today i will introduce the basic concepts
of probability theory so probability it origins
in the games of chance in ah early sixteenth
and seventeenth century when the owners of
gambling houses in europe they became interested
to explore that whether one can find out the
probability of various events which take place
during the gambling game such as tossing of
dies ah rolling of a ah coins ah roll at wheels
etcetera
so they contacted some of the prominent mathematicians
of that time say pascal and fermat and through
the correspondence between this mathematician
the theory of ah probabilities are get to
developed ah one of the fundamental features
of this ah probabilities that the phenomena
which we are interested in r random in nature
so for example if you consider tossing of
dies ah then we do not know whether which
face we are in number one two three 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 six will be one by six then there
it mean is that the probability are the chance
of appearance of six is one by six
similarly in the tossing of a coin we do know
that at each trail whether we will get a head
or a tail but if we toss a large number of
times then we know that nearly fifty percent
of the times there will be a head or fifty
percent of the time there will be tail ah
this long term behavior is known as a statistical
regularity and that is what encourages us
to a study the subset probability a similar
kind of ah observation you can make in the
experiments which are connected with the real
life such as experiments in physics ah experiments
in genetics or virtually any phenomena in
real life considered for example ah birth
of a
now suppose we considered 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 a insurance company while promoting
a new policy would like to know how many of
the improving survive up to the age of maturity
for example if the policy matured at the age
of sixty then it would like to know the percentage
of people in the target group who will be
surviving beyond the age sixty and therefore
may get the benefits which are due to them
now for individual person is not possible
to tell whether he will die at the age of
sixty or not but in the whole population on
can tell the percentage of people dying before
sixty or dying after the age of sixty a similar
kind of ah 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 a country
etcetera ah here what happens that a most
of these cases although the things may look
that they are pre or pre design but predetermined
but actually there will be several conditions
which regulate the which regulate the ah occurrence
of the final phenomena and therefore one can
treat them as random phenomena ah
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 wave definition ah so considered
say rain fall so now in rainfall is a consequence
of several things finally we observed that
there is a rain fall so there is a cloud formation
there is some ah leno occurrence there is
a humidity there are various factors which
we need to there is a rain fall or there its
cloudy or it may not rain at all it may rain
in some other reason
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
crop say weight is there in a particular ah
in a particular filed now this is dependent
upon the seeds the plot of the land where
it is the irrigation procedure and other mechanical
procedure which are used for forming so the
entire process although we are not conducting
but this happening and it is a random experiment
the outcome is recorded as the final yield
yield of the crop ah you must be very well
familiar with lot of experiments which are
done in physical chemical and biological sciences
for example we have various experiment in
chemistry were certain chemicals are mixed
and they result in some compound being made
so broadly speaking we segregate the experiment
into two types of a experiments one 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 class room experiments in physics
chemistry biology etcetera they are like this
for example if i have two molecules of hydrogen
and the molecule of oxygen then we know that
outcome is a water suppose we considered say
ah a we put a ah water in a vessel and heated
then the temperature reaches hundred degree
celsius and that pressure say seven hundred
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 subset of probability
we are concerned with the experiments which
are called random experiments 
in the random experiments although we made
fix the conditions under which the trails
are conducted or the experiments is conducted
but the outcome is still uncertain considered
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 you are tossing but even then
when we toss the coin and it calls the outcome
is uncertain it may be head or tail are in
extreme situation we may considered that is
coins on which side also considers a tossing
of a die so again the condition are similar
it may fix the die in various ways but when
we toss it and if it's a ah if we are really
tossing it then after falling in which face
will be the upwards is not known
suppose we consider drawing of a card from
a deck of cards suppose we considers a birth
of a child suppose we considered 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 grain in a state
suppose we consider the time taken to complete
a hundred meter sprint by an athlete 
all of this phenomena here the conditions
of the experiment are fixed
for example when we look at say the time taken
to completed hundred meters sprint by an athlete
then the conditions are fixed for example
the ground is fixed the starting time is fix
the athletic is in a perfect condition the
person who will directories is prepared the
person who will record the time is prepared
however how much actual time the spender will
take who compute the hundred meter race will
always be uncertain it may be ten seconds
it may be ten point one second it may be nine
point seven seconds etcetera suppose we are
considering say a mechanical instrument such
as a life of a bulb so when you purchase a
bulb from the market and you light it then
the it may work for one hour it may work for
ten hours it may work for hundred hours
so although ah 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 you look at say working
time of a say life of a mechanical instrument
so for example a certain turbine or a certain
engine so although they are all produced by
a certain a standard process but the actual
life of that instrument will not be cannot
be predicted in advance we may consider say
ah so in this cases all of these examples
they relate to certain fix conditions for
the ah conducting of the experiment however
the final outcome is not known in the advance
so all of these are known as ah random experiments
and in the subject of probability we are concerned
only with the discussion of the random experiments
so ah for example if we ah why it is ah type
of events are of interest for example you
considered ah birth of a child or age or death
of a person so now these phenomena are extremely
useful ah in particle ah 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
ah dyeing before the age of maturity he has
to be paid ah full benefits of the policy
and plus some ah assured sum whereas if the
person completes the policy ah that means
he does not circum before the age of maturity
then he face the premium till the maturity
and then he gets certain benefit which are
not that much as much as he would have got
the person had ah and 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 date of the person pre maturely
so the age at death of a person in the target
group is to be estimated and therefore we
will records and we keep the records of the
ages of the persons in that particular ah
target group for which the companies 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 ah the yield
of a crop makes the ah government to be decide
about how much ford grain they are going to
purchase from the former how much they are
going to a store what should be the ah price
which should be given by the formers 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 statistic we do study this type of phenomena
so now let me introduce the certain basic
ah 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 notation
we will use either capital omega or capital
s etcetera could be note the sample space
so let us considered some examples ah suppose
we are considering tossing of a coin in the
tossing of a coin we may be put the the outcomes
we may considered as head or tail and if we
denote by h the occurrence of head upwards
or by t the occurrence of tail then the sample
space can be describe as h t if we considered
tossing of a die then the sample space we
may describe as occurrence of the face upper
most so the sample space will be the set of
the numbers one two three four five six
if we are considering drawing of a card from
a deck of cards now in a deck of cards where
are four denominations heart club spade and
diamond and each card as a value one to thirteen
so if we considered drawing of a cards then
it will be consist of the set for example
diamond one to diamond thirteen club one to
club thirteen heart one to heart thirteen
or a spade one to spade thirteen so the sample
space consist of fifty two points if we look
at ah we may also observe the sample expression
and we like for example you may only cover
the colour of that so in that case we may
describe it differently we may call this one
as omega one and if i recording only the colour
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 express will consist of only
four points this also shows that sample space
is not a unique thing in a given experiment
what we are interested in will decide that
what is a sample space if you are looking
at the birth of a child then it is a various
statement or you can say where we have describing
what is the random experiment now we made
a call whether the child is a male or a female
child we made a called whether the child born
has is healthy or not healthy so we may put
it as healthy or unhealthy we may look at
his ah body weight at the birth so the body
weight may be some number is starting from
say zero to say may be ten pounds it may be
the total life of the child so in that cases
may be something like say zero to hundred
so it depends upon that what is the our actual
interest and we can write the sample space
accordingly ah [vocalize-noise] age at death
of a person so this may be you say zero to
say may be hundred twenty keep in to a counted
there are some people who leave very long
there this unit of the time is years amount
of rainfall may be recorded in say centimeters
yield of a crop may be recorded in some metric
tons the time taken to complete a hundred
meter sprint may be your time from say a nine
seconds to say eleven seconds about i am considering
on international filed life of a bulb it is
a number say which is zero to infinity although
theoretically speaking it is not infinity
but it can be a large number life of mechanical
instruments in the either you can do described
suppose we are looking at the number of defective
items produced by a company
a particular kind of items you are looking
at suppose we are looking at certain boards
then what will happening that we are we will
may define the defectives that if they do
not conformed to certain describes standard
of measurement
so now the number of effectives may be recorded
in terms of percentage so the percentage can
be say zero to hundred percent or it could
be say proportion in that case we may write
the number as zero to one 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 is qualifies
to be an event to be called an event as far
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 you look at omega is equal
to h t we may consider a subset as consist
of only h we may consider as 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 two four six this means occurrence
of an event number 
in the birth of a child suppose we are looking
at say weight and birth in pounds and suppose
we say e is equal to four to eight then it
means that the birth weight of the child is
four to eight pounds if we look at the amount
of rainfall an in centimeter during the particulars
monsoon season and we may put say fifty to
seventy five that means the actual rainfall
is between fifty centimeter to seventy five
centimeters in that geographical area during
that particular monsoon season so any subset
of the sample space can be considered an event
now we have various kind of events for example
impossible event since every subset is a subset
of every event is a subset of the sample space
therefore the mt set phi that is a subset
of omega therefore this is 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 he say that seven occurs so that will
correspond to an impossible event because
seven is not a subset of this and that will
correspond to five as far as this random experiment
is concerned
suppose i am looking at the age at death of
a person and we say thousand years then it's
an impossible event suppose we say time taken
to complete a hundred meter sprint by an athlete
and we may put the time as say five seconds
then in the present circumstances or present
age this is an impossible event similarly
if i put that the time take into complete
a hundred meter sprint by an athlete why did
he complete the race is less than one minute
then this will be a sure event
if we look at the life of a bulb and we says
the positive number then or a nonnegative
number then it's a sure event so these are
two ah you can say types of events which are
possible however there are various set theoretic
operations like unions intersections differences
complementation and therefore given any two
events when we take their unions intersections
differences complementation excreta there
must correspond to certain events and we can
describe them in the form of probabilistic
explanation
so for example union of two events 
so what does union of two events implies that
if i say a and b are true events then a union
b this will mean occurrence of at least one
of a and b that means either a occurs or b
occurs or both occur so in set theoretic representation
a and b a union b means that set of aliments
which are either in a or b or in both in probability
theory the event a union b will indicate that
at least one of a or b as occurred similarly
we may consider union of a n events a one
a two to a n then this will be in occurrence
of at least one ai i is equal to one to n
you may even considered an union of finite
number of events union ai i is equal to one
to infinity this will mean occurrence of at
least one ai i is equal to one to infinity
intersection of two sets denotes set of all
those points which are common to the two sets
now in set theory that is the presentation
in probability theory a intersection b will
mean the simultaneous occurrence of a and
b that means both event a and b are being
to have occurred similarly we can considered
intersection of n events ai i is equal one
to n that is simultaneous occurrence of a
one two a n that means all of the events a
one a two a n occurs and in a single way intersection
of an any countable collection of events a
one a two excreta
here when we considered the unions or the
intersections certain basic properties are
clear for example we may have union of ai
say i is equal to one to n is equal to omega
that means all the points are of omega are
contained in one other of ais such events
are called exhaustive events so if even union
of ai is equal to omega we call a one to a
n to be exhaustive events here in place of
n we may have an infinite collection of events
also
similarly if a intersection b is equal to
phi now the set theory is means joint sets
in probability theory a intersection b is
equal to phi denotes that the event a and
b cannot occur together they are called mutually
exclusive events a and b are called mutually
exclusive events that means happening of one
of them excludes the possibility of happening
of the other
we may also consider something like this that
we have a collection a one a two etcetera
events such that ai intersection aj is equal
to phi then i is not equal to j then we say
that a one a two excreta are pair wise disjoint
or mutually exclusive events given an event
a the a complement 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 complement that
means simultaneous occurrence of event a and
b which means that a a occurs and b does not
occur so it is a simultaneous occurrence of
a and b complement and it is translated to
occurrence of a but not no occurrence of b
