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ah based on this we define we give a first
or you can say a primary definition of probability
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it is called classical or mathematical definition
of probability and
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this definition is due to laplace which are
available on his critize on probability in
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eighteen hundred and twelve now this is based
on certain conditions i have already introduced
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the concept of several events so these all
are subsets of the ah sample space and we
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make all ah event say a one a two a n to be
equally likely now this is some terminology
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which is like circular in nature
let me again explain so i will say events
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a and b are equally likely if a and b has
the same chances of now till now we have not
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defined what is the chance so this definition
it's look circular but anyway this is what
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has been used in the classical definition
of probability suppose a random experiment
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has n possible outcomes which are mutually
exclusive exhaustive and equally likely so
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that means we are looking at the elementary
outcomes of the random experiment which are
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colour collected in the sample experiment
that means the sample space as the total of
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n points which is the finite number and ah
naturally then when we are describing all
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of them they are suppose to be mutually exclusive
and we have exhausted all the possibilities
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so i am making an assumption that they are
equally likely let m of these outcomes be
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favorable to the happening of event a then
the probability of a is defined by probability
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of a is equal to m by n so ah this definition
was given by laplace because the of the earlier
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statements which are that the ah probability
theory has the origin in the given sub chance
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such as a tossing of a coin throwing of a
die ah the numbers coming on a rayleigh field
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etcetera
so all of are driving up a card in a pack
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of cards so all of those experiments had a
peculiar thing that they had a finite number
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of outcomes and assuming that the things game
is fair for example if you toss the coins
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so you assume that it is a pa[ir]- coin if
your die is thrown then you can see whether
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its a pair die etcetera you could say a pack
of card then you assume that it's a pack of
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well shuffled cards ah fifty two cards so
these assumptions seem to be valid there that
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is exhaustive mutually exclusive and equally
likely and therefore these definitions was
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given and this is the one which is used in
the calculation of the probability in the
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classical examples
so main disadvantages or you can say ah drawbacks
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of these definition are that n need not be
finite for example if you are considering
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the number of trails needed for the first
success then we do not know when we will start
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suppose you are considering light of a bulb
suppose you are considering weight of birth
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of a child then all are these outcomes the
collection of outcomes those are either countedly
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infinite or uncountedly infinite set
then second is the more crucial thing that
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we are saying that the outcomes are equally
likely now equally likely thing means that
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we are knowing that ah coin is sphere or the
die is sphere or the packing well shuffled
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etcetera
but that is binding energy in inherent understanding
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of the definition of probability whereas we
are actually defining probability now so it's
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a circular definition the definition is circular
in nature as it uses the term
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equally likely which means outcomes with equal
probability similarly in a given experiment
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we may not be able to express the outcomes
as mutually exclusively outcomes we may not
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be able to exhaust all the possibilities of
the outcomes because the now total number
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of possibilities may be a really large to
describe
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so this definition though quite useful in
the beginning of the development of the subject
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as it's limitations ah later on ah in more
important or you can say a more practical
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or a more applicable definition was developed
which we call relative frequency definition
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of probability which is based on actual conducting
of the experiment so we can also call it empirical
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because it is based on the actual observing
of the outcomes or a statistical definition
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of probability
in the form of this definition is due to von
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mises
suppose a random experiment
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is conducted a large number of times
independently under identical conditions let
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a n denote the number of times the event a
occurs in n trails of the experiment then
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we define the probability of the event a to
be limit of a n by n as n tends to infinity
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provided the limit exists ah
now let me give you an example of the actual
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application of ah these definition let us
consider say a trail of conducting a tossing
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of a coin and we want to find out the probability
of it so this we are doing because we don't
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know whether the coin is sphere or not in
the classical definition we assume that the
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coin is sphere and then you try to find out
probability of a etcetera
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but suppose i don't know what a coin is sphere
i actually want to find out the probability
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of it suppose an experiment of conducting
of ah tossing of a coin results in t h followed
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by n now let us look at the sequence a n by
n here so in the sequence a n by n here if
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you make it in first trail you have a head
and you are interested in the occurrence of
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head so a n by n is one by one in the second
trail again there is a head and therefore
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the ratio a n by n is two by two
so you are looking at the event a is occurrence
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of h so if i look at the third trail then
in third trail tail has occurred so the occurrence
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of head is two and the total number of trails
is three now if you just continuing this direction
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we can write the sequence a n by n like one
by one two by two two by three then it becomes
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three by four four by five four by six etcetera
and now in order to calculate the probability
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of head we need the limit of this particular
sequence
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so in order to do that let us write a proper
mathematical expression for this we can write
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it as say two k minus one by three k minus
two for k equal to one two etcetera we may
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write it as two k by three k minus one for
k equal to one two etcetera
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if the number of the trails is of the form
three k then it is two by three two k by three
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k for k is equal to one two and so on so in
each of the cases you are able to describe
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the ratio a n by n and it is very obvious
now that if i take the limit of this as a
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tends to infinity that is n tends to infinity
when the limit of this is two by three and
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therefore probability of occurrence of head
is two by three that means this is a bias
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point in favor of head
ah now the relative frequency definition seems
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to be the one of the most ah reasonable definitions
of ah probability in the sense that ah it
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is based on the actual experience and that
is what the subject ah probability should
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be all about in terms of a statistics for
example if you are looking at the ah age at
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the death of a person then it should be based
on the experiment that is really how many
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people survive beyond the age sixty beyond
the age seventy that means it should be based
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on the the correct age at death for the various
persons
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if you are if you want to find out the sex
ratio in ah population then we should look
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at how many children are born as male and
female so it should be based on the actual
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data
if you want to say a something about a rain
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fall then you should know in the past ten
years are in past fifteen years what is the
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pattern of the rainfall during monsoon season
therefore this relative frequency definition
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seems to be the most useful definition for
calculation of the probabilities however even
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these definition has certain drawbacks
for example we are making an assumption that
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the experiment is observed but there may be
certain experiments where we may not be able
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to look at the observations the observations
may be in less in number or may be too complex
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too complicated too costly for example if
you are looking at the failure rate of avalanched
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satellites then it's a not on experiment which
can be conducted every day and if we are looking
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on a country like india then the overall launches
of the satellite itself may be limited to
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a very small number and therefore in order
to find out the probability of a successful
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launch may be quite complicated
although we are not saying that this type
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of calculation of probability is impossible
but it is difficult certainly if we want to
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look at the full experiment however the probability
can be calculated using certain rules of probability
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by symmetrically splitting the entire launch
of satellite into various sections or various
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segments and then we combined the probabilities
of those various things
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so the actual conduct of the experiment are
actual observation of the experiment may not
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be possible sometimes also there are certain
experiments which are ah destructive in nature
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so for example we want to look up how many
of the match sticks kept in a match box are
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useful so conducting of an experiment mean
that we actually like a match stick and observe
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whether it is being good burn the holistic
burns etcetera so this kind of experiment
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is destructive in nature because it will total
destruction of the material itself and similarly
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there are various experiments conducted to
test the extent of the materials then it means
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then we apply a certain pressure on the material
the material which is used for making of certain
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mechanical things such as a car or an engine
or a train line
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so let's think of a material very important
how about the testing requires that we put
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some compressible force on that and observe
it to break at a certain force and then we
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estimate that how much actual force will be
required if the actual force ah which is going
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to be applied on that material is less than
that strength then we say that the material
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is all right so however such experiments are
also costly in nature and ah sometimes ah
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it is escaped in all the and it is replaced
by certain above process in a word that is
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strength
sometimes this ah probability ah relative
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frequency definition may give you a result
which may not be very intuitive for example
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if i say that probability of an event is zero
then we must feel that the occurrence the
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number of occurrences in experiment for that
particular event must be zero however real
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sense it is taken as the limit so we may have
say a root n was the number of atoms in n
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trials and if i consider the limit as this
when this is zero
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so although the number of occurrences is not
zero but the probability of the event is zero
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similarly we may have n minus root n by n
which will converge vn so every time the event
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does not occur however the probability is
one which one indicates that this is the shear
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event however in the in the sense that it
mean that the number of occurrences is negligible
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in comparison with the total number of trials
now again this will lead to little bit of
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confusion for example i may consider n to
the power say one minus epsilon divided by
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n where epsilon may be a very small positive
number now here again this will go to zero
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however this will not be a negligible number
so now it is negligible in the sense that
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if i look at the order then in the terms of
order the order of n is more than the order
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of n to the power one minus epsilon so justification
can be given for ah the relative frequency
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definition
now in a now we consider a more rigorous definition
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of the probability ah the first two definition
which i have given they are based on the ah
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you can say basic they were based on the basic
development of the subject of the probability
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itself for example in the mathematical definition
was developed as a consequence of the ah interest
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of ah some of the rambling houses to know
probabilities of certain events and they contracted
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the mathematicians of that time and ah they
looked at the entire thing has a finite set
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of outcomes which may be equally likely and
they of the a probability based on that
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the relative frequency definition are the
statistical definition as a similarly based
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on the experience so when people were really
looking on their statistical point of view
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certain events and they are not necessarily
the events of the type where we have only
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head or tail or coin tossing etcetera then
they looked at that how many times the actually
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event occurs though in certain number of trails
ah and we have seen that the definitions of
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drawbacks and therefore they are not been
eventually applicable in order that ah vary
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the consistent or any applicable we need certain
axioms
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so in nineteen thirty three the russian mathematician
kalmogorov he gave what is known as the axiomatic
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definition of probability and this is based
on ah the set theoretic development which
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we gave on the very beginning that is on algebra
source so now we describe what is axiomatic
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definition of probability ah what we have
is that given a random experiment we have
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a sample space and now certain subsets of
omega are the events which may be of interest
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to us we may not be interested to consider
all subsets of omega
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so what we can consider is we can consider
a sigma field of subsets of omega so a sigma
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field subsets of omega will consist of certain
events when they are unions the complementations
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their intersections their differences in other
words if we are considering certain events
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then all the manipulations of those events
which are of interest to the ah experimenter
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will be included in the set b and therefore
this definition of sigma field is useful in
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00:20:51,630 --> 00:20:55,649
development of this definition or events the
axiomatic definition
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so given a random experiment we are considering
a sample space and b is a sigma field of subsets
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of omega that means ah in the ah terminology
of probability theory this is a set of events
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so omega b is called a measurable space and
now we are interested to define the probability
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so for every event which is included in b
we must be able to define the probability
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function
so the axiomatic definition
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of probability it is due to kolomogorov let
omega b be a probability sorry be a measurable
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space
a set function p from b to r r is a set of
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real numbers is said to be a
probability function if it satisfies the following
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three axioms i will call them p one that is
probability of a a is greater than or equal
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to zero for all a subset of b second of event
which are the probability of the sample space
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is equal to one the third assume is that for
any sequence of pair wise disjoint subsets
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ei one into b probability of union ei is equal
to sigma probability of ei i is equal to one
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to infinity
this ah last axiom is known as the countable
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additivity axiom the ah first axiom is known
as the non negative axiom and the second axiom
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is known as the axima computeness so we will
continuing from where we look at the probability
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ah properties of the probability function
which is given by the kolmogorovas definition
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in the next class
thank you