
and in the subject of probability we are concerned
only with the discussion of the random experiments.
So, for example, if we have this type of events
are of interest, for example you consider
birth of a child or age or death of a person.
So now, these phenomena are extremely useful
in practical. For example, insurance companies
when they propagate a life insurance policy,
they are very much interested; that what premium
they have to fix. Now, how to decide about
the premium the company charges a premium
and in the case of unlikely case of the person
dying before the age of maturity he has to
be paid full benefits of the policy and plus
some assured sum. Whereas, if the person completes
the policy; that means, he does not circum
before the age of maturity then he pays the
premium till the immaturity and then he gets
certain benefits which are not that much as
much as he would have got if the person had
died before the age of maturity.
Therefore, the company has to estimate how
much premium it will be charging and how much
will be actually the cost to the company in
the event of the death of the person prematurely.
So, the age at death of a person in the target
group is to be estimated and therefore, we
will record we keep the records of the age
of the persons in that particular target group
for which the company is trying to sell the
insurance policy.
If you look at the amount of rainfall then
extremely important phenomena because there
are lot of policies of the government: the
policy - agricultural policy, the economic
policy, which are based on actual rainfall
which is going to be there in the country,
that yield of the crop makes government to
decide about how much food grains they are
going to purchase from the farmers; how much
they are going to store; what should be the
price which should be given to the farmers;
what should be the price for the market.
So, all of these events the experiments although
they do not look random beforehand; however,
outcomes are not known and therefore, they
are random and in the subject of probability
and statistics we do study this type of phenomena.
So, now let me introduce certain basic terminologies
of the random experiments. The first of this
is the concept of a sample space. So, the
set of all possible outcomes of a random experiment
is called a sample space; the usual notations
we will use either capital omega or capital
S etcetera, to denote the sample space.
So, let us consider some examples suppose
we are considering tossing of a coin in the
tossing of a coin we may put the the outcomes
we may consider as head or tail and if we
denote by H the occurrence of head upwards
R by T the occurrence of tail then the sample
space can be described as H T.
If we consider tossing of a die then the sample
space we may describe as occurrence of the
base uppermost. So, the sample space will
be the set of the numbers 1, 2, 3, 4, 5, 6,
if we are considering drawing of a card from
a deck of cards. Now, in the deck of cards
there are 4 denominations: heart, club, spade
and the diamond and each card has a value
1 to 13. So, if we consider drawing of a card
then, it will consist of the set for example,
diamond: 1 to diamond 13; club: 1 to club
13; heart: 1 to heart 13; or a spade: 1 to
spade 13. So, the sample space consists of
52 points. If we look at, we may also observe
the sample space in a way like for example,
we may only record the color of the…
So, in that case we may describe it differently;
we may call this one as omega 1 and if I am
recording only the color then, it may be say
black or red or we may only record the denomination
that is whether it is a heart or whether it
is a diamond or it is a club or it is a spade.
So, the sample space will consist of only
4 points; this also shows that sample space
is not a unique thing. It shows that in a
given experiment what we are interested in
will decide that, what is a sample space?
If we are looking at the birth of a child
then it is vague statement or you can say
vague way of describing what is the random
experiment.
Now, we may record whether the child is a
male or a female child; we may record whether
the child born has is healthy or not healthy.
So, we may put it as healthy or unhealthy;
we may look at his body weight at the birth.
So, the body weight may be some number starting
from say 0 to say may be 10 pounds; it may
be the total life of the child. So, in that
case it may be something like say 0 to 100.
So, it depends upon that what is our actual
interest and we can write the sample space
accordingly. The death of a person, this may
be say 0 to say may be 120, keeping into account
that there are some people who live very long;
there this unit of the time is years.
Amount of a rainfall may be recorded in say
centimeters; yield of a crop may be recorded
in some metric tons; the time taking to complete
100 meter speed may be a time from say 9 to
say 11 seconds.
Suppose, I am competing an international field,
life of a bulb it is A number say which is
0 to infinity. Although, theoretically speaking
it is not infinity, but it can be a large
number; life of a mechanical instrument similarly
can be described.
Suppose, we are looking at the number of defective
items produced by a company; a particular
kind of items we are looking at. Suppose,
we are looking at certain bolts then what
will happen that, we will define the defectives
that if they do not conform to certain prescribed
standards of measurement. So, now the number
of defectives may be recorded in terms of
percentage. So, the percentage can be say
0 to 100 percent or it could be say proportion.
In that case, we may write the number as say
0 to 1. So, the sample space will again be
dependent upon the way we want to look at
it.
Next, we define what is an event an event
is any subset of the sample space now this
is a very broad definition and therefore,
any subset of the sample space qualifies to
be an event to be called an event as per as
the probability theory is concerned. So, let
us look at the experiments now and the examples
that we have already done.
In the first case, if we look at omega is
equal to H T, we may consider a subset as
consisting of only H. we may consider a subset
consisting of say only t. So, the set A denotes
that head has occurred the set b denotes that
t has occurred. So, these are events we may
consider say the set e in the case of tossing
of a die we may write 2, 4, 6; this means
occurrence of an even number. In the birth
of a child, suppose, we are looking at say
weight at birth in pounds and suppose we say
e is equal to 4 to 8 then, it means that the
birth weight of the child is between 4 to
8 pounds.
If we look at the amount of rainfall and in
centimeter during a particular monsoon season
and we may put say 50 to 75; that means, the
actual rainfall is between 50 centimeter to
75 centimeters in that geographical area during
that particular monsoon season. So, any subset
of the sample space can be considered as an
event. Now, we have various kinds of events;
for example, impossible event since every
subset is a subset of every event is a subset
of the sample space therefore, the empty set
phi that is a subset of omega therefore, this
will correspond to impossible event. Similarly,
we have sure event since omega itself is a
subset of omega. Therefore, this is denoting
the sure event for example, we may consider
tossing of a die and we say that seven occurs.
So, that will correspond to an impossible
event because 7 is not a subset of this and
that will correspond to phi as per as this
random experiment is concerned. Suppose, I
am looking at the a that death of a person
and we say 1000 years then it is an impossible
event.
Suppose we say, time taken to complete a 100
meter speed by an athlete and we may put the
time as say 5 seconds then, in the present
circumstances or present age this is an impossible
event. Similarly, if I put that the time taken
to complete a 100 meter speed by an athlete
for him to complete the race it is less than
1 minute then, this will be a sure event.
If we look at the life of a bulb and we say
it is a positive number then or A negative
number then it is a sure event. So, these
are 2, you can say 3 types of events which
are possible. However, there are various sectors;
it is operations like unions, intersections,
differences, complementation and therefore,
given any 2 events when we take their unions,
intersections, differences, complementation,
etcetera, they must correspond to certain
events and we can describe them in the form
of probabilistic explanation.
For example, union of 2 events: what does
union of 2 events imply? That, if I say A
and B are 2 events then A union B this will
mean occurrence of at least 1 of A and B;
that means, either A occurs or B occurs or
both occurs.
So, in set theoretic representation A and
B, A union B means that set of elements which
are either in A or in B or in both in probability
theory the event A union B will indicate that
at least 1 of A or B has occurred.
Similarly, we may consider union of A n events
A 1 A 2 A n then this will mean occurrence
of at least 1 a I i is equal to 1 to n. You
may even consider an union of infinite number
of events union A I - A I is equal to 1 to
infinity. This will mean occurrence of at
least 1 A I i is equal to 1 to infinity intersection
of 2 sets denotes the set of all those points
which are common to the 2 sets.
Now, in set theory that is the representation
in probability theory a intersection b will
mean the simultaneous occurrence A and B that
means, both event A and B are known to have
occurred similarly we can consider intersection
of n events A I i is equal to 1 to n that
is simultaneous occurrence A 1 A 2 A n; that
means, all of the events A 1 A 2 A n occurs
and in a similar way, intersection of I mean
countable collection of events A 1 A 2, etcetera.
Here, when we consider the unions or the intersections
certain basic properties are there for example,
we may have union of A I say I is equal to
1 to n is equal to omega; that means, all
the points of of omega are contained in 1
or the other of the a I’s such events are
called exhaustive events.
So, if a 1 union of a I is equal to omega
we call A 1 A 2 A n to be exhaustive events
here in place of n we may have n infinite
collection of events. Also similarly, if A
intersection B is equal to phi. Now, in the
set theory this means disjoint sets in probability
theory A intersection B is equal to phi denotes
that the events A and B cannot occur together.
They are called mutually exclusive events
A and B are called mutually exclusive event;
that means, happening of 1 of them exclude
the possibility of happening of the other
we may also consider something like this that
we have a collection A 1 A 2 etcetera of events
such that A I intersection A j is equal to
phi then I is not equal to j.
Then we say that A 1 A 2 etcetera are pair
wise disjoint or mutually exclusive events
given an event A then A compliment will denote
not happening of a 
in a similar way if I have event a and an
event b then a minus b will denote happening
of A, but not of B this is true because this
is equal to a intersection b compliment. That
means, simultaneous occurrence of event A
and B which means that A; A occurs and B does
not occurs. So, it is a simultaneous occurrence
of A and B compliment and it is translated
through occurrence of A, but not no occurrence
of B.
Based on this, we define we give the first
or you can say a primary definition of probability
it is called classical or mathematical definition
of probability this definition is due to laplace
which was available in on probability in taken
under trial.
Now, this is based on certain conditions I
have already introduced the concept of several
events. So, these are all subsets of the sample
space we may call events say A 1 A 2 A n to
be equally likely now this is some terminology
which is like circular in nature let me again
explain. So, I will say events A and B are
equally likely if A and B have the same chances
of appearing.
Now, till now we have not defined what is
a chance. So, this definition itself looks
circular, but anyway this is what has been
used in the classical definition of probability
suppose a random experiment has n possible
outcomes which are mutually exclusive exhaustive
and equally likely.
So that means, we are looking at the elementary
outcomes of the random experiment which are
collected in the sample space; that means,
the sample space has a total of n points which
is a finite number and naturally then when
we are describing all of them they are supposed
to be mutually exclusive and exhausted all
the possibilities.
So, I am making an assumption that they are
equally likely let m of these outcomes be
favorable to the happening of event a then
the probability of A is defined by probability
of A is equal to m by n.
So, this definition was given by Laplace because,
the of the earlier statement which I gave
that the probability theory has origin in
the games of chance such as tossing a coin
rolling of a die the numbers coming on a roller
wheel etcetera. So, all are drawing of a card
in a pack of cards. So, all of those experiments
had a peculiar thing that they had a finite
number of outcomes and assuming that the things
game is fair for example, if you toss a coin.
So, you assume that it is a fair coin. If
a die is thrown then you consider that it
is a fair die, etcetera. If it is a pack of
cards then you assume that it is a pack of
well shuffled cards fifty-two cards. So, these
assumptions seem to be valid there that is
exhaustive mutually exclusive and equally
likely and therefore, this definition was
given and this is the one which is used in
the calculation of the probability in the
classical examples.
So, when this advantages or you can say drawbacks
of this definition are that n need not be
finite for example, if we are considering
the number of trials measured for the first
success then we do not know when we will stop.
Suppose, we are considering life of a bulb,
suppose we are considering weight at birth
of a child, then all of these outcomes - the
collection of outcomes is an either countably
infinite or uncountably infinite set then,
second is the more crucial thing that you
are saying that the outcomes are equally likely.
Now, equally likely thing means that we are
knowing that the coin is fair or the die is
fair or the pack is well shuffled etcetera,
but that is bringing in a in a inherent understanding
of the definition of probability; whereas,
we are actually defining probability now.
So, it is a circular definition the definition
is circular in nature as it uses the term
equally likely which means outcomes with equal
probability. Similarly, in a given experiment
we may not able to express the outcomes as
mutually exclusive outcomes we may not be
able to exhaust all the possibilities of the
outcomes because now the total number of possibilities
may be really large to describe.
So, this definition though quite useful in
the beginning of the development of the subject
has its limitations. Later on, a more important
or you can say a more practical or a more
applicable definition was developed which
we call Relative frequency, technique of probability
which is based on actual conducting of the
experiment.
So, we can also call it empirical because
it is based on the actual observing of the
outcomes or statistical definition of probability.
The form is 
this definition is due to. Suppose, a random
experiment 
is conducted a large number of times 
independently 
under identical conditions let A n denote
the number of times the event a occurs in
n trials of the experiment then we define
the probability of the event A to B limit
of A n by n as n tends to infinity, provided
the limit exists let me give an example of
the actual application of this definition.
Let us consider say trial of conducting tossing
of a coin and we want to find out the probability
of that. So, this we are doing because we
do not know whether the coin is fair or not
in the classical definition we assume that
the coin is fair and then we try to find out
probability of etcetera.
But suppose I do not know that my coin is
fair I actually want to find out the probability
of it suppose an experiment of conducting
of of tossing of a coin results in 2 heads
followed by a tail now let us look at the
sequence A n by n here. So, the sequence A
n by n here if you make it in first trail
you have a head and you are interested in
the occurrence of head. So, A n by n is 1
by 1 and in the second trial again, there
is a head and therefore, the ratio A n by
n is 2 by 2.
So, we are looking at the event is occurrence
of head. So, if I look at the third trial
then in third trial tail has occurred. So,
the occurrence of head is 2 and the total
number of trials is three now if we just continue
with this direction we can write the sequence
A n by n like 1 by 1 2 by 2 2 by three then
it becomes three by 44 by 54 by 6 etcetera.
Now, in order to calculate the probability
of head we need the limit of this particular
sequence. So, in order to do that, let us
write a proper mathematical expression for
this we can write it as say 2 k minus 1 by
3 k minus 2 for k equal to 1 2 etcetera, we
may write it as 2 k by 3 k minus 1 for k equal
to 1 2 etcetera. If the number of the trial
is of the form 3 k then it is 2 by three 2
k by 3 k for k is equal to 1 2 and so on.
So, in each of the cases we are able to describe
the ratio A n by n and it is very obvious
now that if I take the limit of this as k
tends to infinity that is n tends to infinity
then the limit of this is 2 by 3 and therefore,
probability of occurrence of head is 2 by
3; that means, it is a biased point in favor
of head.
Now, the relative frequency definition seems
be the one of the most reasonable definitions
of probability. In the sense that, it is based
on the actual experience and that is what
the subject probability should be all about.
In terms of statistics for example, if we
are looking at the age at death of a person
then it should be based on the experiment
that is really how many people survive beyond
the age 60 beyond the age 70. That means,
it should be based on the recorded age at
death for the various persons.
If you are, if you want to find out the sex
ratio in a population then we should look
at how many children are born as male and
female. So, it should be based on the actual
data if you want to say something about rainfall
then you should know in the past 10 years
or in past 15 years, what is the pattern of
the rainfall during monsoon season.
Therefore, this relative frequency definition
seems to be most useful definition for calculation
of the probabilities. However, even this definition
has certain drawbacks for example, we are
making an assumption that the experiment is
observed, but there may be certain experiments
where we may not be able to look at the observations
the observations may be less in number or
they may be too complex too complicated too
costly.
For example, if we are looking at the failure
rate of launched satellites then it is not
an experiment which can be conducted every
day and if we are looking at from a country
like India, then the overall launches of the
satellites itself may be limited to a very
small number. And therefore, in order to find
out the probability of a successful launch
may be quite complicated.
Although, we are not saying that this type
of calculation of probability is impossible,
but it is difficult certainly if we want to
look at the full experiment; however, the
probability can be calculated using certain
rules of probability by splitting the entire
launch of satellite into various sections
or various segments and then we combine the
probabilities of those various things. So,
the actual conduct of the experiment are actual
observation of the experiment may not be possible
sometimes also there are certain experiments
which are destructive in nature for example,
you want to look at how many of the match
sticks kept in a match box are useful.
So, conducting of an experiment means that
we actually light a match stick and observe
whether it is doing good result; that means,
it burns - the full stick burns etcetera.
So, this kind of experiment is destructive
in nature because it will lead to the destruction
of the material itself and similarly, there
are various experiments conducted to test
the strength of the materials then it means
that you apply a certain pressure on the material.
The material which is used for making of certain
mechanical things such as say a car or an
engine or a train line. So, the strength of
the material is very important.
However, the testing requires that you put
some compressible force on that and observe
it to break at a certain force and then you
estimate that how much actual force will be
required if the actual force which is going
to be applied on that material, is less than
the breaking strength. Then we say that the
material is alright.
So however, such experiments are also costly
in nature and sometimes it is skipped in order
and it is replaced by certain other process
to measure that strength sometimes this probability
relative frequency definition may give a result
which may not be very intuitive for example,
if I say the probability of An event is 0
then you must think that the occurrence the
number of occurrences in experiment for that
particular event must be 0.
However, here since it is taken as the limit,
we may have say, root n as the number of happenings
in n trials and if I consider the limit of
this then this is 0. So, although the number
of occurrences is not 0, but the probability
of the event is 0.
Similarly, we may have n minus root n by n
which will convert to 1. So, every time the
event does not occur however, the probability
is 1 which indicates that this is sure event.
However, we may treat it as in the sense that
it means, that the number of occurrences is
negligible in comparison to the total number
of trials again this may lead to little bit
of confusion.
For example, I may consider n to the power
say 1 minus epsilon divided by n where epsilon
is a very small positive number or here again
this will go to 0. However, this will be not
negligible number. So, now, it is negligible
in the sense that if I look at the order then
in the terms of order, the order of n is more
than the order of n to the power 1 minus epsilon.
So, this type of justification can be given
for the relative frequency definition. Now,
we may now we consider a more definition of
the probability. The first 2 definitions which
I had given, they are based on the… you
can say basic - they were based on the basic
development of the subject of the probability
itself.
For example, the mathematical definition was
developed as a consequence of the interest
of some of the gambling houses to know probabilities
of certain events and they contracted the
mathematicians of that time. And, they looked
at the entire thing as a finite set of outcomes
which may be equally likely and they gave
the rules of the probability based on that.
The relative frequency definition or the statistical
definition is a similarly based on the experience.
So, when people are really looking from statistical
point up to certain events and they are not
necessarily the events of the type where you
have only head or tail or coin tossing etcetera,
then they looked at that. How many times the
actual event occurs during a certain number
of trials? As we have seen that the definitions
of drawbacks and therefore, they are not universally
applicable in order that a theory be consistent
or universally applicable; we need certain
axioms.
So, in 1933 the Russian mathematician, he
gave what is known as the arithmetic definition
of probability and this is based on the set
theoretic development which we gave in the
very beginning that is on algebra of sets.
So, now we describe what is axiomatic definition
of probability. What we have is that, given
a random experiment we have a sample space
and now certain subsets of omega are the events
which may be of interest to us we may not
be interested to consider all subsets of omega.
So, what we can consider is, we can consider
a sigma field of subsets of omega. So, a sigma
field of subsets of omega will consist of
certain events then their unions, their complementations
their intersections, their differences.
In other words, if we are considering certain
events then all the manipulations of those
events which are of interest to the experimenter
will be included in the set B and therefore,
this definition of sigma field is useful in
development of this definition or you can
say axiomatic definition. So, given a random
experiment here, considering a sample space
and b a sigma field of subsets of omega; that
means, in the terminology of probability theory
this is a set of events.
So, omega B is called a measurable space and
now we are interested to define the probability.
So, for every event which is included in B,
we must be able to define the probability
function.
So, the axiomatic definition of probability
it is due to let omega be a probability sub
be a measureable space, a set function P from
B to r r is the set of even numbers is said
to be a probability function. If it satisfies
the following three, assume I will call them
P 1 that is probability of A is greater than
or equal to 0 for all a subset of B, the second
assume is that the probability of the sample
space is equal to 1.
The third assumption is that, for any sequence
of pair wise disjoint subsets e I belonging
to B probability of union e I is equal to
sigma probability of E I i is equal to 1 to
infinity; this last axiom is known as the
countable additivity axiom.
The first axiom is known as non negativity
axiom and the second axiom is known as the
axiom of. So, we will continue from where
we look at the properties of the probability
function which is given by the definition
in the next class, thank you.
