In this particular module, we will we will
be discussing fundamentals of probability,
and statics. Nowadays probability, and statics
plays a very important rule in all walks of
life, all engineering, and science disciplines
in social sciences. In fact, we can say that
there is no area of human activity, where
probability or statics is not used. The subject
of probability as old as the civilization
itself; however, the modern theory of probability
as we know today as its routes in games of
chance, and particularly in 15th and 16th
century Europe, when the games we are been
played in the form of dice games, cards games,
etcetera.
Then some of the people got interested in
knowing the that what are the chances, that
if the bit on a certain event, then whether
it will be more likely or less likely. For
example, in a grame of cards, they would like
to know whether a particular player will get
all the 4 asses or all the for kings or whether
you will get the top 4 cards of a particular
denomination.
This let us to the certain correspondence,
and activity among the mathematician especially
2 of the... So, today we will discuss basically
the laws of probability - the basic laws of
probability. The some of the famous mathematicians
of the day for example, Fermat Pascal, they
had famous correspondence in which we discuss
the elementary laws of probability. However,
the first publish to our is probably by Huygens
in 19 in the year 1657, in his book; there
is also contribution by James Bernoulli around
the same time.
The landmark are you can say the first milestone
in the development of the subject probability
can be considered by the monumental work by
Laplace, the French mathematician in his book
theory analytique des probabilities Laplace
covered all the development of the subject
of probability, which was known till that
day. So, that he added is a own theories,
and this gave a foreign foundation and a mathematical
treatment to the subject probability.
Some of the famous contribution to the subject
of probability are by von mises probability
is statistics, and truth. The model probability
theory that is after the development of that
any mathematical theory should have an arithmetic
setup, and from there the entire theory should
be able to derived. This let to the arithmetic
development of the probability theory, and
A N Kolmogorov the Russian mathematician published
a book in 1933, foundations on the theory
of probability.
Let us see that, what are the fundamental
units that wants should know or the fundamental
definitions for the development of the subject
of probability. So, when we give a typically
statement, that it is likely that it will
rain today or I may miss by train today or
my for example, if I am doing a study, I may
say that my study is likely to be successful
or unsuccessful. This type of statements or
in a sense giving as a measurement of how
much likely on event is, but in the first
place we may not actually give number, we
may just say that it is likely that it will
rain today without quantifying it. However,
if we say that there is a 75 percent chance
that it will rain today, then we are putting
measurements with that.
So, the subject of probability is related
to giving numerical values to the probabilities
of various statements. And therefore, it deals
with certain phenomenon, where things are
uncertain. For example, let us look at two
types of experiments: One experiment is that
we take to molecules of a hydrogen, and molecule
of a oxygen and we mix them. We know that
the reaction of it will be lead to the water.
Now, this is an experimented chemistry, and
similarly there are various experiments in
science and engineering, where we carry out
of experimental under certain conditions,
and we come with this certain conclusion.
We know that this will be the outcome of the
experiment, such experiments are known as
deterministic experiments. So, fundamental
unit you can say in the study of the subject
probability, and statistics is an experiment.
So, an experiment can be a deterministic experiment,
I give the example of a deterministic experiment
just now. Another experiment could be where,
if we contact the experiment we are not sure
of what outcome will be there. For example,
if we task a queen, then we may get a head
or tail or we may also include the possibility
that it may stand on its side. If we toss
a dice, then we may get any face up for example
1, 2, 3, 4, 5, 6, if we talk about the weather
tomorrow, then the weather could be sunny,
it could be cloudy, it could be rainy, there
could be thunderstorm.
If we are talking about say next 10 years,
say it states of the number of earthquakes
in a particular seismic zone, then there may
be one earthquake, there may be no earthquake,
there or there may be 3 earthquakes or there
may be 15 earthquakes and so on. Now, these
are the kind of events, where we are unable
to know the outcome of the experiment before
and, now we say experiment - experiment does
not mean that only laboratory experiment;
experiment means observing or contacting something
under certain conditions. So, when we say
whether then we are only observing the weather,
because as we know that there will be is spring,
there will be some are there will be raining
season, then there will be winter and so on.
So, the weathers happen due to certain natural
phenomena; however, when we want to observe
that then it is considered as an experiment.
And if we are looking at say day today or
behaviour over a period of time, then it is
a random experiment. A non deterministic experiment
is known as a random experiment, because we
are not sure of what outcomes will be there
in the beginning. So, random experiments are
the one, here the outcome is not certain beforehand.
Now, one may just question that if we are
not knowing the outcome before, and then why
do we actually study the subject of probability.
For example if we are looking at the tossing
of a queen, then suddenly it can be head or
tail, but it is not known at each toss of
coin, whether we will get head or tail. But
there is other feature of such experiments,
which allows as to study the subject of probability,
and a more theoretical basics. For example,
if we contact the experiment say 100 times,
suppose the coin is the fear coin, we may
observe something like say 48 heads, and say
52 tails; out of 100 times.
Suppose, we conduct the experiment of tossing
of the coin 1000 times, we may observe the
there are say 505 heads, and say 495 tails.
That means, if we are contacting at a large
number of times, we may observe a pattern;
and that pattern gives as the probability
of individual events. For example, we may
safely say that the probability of occurrence
of a head is half here or the probability
of the occurrence of tail is half here. Similarly,
if we today occur that my contacting of this
experiment would have l yielded. For example,
in the toss of 100 times, suppose we have
60 heads, and say 40 tails.
Similarly, suppose in 1000 we may get something
like say 602 heads, and say 398 tails. Then
we may be more and clime to say that it say
bias coin in the favour of head, and we may
put say probability of head as say 2 by 5
sorry 3 by 5, and probability of tail as 2
by 5. Therefore, what we observe here is that
individual experimental outcomes are not known,
but when, but there is a long term its statistical
regulatory; this term is known as its statistical
regulatory. That is the long term behaviour
that one can say.
So, for example, in a 1000 births in a city
hospital in a period of say 6 months. We may
except that there are 500 girl child s, and
500 boy child s born. If we observe the weather
pattern over several years in a monsoon region,
then we may say that during the amount of
rain fall may be say 100 centimetres or if
we are observing the weather pattern of drought
and rains, we may say that after every 10
years there is a likelihood of a drought year.
So, for example, we may say that the probability
of a drought is1 by 10.
So, this long term behaviour that we can make
out, that we can predict out a uncertain event;
justifies the a study of the subject probability.
Now, we introduce the examples of random experiments.
So, we may consider various kinds of phenomena.
For example, I just mentioned about say tossing
of a coin throwing of a die, ignore of a car
form a car etcetera.
However, these are not only text book kind
of example that are random experiments, there
can be many more. For example, we may observe
observing the number of vehicles crossing
at a busy traffic signal during an hour. So,
here for example, the number of the vehicles
could be 0, 1, 2, and so on. The amount of
rainfall in a certain say geographical region
during a year. So, depends upon the region
suppose the region is play a such that, this
is lot of rain then this amount could vary
from say 20 centimetre onwards to say 200
centimetre in have interval like this. The
weight of the child is recorded at the birth,
so different children will have different
weight at birth. So, then a vary from a few
100 grams to few kilograms kind of thing.
The number of times a child contacts a cold
infection during a year. The number of students
scoring more than qualifying marks in an examination.
So, for example, the total marks are 100,
and the qualifying marks are fixed as a 60;
therefore, how many students out of the total
number of students which are appearing, how
many of them will cross that qualifying marks
that will become a random experiment. You
are going on a route and you are a traffic
signal, so the traffic signal may be green.
So, you may just cross or it may be red in
that case you may have to wait depended upon
the total duration of the signal. For example,
it may vary from 0 to 3 minutes.
So, the waiting time at a traffic signal,
so suppose we are according in seconds, then
it could be vary from 0 to180 second; the
maximum limit till which the signal may be
a having a particular sign. The time to decay
of a page in a text book. So, this could be
the time recorded in years; for example, it
may take say 20 years to 100 years for the
periods to decay or may be 20 years to 200
years for the page to decay. The time for
onset of a disease since the infection, so
for example, one is infected with say HIV
virus, and then the time when the aids disease
develops in the person. So, that is a random
experiment.
The total food grain produced during a harvesting
season. The number of units of an item sold
from a store during a day. So, you can see
that the examples of random phenomena as very
yield as possible, and the examples are ranging
from engineering physics, medical, social
sciences, economics.
So, almost there is no area human activity,
where the random phenomena is not there. In
fact, the quantum mechanics assume that the
moment of the electrons is random. And that
is why we the modern theory of physics are
you as it called as statistical physics is
there.
Now, we look at some basic terminology which
is used in the subject of probability, and
then we will find the probability. So, let
me take a basic unit of a random experiment
is a sample space. So, a sample space is the
set of all possible outcomes of a random experiment.
So, the usual notations one can be use S,
omega, theta, etcetera. So, let me look at
the example that we discuss just now.
So, for example, if we are looking at the
number vehicles crossing at a busy traffic
signal during an hour. And here the sample
space we may write as the numbers 0, 1, 2,
and so on. If we are looking at the amount
of rainfall in a certain geographical region
during a year, and as I mention that the area
is such that it receives lot of rain, then
my sample space can be expressed as a an interval
22, 200 where the unit of measurement use
in centimetres. The weight of the child at
birth, so it may be as it small as a few hundred
grams to say 500 grams to say 5000 grams;
it may vary little more also depending upon
what features we are study here. The number
of times a child contact cold infection during
a year; once again the number could be 1,
2, and so on. And it may end up at a finite,
because the total number days in year is 365.
So, suddenly the number cannot be very large,
you may put say 0, 1 to 50. The number of
a student is scoring more than qualifying
marks in an examination.
So, suppose the total number of students are
N, which are taking the exam, then the sample
space could be the waiting time at a traffic
signal in seconds for example, it would be
0 to 180, the time for decay. So, in un experiment
one, define what is the sample space depended
upon what is our area interested.
Then, what is an event? Any subset of the
sample space is called an event. And we usually
imply English letters in capital to denote
the events. So for example, if we look at
the number vehicles crossing at a busy traffic
signal. I may define the event A as the number
of vehicles is less than 10; the naturally
A is consisting of 0, 1, 2 10. This is the
subset of S. I may consider say the number
of times a child contacts cold. So, you may
say the number of times the child contacts
the cold.
So, we may a put is say more than 3; in that
case the set B will be 3, 4 and so on. Suppose,
we are putting the upper bound . So, this
is subset of& Here if I it denoted by say
omega, then this is the subset of omega. So,
event is the subset of the sample space; however,
we may have extreme cases, then we say subset,
then empty set is also a subset, the full
set is also a subset. So, if we say full set.
So, S is a subset. So, this is called sure
event, and phi is the subset of S this is
known as impossible event.
For example, if we say that the weight of
a child at birth minus 32, then it is impossible
event. Suppose, we say that the number of
vehicles crossing at a busy traffic crossing
is say one million, then there is will be
a impossible event, because it cannot cross
the total number of vehicles which are available
there, and so on. Now, when we associate sets
with the events, then there are set operations
like union, intersection, complementation,
etcetera. So, in terms of a events they have
various interpretations I would like to explain
this now.
So, we may consider say A union B; now if
there are 2 events A and B, A union B represents
occurrence of either A or B or both. Now,
we can generalize notation for example, if
I have A1 union A 2 union A n, then we may
say this is occurrence of at least one A i
for i is equal to1 to n. We may even talk
about, and in finite union i is equal to1
to infinity; the interpretation of this will
also the same, occurrence of at least one
A i, occurrence of at least one A i; now here
i will be 1, 2, and so on.
Similarly, if we consider the concept of intersection
of sets, then here intersection of the sets
will denote the common elements belonging
to A and B. So, this means that both A and
B occur. So, this we can say simultaneous
occurrence of events A, and B. Now, this concepts
can further we generalized to n events or
infinite number of events also. So, this is
simultaneous occurrence of A i, that is all
A i s occur together. Then there is a concept
of complementation for an event A - A complement
denotes not happening or no occurrence of
A; similarly we may interpret A minus B. A
minus B in the set theory denotes the set
of elements which are in A, but not in B.
So, this will become A intersection B complement;
that means, occurrence of A, and not occurrence
of B. Then there is a concept of , because
when we consider the concept of intersection,
the intersection could be phi also.
If the intersection is phi; if A intersection
B is phi, then they are known as disjoint
set. So, here we call them mutually exclusive
events; the meaning is that if A occurs B
cannot occur, and if B occurs then A cannot
occur. So, this is events A and B are said
to be mutually exclusive events; that is occurrence
of one excludes the possibility of occurrence
of the other. There is also a chance that
some of the events for example, A union B
is equal to the full sample space; if A union
B is equal to omega, then all the possibility
of the sample space are considered by A and
B, then we say A and B are exhaustive events.
This can be generalized, we may have i is
equal to1 to n is equal to omega or we may
say union A i, i is equal to 1 to infinity
is equal to omega. See for example, if we
are considering tossing of a die, if we are
tossing of a die, and we consider A i as i.
Then A1, A 2, A 6; they will exhaust all the
possibilities of 1 to 6. So, they are exhaustive,
if we are looking at say weather on a day,
and we define the events say E as sunny, F
as say cloudy, and G as say rainy. Then all
the possibilities of the weather are exhausted,
and we may say E, F, G are exhaustive.
Now, we are ready to look at the one of the
preliminary definitions of probability, which
we call as a classical definition 
of probability. This can be attributed to
Laplace; he was the first one who gave it
in this particular form. Let a random experiment
be contacted; suppose it has n possible outcomes,
and these outcomes are exhaustive. That means,
we have consider all the possibilities, equally
likely equally likely means that each of them
has the same chance of a appearing, and mutually
exclusive. That means, occurrence of one will
be excluding the possibility of the occurrence
of the other. Let E be an event for which
m of these outcomes can be considered to be
favourable. Then the probability of E is defined
as probability of E is equal to m by n.
So, that definition is as you can see, it
is applicable to the experiments, where we
have a finite number of outcomes all of which
we can an numerate, and we are putting additional
restrictions such that they are exhaustive,
they are mutually exclusive, and also they
are equally likely, in those cases this definition
can be applied. Let us look at very simple
example say tossing of a fair die, your sample
space consist of 6 possibilities, and we associate
say event A by saying the number is even.
Now, if we want to find out the probability
of A, then there are 3 favourable cases 2,
4, and 6; total number of possibility is 6.
So, you get half, suppose we say the number
is less than 5, then number is less than 5
has 4 possibility here. So, probability of
B will be equal to 4 by 6; that is equal to
2 by 3.
Suppose, we consider say tossing of 2 fair
dies, and we say E is the event, that the
sum is say 7. Then what are the possibilities
here? Then we toss 2 fair dies, the total
number of possibility is 36, 1,1,1, 2, 2,1,
2, 2, 2, 2, 6, 3,1, 3, 2, 3, 6, and so on.
There will be total 36 possibility, and if
we assume that the dies are fair, then each
of them will be. So, the set E will be represented
by (1,6), (2,5), (3,4), (4,3), (5,2), and
(6,1); we are 6 possibilities, which we will
lead be the sum 7.
So, probability of E will become equal to
6 by 36, that is equal to1 by 6. Suppose,
we take another example here.
4 players A, B, C, D are distributed 13 cards
each at random from a complete duck of 52
cards. what is the probability, that the players
C has all 4 jacks. Now, here we look at this
problem, the total number of possibilities
for each player. So, there are 4 players,
and they are distributed 13 cards each. So,
the total number of possibilities for a player
C, because we are interested in the event
for the player C, the total number of cards
that the player C get 13, out of 52.
So, if we consider the possibility, that it
will be 52 C 13. Now, he is getting all the
4 jacks; that means, out of a 13 cards, he
is now total there are 52 cards out of its
they are 4 jacks, and he gets all the 4. So,
4 C 4 and from the remaining 48 cards, we
get any 9 cards. So, this will be the probability
of player C has all 4 jacks. So, this answer
comes by direct counting; assuming that all
the cards are equally likely to be distributed
to all the players. This definition is helpful
for answering questions of this nature we
are the sample spaces are finite, and we are
having the facility observing are you can
say enumerating being able to enumerate all
the possibilities; however, this as some practical
difficulties.
However, this has some practical difficulties;
for example, we may have 
this total number n need not be finite or
determined; we may not event able to determined,
what is n. Another thing is that we are assuming
here that the events are or the outcomes are
equally likely; the meaning of equally likely
which I mentioned is that they have the same
chance of occurrence. Now chance is associated
with that our probability; we are defining
probability here; in that sense this definition
is circular. The definition fails. We are
already assuming that each outcome has a equal
probability; therefore, this particular type
of definition is applicable only to theoretical
kind of exercises.
The definition of probability here uses the
term equally, likely, meaning outcomes, which
have the same probability of occurring. Thus
the definition is circular in nature. A more
practical definition was developed, and this
is based on the empirical evidence. When we
made usually statements, it is likely that
it may rain today; for example, if we have
observed three days of intense heat and humidity
in a region, then we say that on the fourth
day evening, we may say in the morning, we
say today evening it may rain. Now, this is
based on our experience. Similarly, when we
are observing the performance of the students,
and we make a statement, we fix up the qualifying
marks 60, and then we say that nearly 50 percent
of the students will qualified. Now this is
based on our previous experience or previous
experimentation for the same event.
As I already mentioned that the subject probability
itself is of interest, because of the feature
of a statistical regularity or long term prediction.
Therefore, another definition which is a more
statistical definition is based on the empirical
observations. So, we give empirical definition
of probability; this is also called relative
frequency definition relative frequency definition.
Now, we are looking at the outcomes of a random
experiment, based on that certain event is
there, for which we are looking at the probability.
So, for example, whether it will rain after
three days of intense heat etcetera; what
is the probability of the students qualifying
in a given examination; what is the probability
of a patient recovery from a certain disease
if he is given a certain meditation.
So, in all these cases, we are observing so
for example, patients are being giving certain
medicine for certain disease, and then we
have the data that how many of them may be
recovering, so may be 90 percent are recovering,
80 percent are recovering and so on. Now each
unit of observation; so, for example, one
patient is being giving the medicine or meditation
for a certain disease, then the second patient
is given, and this we are observing over a
period of time. This is considered as repeatedly
contacting the experiment, and we may say
roughly that it is contacted under identical
conditions.
So, and also we may assume that occurrence
are happening of one experiment that means,
whatever be the outcome of one experiment
does not affect the outcome of the next time.
So, for example, one patient is given a certain
medicine, he may recover from there; and now
a second patient comes with the same disease
and the same meditation is given, he may not
recover; that means, effect of one occurrence
should not be there on the other one. So,
we say that the random experiment is repeated
under identical conditions, and also independently.
Now, we see how many times over a period of
time or how many times over a certain number
of trails, this particular event occurs; now
this ratio is some value, and over the long
range when we are observing over a period
of time, this will stabilize; this is known
as the probability of that event. So, let
me write it here. Consider a random experiment
and it is repeatedly contacted under identical
conditions and the repeated trails are performed
independently. Let say p n be the number of
occurrences of a particular event A in n such
repeated trails of the random experiment.
Then we define the probability of the event
A as limit p n by n as n tends to infinity;
that means, over the long term, what is the
ratio of occurrence of the event, in which
we are interested to the total number of trails
of the experiment.
Let me explain through an example here. Consider
say 
let us consider weather report of a region
for say 1000 randomly selected days from say
30 years later. Suppose the weather pattern
is say sunny day, sunny day, cloudy day, rainy
day, sunny day, sunny day, cloudy day, rainy
day and so on. The pattern is somewhat fixed
that is two sunny days followed by a cloudy
day, and then a rainy day. Now we are interested
in finding out the probability of, we want
the probability of a sunny day. So, we look
at this ratio p n by n.
Now this p n by n, when the first trail was
there, we observed sunny day, so the ratio
becomes 1 by 1. In the 2 trails, 2 days it
was sunny, 2 by 2. Now the third turned out
to be cloudy. So, the number of sunny day
in the 3 trails is also 2. The next turn out
to be rainy, so the number of occurrences
of the event S was 2 out of 4 also, then 3
by 5, 4 by 6, 4 by 7, 4 by 8, 5 by 9, 6 by
10, 6 by 11, 6 by 12 and so on. Now, we want
to find out the limit of this; if we just
observed like this, it is difficult to find
out the limit, because number in rapidly.
So, we put in a more mathematical form; we
can like write like this. It is equal to if
we observe each fourth occurrence here, this
is of the form 2 by 4, 4 by 8, 6 by 12 and
so on; that means, we can write as 2 k by
4 k, whenever n is the form 4 k. If we observe
here it is 2 k by 4 k minus1 that is whenever
n is of the form 4 k minus1. Here it is equal
to 2 k by 4 k minus 2, whenever n is of the
form 4 k minus 2; it is of the form 2 k minus1
by 4 k minus 3, whenever n is of the form
4 k minus 3, for k equal to1 to and so on.
Clearly you can see here that each of this
subsequence converges to half, this is equal
to half; as k tends to infinity, this goes
to half; as k tends to infinity, this goes
to half; as k tends to infinity this goes
to half k.
So, clearly you can say limit of p n by n
as n tends to infinity is equal to half. So,
probability of a sunny day is equal to half;
so this is the experimental, you can say demands
station of this method of the relative frequency.
So, here you see that 2 sunny days followed
by 2 non sunny days, if that is the pattern
then definitely the probability of a sunny
day over a period of time, should be half.
So, relative frequency definition is based
on the experience, and this is the most widely
applicable definition of the probability today.
The definition which I give earlier as a mathematical
definition is more applicable for theoretical
problems, where we can see that the conditions
of the are satisfied. Now, in this also there
may be some discrepancy; for example, we are
taking the ratio, now that ratio will have
a limit, now the limit could be 0 or 1 also.
For example, you may say p n is equal to say
n to the power1 by 3; now n to the power1
by 3 is not a negligible number, but if I
consider p n by n then that will become1 by
n to the power 2 by 3. So, that will goes
to 0. So, probability of a non null event
may be 0; which looks little counter intuitive.
Although in the long run it has a meaning,
what it mean that as n becomes large, n to
the power 1 by 3 becomes much smaller compared
to n. Similarly, we may have say p n is equal
to n minus root n, in that case p n by n that
will converge to 1.
Now, here you can see that the event is not
a sure event. So, probability of a non sure
event may be one, which is again little counter
intuitive. Although from the is statistical
distribution of the probability this is alright,
because what did says that if we have n large,
then out of that the number of occurrence
of the event is almost full. That means, sometimes
it may not occur, but that number is negligible;
however, because of these drawbacks, we cannot
use these definitions as the you can say arithmetic
definitions, because they do not satisfy all
the conditions. There are certain other problems
also for example, this requires that the experiment
be performed or we should be able to observe
the occurrences, and the outcomes of the occurrences.
Now, they can be various experiment, where
this is not possible. For example, rare phenomena
or suppose we are looking at industrial experiment;
and an industrial experiment suppose it is
a large scale industrial experiment, in that
case if we have to look at whether the system
will fail at what time it will fail. Then
suddenly, we are wait till the time in the
system actually fails. So, in many of these
conditions, the direct application of the
definition is not possible. Based on this
the Russian mathematician gave a the now the
well known arithmetic definition of probability.
So, we have the sample space, and we consider
a class of subsets of a class of subsets of
omega; that means, these are events. Now,
this should satisfy satisfying the following
2 conditions. One that for every E belonging
to omega E complement also belongs to omega;
that means, it is close under the complementation.
And second is that for any sets A1, A 2, and
so on belonging to B union of A i also belongs
to B. That means, it is under closed closed
under the infinite union also.
Such a B is actually called, this is called
a sigma field or sigma algebra. This is a
algebra is a structure; however, we are not
getting into this. The main purpose is that,
when we are considering a random experiment
and its outcomes, then all the events should
be included in the subject under study. That
means, whatever set we are considering it
should include all the events, and that is
why we make it closed under the operation
of complementation, and the infinite union.
As we know that, this will further allow infinite
intersections, it will allow the differences;
that means, all the possible said theoretical
manipulations will be included in the relevant
is space.
And therefore, and therefore, we will be able
to contact the study of the probability; that
means, we can find out the probabilities of
the related events provided, a certain probabilities
are known to us in advance either by the first
definition, that is the classical definition
or by the second relative frequency definition.
That means, probability of certain events
may be known to us, and there after we can
use them today the probabilities of various
other events. So, in the following class,
I will be discussing the axiomatic definition
of the probability its ramification, and then
various important results of probability.
