Welcome students to the first lecture on advanced
probability theory.
As I said in my introduction, that probability
is something that we often use very loosely
in our daily conversation.
Say for example, today there is high probability
of having a rain.
How is that probability computed, like any
mathematical science, there should be a solid
mathematical theory behind it and you can
compute probability of an event, if and only
if that particular event has occurred a large
number of times, so that we can compute its
probability.
Therefore, such use of the term probability
in my opinion is not at all scientific, it
is highly subjective.
And therefore, for any scientific calculation,
we need to understand how the probability
needs to be calculated.
And to do that we need to have a firm grip
on the theory of probability.
So, with that small introduction, let me now
come to the subject.
So history, probability some of you may know
that is something that people are talking
about for centuries.
In fact, in the mid 17 th century, people
have started talking about probability, say
people like Blaise Pascal, Fermat.
They started using the concept of probability.
Later in 18th century 
La Place brought the notion of mathematical
formulation 
of probability.
And he suggested that probability of an event
is the ratio of number of favorable events
upon total number of events.
But, such a definition, probability of the
event is defined as the number of favorable
outcomes, total number of outcomes.
But such a definition almost invariably needs
to define certain things such as what is event,
what is total number of events, what is favorable
events.
And therefore, we need to define these terms
before we go into the more basic fundamentals
of probability.
So, we can think of 
computing probability of an event only if
the corresponding 
experiment is random.
So, what is an random experiment?
So, let us start with that first, an experiment
is 
something 
or some action 
that can be conducted any number of times
under the same setup for example, simplest
one is tossing a coin.
If you have a coin, you can toss it any arbitrary
number of times and you can get the output
to be head or tail.
Similarly, throwing a die 
we can keep on throwing a die any number of
times and we would expect one of the possible
6 outcomes 1, 2, 3, 4, 5 and 6.
There can be many other examples say for example,
there is a deck of cards of 52 cards, all
of us know that there are 4 suits clubs, hearts,
spades and diamonds and under each suit there
are 13 from Ace to King.
So, there are 52 possible outcomes.
If we take a card from complete deck.
Some other standard example are, suppose a
box contain say m white balls and n black
balls and our job is or the experiment is
to pick a ball from the 
box and note its color 
and put it in the back in the box.
As you can understand, this can be continued
any number of times, this experiment is called
with replacement, why it 
is called with replacement?
Because the ball that is picked up from the
box is going to be put back into the box again.
Hence it is called with replacement.
In a very similar setup 
if the ball is not put back into the box 
then it is called 
without replacement.
Quite obviously, this cannot be executed infinite
number of times because the box will be empty
after all the balls are taken out of the box,
but still without replacement sampling or
picking up the ball from the box is often
very important for our studies, as we will
see later.
An experiment 
will be random.
If we know the total number of 
possible outcomes and that set is 
often denoted by omega for example, tossing
coin omega is equal to head or tail, tossing
a coin 2 times.
Then omega will be, if you are thinking it
can be HH the outcomes of the 2 events.
It can be outcome of the 2 trials.
HT, TH and TT, head head, head tail, tail
head and tail tail and throwing a die 
will have omega is equal to 1, 2, 3, 4, 5,
6.
So, I hope you understand what is the omega
or the total number of possible outcomes is.
Now in a random experiment 
the omega will be known but the outcome that
is coming 
will be unknown with each trial 
of that experiment, now that what is a trial,
each time an experiment is carried out, we
call it a trial.
So, if I toss a coin 3 times, then we mean
there are 3 trials but suppose I take a pair
of coin I toss them together and then what
I am getting HH, HT, TH or TT and that itself
is one trial and that if I do 5 times, we
will get 5 pairs of outcome.
Event, any subset of omega is called an event.
For example, say throwing a die 
and getting an even numbers.
So, if we consider the omega is equal to 1,
2, 3, 4, 5, 6 these are the 6 points of the
omega and the event is getting an even number,
so that event will occur if the outcome is
2 or 4 or 6.
So, this event can occur in 3 different ways.
And there are 3 other ways when the event
will not occur.
Therefore, by Laplace formulation, probability
of getting and even numbered is number of
favorable outcomes divided by total number
of outcomes.
Alright or write it at number of total outcomes.
Now, in this case, the favorable outcome is
3 if it is 2, if it is 4 and if it is 6 and
the total number of outcomes is cardinality
of omega, which is equal to 6 and therefore,
half thus probability of getting an even number
will be computed as half.
Similarly, with respect to 
throwing a die, we can think of many events.
For example, getting a 6 that is an event,
getting a numbers 
more than 3 that is an event, getting a numbers
between 2 and 5 that can be an event or in
other words, the way we take a subset of omega,
we can define a particular event.
Let us take the example of picking a card
possible events are getting spade, what will
be the probability?
There are 13 cards of the suit spade in a
complete deck which has 52 elements.
Therefore, this probability is going to be
1 by 4, getting an ace there are 4 aces corresponding
to the 4 suits.
Therefore, number of favorable outcomes is
4 total number of outcomes is 52.
Therefore, this probability is going to be
1 by 13, getting a red queen or black numbers
between 2 to 10.
Now, we know that there are 2 queens of red
colored, queen of hearts and queen of diamonds
and 2 to 10 there are 9 numbers and this 9
can be black, if it is coming from card or
if it is coming for the clubs or coming from
the spade.
So, there are 18 ways of getting a black number
between 2 to 10.
Therefore, number of favorable outcomes is
2 plus 18 is equal to 20, total number of
outcomes is 52.
So, it is 20 upon 52 is equal to 5 up on 30.
So, now you can understand how we are defining
an event and how we are computing its probability.
However, things are not so straightforward
when we are 
working on an experiment where the omega is
continuous say for example, what is the probability
that today the temperature will be between
15 degrees Celsius to 20 degree Celsius?
Now, suppose on a particular day, the possible
temperature can range between say 10 degrees
Celsius to 25 degree Celsius.
Therefore, there are uncountably many points
in omega, because on the real line between
10 to 25 there are uncountably many real numbers
and each one of them can happen to be the
temperature of a particular day.
Similarly, there are uncountably many different
ways for the favorable event 
to happen, because our favorable event is
that our temperature will be between 15 to
20 degree Celsius.
Therefore, both the number of favorable events,
favorable outcomes divided by number of total
outcomes.
This is actually infinity by infinity and
therefore not defined 
and if it is not defined, therefore, we cannot
calculate probability as the ratio of 
these 2 numbers.
Hence probability 
has to be defined in a more mathematical way.
The corresponding concept is called probability
measure.
Say for example, measure can be something
that gives an idea 
of the relative size or volume of the set
of favorable outcomes to the size or volume
of the total number of outcomes.
This can be 
say length for example, 15 to 20 degree upon
10 to 25 degree.
So, this is the possible outcomes and this
is favorable outcomes.
Therefore, the length is say 5 here the length
is 15.
So, we could say the probabilities something
like 1 by 3 in a similar way, say sometimes
we can use area.
Suppose, we are looking at event, what is
the probability 
that a point picked randomly from a unit circle
has distance less than half from the center.
So, let us consider a unit circle, the radius
is 1 and suppose this is the center and consider
this circle of radius half therefore we shall
say a favorable outcome will be, if we choose
the point from inside the circle, this circle
therefore its distance from the origin will
be less than half.
Therefore, area of consider this is B and
let us call it A, B upon area of A, that can
give us the probability of that event to occur.
That is, the randomly picked up point is at
a distance less than half from the origin.
Now, area of B is going to pi half square
area of A is going to be pi 1 square.
Therefore, these probability is 1 by 4.
I hope I could make it clear how the probability
is calculated not just merely counting the
number of favorable outcomes to the total
number of outcomes.
Rather, we look at the subset that is depicting
the event and we will compare its measure
in comparison with the total measure of omega
and that is what we are trying to drive at.
Before we go further, we give some basic definitions.
Exhaustive events the total number of possible
outcome is called as exhaustive events.
Therefor all the members of omega together
gives us the set of exhaustive events.
Elementary event, the outcome obtained after
each trial is an elementary event.
For example tossing a coin 2 elementary events
H and T. Suppose my experiment is toss a coin
and throw a die and not the pair of observation.
Then there are 12 elementary events H1, H2,
H3, H4, H5, H6, T1, T2, T3, T4, T5 and T6.
So, this set of 12 pairs is my set of elementary
events and together they are the exhaustive
event.
Equally likely events, if taking into consideration
all relevant evidences there is no reason
to believe 
that one event has more preference to other
then the set of events are called equally
likely.
Consider for example, tossing a coin 3 times
and note the outcomes.
So, let us look at it first from the perspective
of the following, the set of possible outcomes
are HHH, HHT, HTH, HTT, THH, THT, TTH and
TTT.
If the coin is unbiased that is both head
or tail have equal chance to occur, then all
these 8 possible observations may be considered
equally likely.
On the other hand 
suppose under the same experimental setup,
we count the number of heads then the set
of possible outcomes are 0, 1, 2 and 3.
However, we cannot say that they are equally
likely, because 0 can happen only if the outcome
is TTT, the 3 can happen if only the possible
outcome is HHH, because all 3 are heads therefore,
count of head is 3 here none of them is head
therefore count of head is 0 and therefore
as you can understand that 2 can happen for
this, this and this.
On the other hand, one can happen for this,
this and this.
Thus, we can understand that there are more
chance for 1 or 2 happening instead of 0 and
3 hence not equally likely.
Mutually exclusive event, two events are said
to be mutually exclusive if the occurrence
of one of them precludes the occurrence of
the other.
For example, picking a card from a deck there
can be 4 mutually exclusive events, getting
a club, getting a spade, getting a hearts
and getting diamonds.
So, if I get a club, we know that we cannot
get a spade or a heart or a diamond and similarly
for other events.
Now any subset of omega can be an event.
Therefore, if A is an event A compliment with
respect to omega is also an event because
A compliment is also a subset of omega or
in other words this is my omega and this is
A, so which consists of this elementary points
then A compliment 
is also an event.
If A1, A2, An are events then A1 intersection
A2 intersection An is also an event.
For example, suppose this is my omega and
this is the event A, this is the event B,
this is my event A, this is my event B and
this is the event C, then consider this part
which is A intersection B intersection C and
therefore, this can be an event because it
is a subset of omega.
Thus in general we can think of finite or
countable union intersection of events or
their compliments to be events.
And whenever we talk about events, we need
to think about assigning probability to all
possible events that we are looking at, under
the random experiment that we are pursuing
for studying our Probability.
Question is how to do this?
The basic mathematical trick is said to be
that of sigma field.
I am not sure how many of you are familiar
with the term sigma field, but this is the
basic mathematical notion that we shall need
along with something which is called a probability
measure that is helpful for us for defining
probability.
Okay students, I stopped here now, in the
next class, I shall start with these mathematical
concepts for our study of probability theory.
Okay friends, thank you.
