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Welcome to the second lecture for the course
probability methods in civil engineering.
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This is basically the starting of module two,
in this module two, I will cover the concept
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of random events which is very useful in the
probability methods.
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In this module, there
are couple of lectures, 4-5 lectures will
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be there; in the very first lecture I cover
the
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random events and probability concept.
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.
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Basically we have to know the concept of random
experiment first, and before that, what
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is an experiment?
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It says a set of conditions under which behavior
of some variables are
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observed, and when you say that a particular
experiment is random, that is random
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experiment, in those type of experiment, where
it is not possible to ascertain or control
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the values of certain variables, and results
varies from one performance to the other
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performance.
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For example, if I say that if I threw a coin,
and I want to see what is the
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.outcome of the throwing of a coin, I cannot
say with full certainty that what should be
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outcome whether it will be the head or it
will be a tail.
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So, this kind of example that is tossing of
a coin, and similarly a rolling of a dice,
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where
I cannot say the particular outcome from that
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particular experiment, then the outcome of
that experiment becomes random.
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And this kind of experiments is known as random
experiment.
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.
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When we take this random experiment, there
are different types of sampling techniques.
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One is with replacement and another one is
without replacement, in the case of with
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replacement, each item in the sample space
is replaced before the next draw.
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Say for
example, if I say in a box there are few balls,
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and the balls are of different colors and
I
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want to take out one ball in random, and I
want to observe that what is its color.
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Now, if I
take out one ball and place it back to the
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box again, and do the next trial after replacing
the ball that is known as the with replacement.
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If I do not go for this example, and if I
give the example of the tossing of a coin,
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this also
can be treated as with replacement sampling
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technique, because if in the first trial I
get
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one head, and I am going for the next trial
that means all the options that is all the
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possible outcome head and tail both are with
me.
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.So, this kind of experiment is known as the
with replacement experiment; there is also
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possible that we can do the experiment without
replacement also in the example of
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drawing walls from a box, we can just take
out one after another a particular color of
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ball
and keep it outside.
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So, that we will know later gradually in this
course, that this two
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kind of sampling techniques is different when
we are going for the definition of or the
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computation of probability for that particular
experiment.
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.
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There are different types of this kind of
experiment, one is that independent or simple
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experiment, and second one is compound experiment.
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In this independent and the simple
experiment it says that the outcome of one
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experiment has no influence on any such
other experiments or, it is also not dependent
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on whatever the outcome just previous
trial.
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Say for example, if I say again the tossing
of a coin, then if the outcome for a
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particular trial, if the outcome is head it
does not have any influence on the outcome
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for
the immediate next trail.
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So, this kind of experiment is known as independent
experiment.
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Second one is the
compound experiment, it says a new experiment
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by performing N numbers of
experiments in sequence; for example, tossing
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two coins one after another, and getting
head for both the coins.
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So, I just want to show experiment here is
that I throw two coins, and I want to see
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the
outcome for both the 4 coins it comes head,
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if I give the example of rolling a dice where
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.the possible outcome is 6, that means the
number 1 to 6, I can say that what is the
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possibility that the summation of the outcome
of both the dies will be greater than or less
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than a particular value.
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So, this kind of experiment it says that the
outcome of one particular - one particular
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experiment depends on the total performance
or the success rate of the experiment
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depends on the mutual performance of both
the trials that we are performing, so this
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kind
of experiment is known as compound experiment.
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.
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There are different possibilities of the outcome
for a particular experiment and based on
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the number of possible outcomes, it can be
divided into two different parts.
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The first one
is the Bernoulli’s trails it says, a if
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a simple experiment has only two outcomes
for
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example, as I said that in case of tossing
a coin there are only two possible outcome
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head
and tail and independent replication of the
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same experiment, then this kind of trials
as
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known as Bernoulli’s trials.
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On the other hand, if a simple experiment
has k numbers of outcome, and independent
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replication of this experiment this kind of
trials is known as multinomial trials.
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So, these things are important to know the
basics of these random events which will be
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useful when we are going to give the concept
of probability, so if we know the basics of
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this experiment and based on this experiment,.
if we know what are the all possible
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.outcome then this will be helpful to understand
and interpret the results of the probability
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and analysis, and if we assign some particular
probability, it will help us to infer from
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the
results in hand.
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So gradually I will show, how this kind of
concept, this kind of
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interpretation of the probability is based
on this kind of random experiment.
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.
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So, first one important terminology that we
should understand clearly is sample spaceIt
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is set of – a set it is generally a designated
as a capital S, a set that consists of all
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possible
outcomes of a random experiment is called
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a sample space and each outcome is called
a
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sample point.
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Example; if a dice is tossed one sample space
or the set of all possible
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outcome is given by the numbers from 1 to
6, so this scan be treated as a pictorial
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representation of the sample space, and this
black dots are the sample points.
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So,
collection of all these sample points, collection
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of this entire possible outcome for a
particular experiment, is known as the sample
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space.
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..
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Second thing just after, we know what is sample
space it comes event and an event is a
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subset of the sample space or in other words
a subset of all possible outcomes.
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So, I
know what is my sample space, if I want to
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show it here the black circle or there red
area.
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here is your sample space and in this sample
space, if I select any combination of
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the sample points then each combination or
each each feasible set, each feasible subset
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of
this sample points is one event Tthere are
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other terminologies like elementary event.
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It is very simple,in the sense that if a particular
sample consists of only one possible
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outcome then that is known as simple event
or the elementary event, If I give the
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example again of this throwing a dice here,
so here the sample space, as I told just in
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the
previous slide, that the sample space is any
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integer number starting from 1 to 6, its event
can be anything.
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If I say that 1 2 3, so this is one event,
I can say that the number is less
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than 4, so this is one event.
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I can say the number is exactly equal to 4
then that is also
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one event.
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So number is exactly equal to 4 that means
there is only one possible outcome, and if
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I
say the outcome is one even number then there
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are 3; possibilities – 1, 4, and 6.
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Similarly, if I say the outcome is less than
5, so these are all events and one particular
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event may have one or more than one possible
sample points.
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So concept of this sample
space and event is very important that will
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help us to understand the concept of
probabilities.
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..
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Another thing, another important thing to
know here is the discrete and continuous
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sample space., So discrete and continuous
sample space can be expressed as follows:
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S,
which denotes the space, S is discrete if
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all subsets corresponds to events; on the
other
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hand S is continuous, if only special subsets
or measurable subsets corresponds to events.
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.
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Coming to the civil engineering here, so the
concept of the sample space, event and the
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probability of that particular thing, this
things should be clear first to understand
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the
probability associated with particular event.
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If I give one example, say, in case of
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.reservoir storage the range from 0 to maximum
capacity of the reservoir forms the
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sample space, and any value can be taken from
this 0 to this maximum capacity., if I say
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it is capital C, so 0 to C, any value is possible,
and this full range is our sample space.
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Now, reservoir storage above the dead storage
or reservoir storage below the 50 percent
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of the total capacity are the example of events,
so these are – for example, the reservoir
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storage above the dead storage, is a particular
subset of the total feasible, total possible
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sample space.
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So, this particular thing is one event, so
if I know this is one event,
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generally the probability are given to this
particular event.
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So, I can ask, what is the
probability that the reservoir storage is
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above dead storage, or I can ask what is the
probability the storage is below 50 percent
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of the total capacity of the reservoir.
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Similarly, suppose if we take the example
of another random experiment, that is the
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traffic volume, in transportation engineering
this is mostly used, so in case of traffic
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volume, if I ask that does all possible types
of vehicle consist the population?
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The
answer is no, because the traffic volume means
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the total number of different types of
vehicle moving across a particular stretch
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of the road.
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So, this is the number that I want
stress here, so that number consists of by
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definition that number is your traffic volume.
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So, here the population or here the total
sample space for the traffic volume consist
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of
any number between 0 to infinity and any range
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of this real numbers.
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So, again here this thing is one example of
the discrete sample space, because these
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numbers cannot be any real number this should
be an integer number, so these sample
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space is a discrete sample space, and any
range of real numbers can be treated as an
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event.
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So, I can say what is the probability of the
traffic volume less than 10 or what is
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the probability that the traffic volume is
greater than 100, this kind of thing.
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So any
subset of this 0 to infinity and it must be
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a real number and that consist of the sample
space for this experiment.
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Coming back to this particular thing or before
that, so I hope you understood that in this
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experiments the reservoir storage this sample.
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space is a continuous, sample space.
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So
the first example is the example of one continuous
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sample space, and second one is one
example of discrete sample space.
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Coming to again this particular question,
so far in case
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of this traffic volume., all possible types
of vehicle are not, the population of this
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one.
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Now, if I ask you that what is then, what
is this?
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What is the collection of all possible
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.types of vehicle?
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Is this not, can this not be a population?
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The answer is yes, this can also
be a population, so you can ask yourself what
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is the experiment involved if this is the
population.
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We will come back to this particular question
again and this is very
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interesting and this is very important concept
at the starting of the understanding of
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random variable, and which we will cover may
be in the successive after some lectures
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that this random variable, the concept of
random variable comes, that it does not mean
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that random variable is a variable which is
which is random.
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This is a completely wrong
definition of this random variable and when
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we discuss about this definition of random
variable we will come back to this particular
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question once again, and we will say that
what is that the particular, what should be
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the exact definition of this random variable.
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Now, we will try, whatever we have understood
so far about it about the sample space
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and its event we will try to tell them, we
will try to understand the concept of the
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probability here.
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.
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The concept of probability was proposed to
explain the uncertainty in the random events
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As I told that the outcome of a random experiment
cannot be ascertained.
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So, I have to
give some numbers some belief, and I have
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to assess that particular belief I have to
assess that particular number, and this concept
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of probability is generally developed to
give some explanation to that uncertainty
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in the random events.
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.Such random events may occur sequentially
or simultaneously.
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As we told in just
previous few previous lectures, few previous
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slides, that any random event, which is can
be either sequentially or this can be in a
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simultaneous fashion.
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Now, in the sequentially,
we will at a later part of this course, we
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will also understand this sequence can be
either
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in the temporal direction or in the spial
direction.
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That is known as the spatio temporal
thing so any particular random variable, we
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can observe it in the time sequence or in
the
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spatial sequence and there are different types
of analysis that we can prove and we will
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take you through all these things.
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So, for the time being we should understand
that this random events can happen either
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sequentially or simultaneously.
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For example, here the simple example that
I can say now,
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is occurrence of road accident in transport,
and safety analysis or occurrence of extreme.
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that is very high or very low rainfall, which
is in case of very high of course, it is that
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which is beyond the capacity of the drainage
network or occurrence of the very low
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rainfall which may affect the agricultural
production or so.
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So, this kind of occurrence, this kind of
random events, is very important for our civil
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engineering purpose, and we will know that
what are the ways that we can assess the
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probability, and what are the ways, different
methods of the probability can be applied,
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this particular oncept of the probability
can be used in that kind of analysis.
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.
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.In this concept of probability again generally
for a particular system, occurrence of such
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event approaches to a constant number; with
the increase in the number of observation,
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this number remain constant; so for example,
if I take a particular experiment in hand
223
00:20:25,460 --> 00:20:33,600
and I just see for a very long record very
long historical record, and if I see that
224
00:20:33,600 --> 00:20:36,080
what is
the chance or what is the frequency of occurring
225
00:20:36,080 --> 00:20:42,730
of that particular event, we can say that
it is, generally, we classify into success
226
00:20:42,730 --> 00:20:48,080
and failure, and we see that what is the chance
of, what is the frequency of the success of
227
00:20:48,080 --> 00:20:51,600
a particular thing in the historical record
and
228
00:20:51,600 --> 00:20:54,050
this historical record is very important here.
229
00:20:54,050 --> 00:20:59,250
Tthis should be sufficiently long to assess
some probability of this.
230
00:20:59,250 --> 00:21:04,400
Before coming to this real life civil engineering
problem, I can
231
00:21:04,400 --> 00:21:09,630
say again a simple example, this example of
tossing a coin again.
232
00:21:09,630 --> 00:21:16,450
So here the tossing a coin and observing its
outcome is your experiment, and counting
233
00:21:16,450 --> 00:21:22,690
the outcome as head, say for a particular
outcome head, or it can be tail, so this is
234
00:21:22,690 --> 00:21:23,930
a
particular event.
235
00:21:23,930 --> 00:21:26,410
So, if I say that
236
00:21:26,410 --> 00:21:29,330
tossing a coin and observing the head in this
237
00:21:29,330 --> 00:21:33,260
experiment, if the head comes then it is a
success and if head does not come that is
238
00:21:33,260 --> 00:21:39,660
failure, so that is one event and if I do,
if I follow this approach, that is, what is
239
00:21:39,660 --> 00:21:43,320
the
occurrence of such a particular event approach
240
00:21:43,320 --> 00:21:49,830
to a constant number, and by our day to
day experiment, I can say, we can say, that
241
00:21:49,830 --> 00:21:58,480
this number for this particular experiment
will approach to 0.5, so this 0.5, generally
242
00:21:58,480 --> 00:22:01,930
we assign this particular number, generally
we
243
00:22:01,930 --> 00:22:09,500
say, this is the probability of getting head
of one experiment having a coin.
244
00:22:09,500 --> 00:22:12,700
We will
come to this real life problem of this cyclone
245
00:22:12,700 --> 00:22:15,880
event or this thing a little bit later.
246
00:22:15,880 --> 00:22:16,880
.
247
00:22:16,880 --> 00:22:25,070
.So, that it will be clear in the in the area
of civil engineering problem also.
248
00:22:25,070 --> 00:22:29,010
Before that
we should know the definition, the classical
249
00:22:29,010 --> 00:22:30,010
definition of probability.
250
00:22:30,010 --> 00:22:33,990
As I told that we
should have a very long record available with
251
00:22:33,990 --> 00:22:40,320
us or if we are conducting the particular
experiment, we should conduct the experiment
252
00:22:40,320 --> 00:22:48,920
for a very long time, so that the number
will approach to a particular constant value.
253
00:22:48,920 --> 00:22:52,100
Here, based on the relative frequency, if
an
254
00:22:52,100 --> 00:22:59,600
experiment is performed n times and an event
A occurs nA times then with a high degree
255
00:22:59,600 --> 00:23:06,770
of certainty, all this terms are very important,
so with a high degree of certainty, the
256
00:23:06,770 --> 00:23:14,960
relative frequency nA by n of the probability
of occurrence of A which is designated as
257
00:23:14,960 --> 00:23:19,990
P
in bracket A is close to nA by n.
258
00:23:19,990 --> 00:23:26,680
So, that probability of occurrence A is the
ratio of nA by n.
259
00:23:26,680 --> 00:23:32,820
Here you can see that this nA
is nothing but the success, the number of
260
00:23:32,820 --> 00:23:36,200
success, and this is the total number of trial.
261
00:23:36,200 --> 00:23:40,130
So
as I told that, this should be sufficiently
262
00:23:40,130 --> 00:23:44,270
long, this should be, this should be, basically
this
263
00:23:44,270 --> 00:23:51,910
should be, infinite number of series, of sequence
should be tested to approach to that
264
00:23:51,910 --> 00:23:53,050
particular number.
265
00:23:53,050 --> 00:23:59,830
So, this definition, so if I want to know
exactly what is this
266
00:23:59,830 --> 00:24:04,470
probability based on this relative frequency
concept, then this n should be sufficiently
267
00:24:04,470 --> 00:24:12,550
large and that is why we write the probability
of A is equals to limit n tends to infinity,
268
00:24:12,550 --> 00:24:19,860
nA by n, so this definition is generally based
on this relative frequency concept.
269
00:24:19,860 --> 00:24:26,180
Other
definitions of the probability are also there.
270
00:24:26,180 --> 00:24:30,830
We will cover, we will see it shortly.
271
00:24:30,830 --> 00:24:31,830
.
272
00:24:31,830 --> 00:24:39,140
.Before that, to assign the probability of
a particular event; there are 3 different
273
00:24:39,140 --> 00:24:41,910
ways that
we can assign a particular probability, for
274
00:24:41,910 --> 00:24:45,600
a particular event, say for example, one of
the
275
00:24:45,600 --> 00:24:49,700
event is Ai; to assign the probability P(A)
of a certain event in a in exact way.
276
00:24:49,700 --> 00:24:52,900
Suppose
this way, this way of assigning the probability
277
00:24:52,900 --> 00:24:57,740
– I have to see the historical record of
a
278
00:24:57,740 --> 00:25:04,470
particular experiment, and based on the historical
success rate of that particular
279
00:25:04,470 --> 00:25:08,170
experiment, I have to assess some number.
280
00:25:08,170 --> 00:25:16,900
So as this historical record is never infinite
series, that is why we say it as the inexact
281
00:25:16,900 --> 00:25:17,900
way.
282
00:25:17,900 --> 00:25:28,060
Here one example is that if out of 365
days in a year, there are 73 days recoded
283
00:25:28,060 --> 00:25:32,520
to have above average rainfall, then we say
the
284
00:25:32,520 --> 00:25:38,370
probability of the event above average daily
rainfall is 73 by 365.
285
00:25:38,370 --> 00:25:50,260
So, this 73 for this
experiment is the success, and the total number
286
00:25:50,260 --> 00:25:58,760
of trial, total number of days, total
number of experiment is your 365.
287
00:25:58,760 --> 00:26:05,520
So this is an inexact way of assigning the
probability
288
00:26:05,520 --> 00:26:07,370
to a particular event A.
.
289
00:26:07,370 --> 00:26:15,210
The second way that is to make an analytical
reasoning for the event, for which
290
00:26:15,210 --> 00:26:16,470
probability is to be assigned.
291
00:26:16,470 --> 00:26:20,510
So, this analytical reasoning again is one
of the most
292
00:26:20,510 --> 00:26:25,850
important things to understand with respect
to the different already established concept,
293
00:26:25,850 --> 00:26:32,020
that is, I generally follow some norms and
to develop some particular, say, particular
294
00:26:32,020 --> 00:26:39,970
material in civil engineering, or say I have
some particular belief in me.
295
00:26:39,970 --> 00:26:44,560
So, in this way if
I know that the way the definition is given
296
00:26:44,560 --> 00:26:50,710
for a particular attribute I should say, and
if I
297
00:26:50,710 --> 00:26:57,590
know what was the background to give that
particular event then I can assess that
298
00:26:57,590 --> 00:27:01,870
probability related to the that particular
event.
299
00:27:01,870 --> 00:27:02,870
Here,
300
00:27:02,870 --> 00:27:06,390
.one example is given.
301
00:27:06,390 --> 00:27:12,630
If you already know the basics of the civil
engineering, then I hope
302
00:27:12,630 --> 00:27:19,610
you know that M20 grade of concrete; if you
do not know – this is a particular grade,
303
00:27:19,610 --> 00:27:27,570
different grades of the concretes are available,
say M15, M 20, M25M35 and M40.
304
00:27:27,570 --> 00:27:31,260
So,
different grades of the concrete arethere.
305
00:27:31,260 --> 00:27:36,530
By definition of these particular grades of
concrete, it says that this is a characteristic
306
00:27:36,530 --> 00:27:44,140
strength and it is defined as – if a concrete
cube and that concrete cube dimension is 15
307
00:27:44,140 --> 00:27:48,980
centimeter by 15 centimeter by 15
centimeter, and if we say that this is made
308
00:27:48,980 --> 00:27:57,200
of M20 grade of concrete and this cube is
subject to a pressure of 20 mega Pascal (Newton
309
00:27:57,200 --> 00:28:03,400
per millimeter square), then the
probability of the event that a cube will
310
00:28:03,400 --> 00:28:05,010
fail is 0.05.
311
00:28:05,010 --> 00:28:12,190
Now where from we get this 0.05 is
that is lying in the definition of the characteristics
312
00:28:12,190 --> 00:28:13,190
strength.
313
00:28:13,190 --> 00:28:17,650
So, this characteristic strength, the definition
of characteristics strength says that what
314
00:28:17,650 --> 00:28:23,580
is
the stress, what is the strength at which
315
00:28:23,580 --> 00:28:27,090
I can assure that at least the failure of
that
316
00:28:27,090 --> 00:28:36,370
particular concrete cube will not exceed 5
percent, so that is why, so if I take a M20
317
00:28:36,370 --> 00:28:42,540
grade concrete and the pressure is 20 Pa,
so that the probability that the event the
318
00:28:42,540 --> 00:28:47,480
cube
will fail is point nought 5.
319
00:28:47,480 --> 00:28:50,980
So this is one analytical reasoning and example
of this
320
00:28:50,980 --> 00:28:54,490
analytical reasoning of assigning probability
to this particular event.
321
00:28:54,490 --> 00:28:55,490
.
322
00:28:55,490 --> 00:29:06,580
Then the third one is that a kind of deductive
approach.
323
00:29:06,580 --> 00:29:11,450
Assume that the probability
follows certain axioms, will come to this
324
00:29:11,450 --> 00:29:15,141
one the axioms of probability, and then by
325
00:29:15,141 --> 00:29:21,650
.deductive approach determine the probability
of an event using the probability of other
326
00:29:21,650 --> 00:29:22,650
events.
327
00:29:22,650 --> 00:29:35,500
So, there are different kinds of events and
somehow, say, you assume for the time being,
328
00:29:35,500 --> 00:29:39,810
that somehow I know the probability of a particular
event, now using this information,
329
00:29:39,810 --> 00:29:44,060
that I know the probability of this particular
event is this, then the questions come that
330
00:29:44,060 --> 00:29:49,970
what is the probability of some other particular
event, then I should be able to assess, I
331
00:29:49,970 --> 00:29:54,950
should be able to compute the probability
of this particular event using the information
332
00:29:54,950 --> 00:30:01,100
of the probability of the available information
of the probability or some other events.
333
00:30:01,100 --> 00:30:06,450
So, this is a deductive approach.
334
00:30:06,450 --> 00:30:10,250
Suppose that here, because this will be more
clear when
335
00:30:10,250 --> 00:30:17,180
we cover this axioms of probability, but before
I cover this one this is a very easy
336
00:30:17,180 --> 00:30:21,260
example that you should be able to understand.
337
00:30:21,260 --> 00:30:26,930
The probability that a testing device,
there are different types of testing device
338
00:30:26,930 --> 00:30:30,110
available in civil engineering, suppose that
the
339
00:30:30,110 --> 00:30:38,220
testing device a company has given it to the
market, and based on the customer
340
00:30:38,220 --> 00:30:44,360
satisfaction, that customer will categorize
that particular testing device as very poor,
341
00:30:44,360 --> 00:30:47,190
poor, average, satisfactory, and excellent.
342
00:30:47,190 --> 00:30:57,560
So based on the market study or some other
way the company knows that a particular testing
343
00:30:57,560 --> 00:31:04,420
device to be categorized as very poor,
the probability of this to be categorized
344
00:31:04,420 --> 00:31:08,090
as very poor is 0.05, to be categorized as
poor4 is
345
00:31:08,090 --> 00:31:15,610
0.15, average 0.68, satisfactory 0.1, and
excellent point naught two, if that question
346
00:31:15,610 --> 00:31:18,340
is
that, what is the probability of the same
347
00:31:18,340 --> 00:31:23,570
device that it will be rated as above average;
so
348
00:31:23,570 --> 00:31:34,830
if I say it is above average, that means it
should be either satisfactory or excellent.
349
00:31:34,830 --> 00:31:41,390
Now, there are some assumption background
to this that a particular customer cannot
350
00:31:41,390 --> 00:31:44,460
categorize it as both satisfactory and excellent.
351
00:31:44,460 --> 00:31:49,560
So, he is having only one choice, so
these two events of categorizing as satisfactory
352
00:31:49,560 --> 00:31:51,360
or excellent is mutually exclusive.
353
00:31:51,360 --> 00:31:56,670
Again
the definition of mutually exclusive– it
354
00:31:56,670 --> 00:32:05,730
says that the occurrence of a particular event
ensures that non occurrence of the other event.
355
00:32:05,730 --> 00:32:08,830
So, for example, the tossing of a coin, if
I
356
00:32:08,830 --> 00:32:15,330
say that the outcome is head, then it automatically
implies that the tail has not occured,
357
00:32:15,330 --> 00:32:22,160
so in other words, the head and tail, both
the outcomes cannot happen simultaneously;
358
00:32:22,160 --> 00:32:30,830
this kind of events are known as mutually
exclusive, and again if I say that this two
359
00:32:30,830 --> 00:32:31,960
things are independent.
360
00:32:31,960 --> 00:32:37,610
So, if one customer has categorized it as
satisfactory, and other one hascategorized
361
00:32:37,610 --> 00:32:39,510
it as
excellent, then these categorization by different
362
00:32:39,510 --> 00:32:50,470
customers is independent to each other,
then if I want to know what is the probability
363
00:32:50,470 --> 00:32:53,050
that the particular device is rated as above
364
00:32:53,050 --> 00:32:58,800
.average, this will be simply the summation
of the probability, for satisfactory, and
365
00:32:58,800 --> 00:33:05,221
excellent, that is point one plus point zero
two so here, in this way of assigning the
366
00:33:05,221 --> 00:33:15,380
probability of a particular event, here the
event is above average, this is not based
367
00:33:15,380 --> 00:33:16,980
on any
particular experiment, this is rather based
368
00:33:16,980 --> 00:33:22,310
on the information that is available on the
probability of some other event.
369
00:33:22,310 --> 00:33:34,309
So, this is a deductive approach, that is
why this is deductive approach, and it is
370
00:33:34,309 --> 00:33:41,440
determined by the probabilities of the other
events for which the probability is known.
371
00:33:41,440 --> 00:33:42,440
.
372
00:33:42,440 --> 00:33:48,640
So, these are the three different ways that
we can assign probability to a particular
373
00:33:48,640 --> 00:33:50,840
event,
so just in the previous slide we discuss about
374
00:33:50,840 --> 00:33:51,860
the axioms of probability.
375
00:33:51,860 --> 00:33:55,870
So, different type
of concept of this probability can also have
376
00:33:55,870 --> 00:33:56,890
the different way.
377
00:33:56,890 --> 00:34:02,090
Just now, we have seen
that what is the relative frequency, the concept
378
00:34:02,090 --> 00:34:06,880
based on the relative frequency, there are
other concepts also, for example, the axiomatic
379
00:34:06,880 --> 00:34:07,880
definition.
380
00:34:07,880 --> 00:34:13,329
So, there are few axioms that
is there for a particular probability event,
381
00:34:13,329 --> 00:34:17,270
there are basically 3 axioms, probability
of A,
382
00:34:17,270 --> 00:34:22,790
P(A) is an non negative number, so this P(A)
is always greater than equivalent to 0, this
383
00:34:22,790 --> 00:34:24,450
is the first axiom.
384
00:34:24,450 --> 00:34:28,250
Second is that the probability of all events
in the set S is unity, so the
385
00:34:28,250 --> 00:34:34,850
probability of all possible outcomes of a
particular event is equals to one, and the
386
00:34:34,850 --> 00:34:40,310
third
one is the probability of event A union B
387
00:34:40,310 --> 00:34:47,490
is the addition of the probability of A and
probability of B that is probability A union
388
00:34:47,490 --> 00:34:50,600
B equals to probability A plus probability
B
389
00:34:50,600 --> 00:34:59,810
.here, this symbol that is A union B is the
union of two events - A and B , defined as
390
00:34:59,810 --> 00:35:04,980
an
outcome when A or B or both has occurred simultaneously.
391
00:35:04,980 --> 00:35:05,980
.
392
00:35:05,980 --> 00:35:16,490
In the concept of the Venn diagram you know
that if this is your probability A, and if
393
00:35:16,490 --> 00:35:21,420
this
is your probability B, so the first circle
394
00:35:21,420 --> 00:35:27,420
I am showing as a black shaded.
395
00:35:27,420 --> 00:35:34,500
If red shaded
area, if this your B, then A union B is the
396
00:35:34,500 --> 00:35:37,810
summation of both.
397
00:35:37,810 --> 00:35:52,670
So, this total area is your A
union B. So these are two different events,
398
00:35:52,670 --> 00:35:57,700
and so if they are unions of the probability
of
399
00:35:57,700 --> 00:36:03,370
this total event is nothing but probability
of A plus B, but there is a condition here,
400
00:36:03,370 --> 00:36:06,579
that
this two events are mutually exclusive, why?
401
00:36:06,579 --> 00:36:10,420
because here if you again see this one.
402
00:36:10,420 --> 00:36:16,410
That
if I say that this area is your probability
403
00:36:16,410 --> 00:36:18,600
A and this area is your probability B, then
if I
404
00:36:18,600 --> 00:36:21,560
just simply add this one then this area obviously
which is common in A and B will be
405
00:36:21,560 --> 00:36:28,320
added twice; that is why, when it is mutually
exclusive then I can say suppose, that this
406
00:36:28,320 --> 00:36:31,119
is
your A, and this is your B, then probability
407
00:36:31,119 --> 00:36:35,710
A union B is nothing but probability of A
plus probability of B.
408
00:36:35,710 --> 00:36:42,630
Now, based on this 3 axioms that we have discussed
with different numbers, we can
409
00:36:42,630 --> 00:36:49,000
deduce different numbers of equations where
we can use it for the detection of the
410
00:36:49,000 --> 00:36:53,950
probability of the others; for example, in
case of this, when two events are overlapping
411
00:36:53,950 --> 00:36:59,770
each other then what should be the probability
of A union B, that you can easily say, that
412
00:36:59,770 --> 00:37:07,089
probability of A plus probability of B. In
this process we have taken this area twice,
413
00:37:07,089 --> 00:37:09,270
so
this should be minus, this area should be
414
00:37:09,270 --> 00:37:13,750
given as minus, and that is probability of
A,
415
00:37:13,750 --> 00:37:21,770
.another new way - it is the intersection
of B, so probability this symbol, this probability
416
00:37:21,770 --> 00:37:29,910
A intersection B, is nothing but this particular
area, where it says that this is common in
417
00:37:29,910 --> 00:37:31,570
both the events.
418
00:37:31,570 --> 00:37:39,460
So in this process, we have added this area
twice, so this should be
419
00:37:39,460 --> 00:37:40,460
deducted here.
420
00:37:40,460 --> 00:37:44,420
So, this is second one, these are the 3 axiomatic
definition, where we can
421
00:37:44,420 --> 00:37:52,670
follow the detective approach, we can get
the probability of the other events if the
422
00:37:52,670 --> 00:37:59,960
probability of the particular events, of some
other events, is known to us.
423
00:37:59,960 --> 00:38:00,960
.
424
00:38:00,960 --> 00:38:07,300
The third one is the classical definition,
the classical definition the probability of
425
00:38:07,300 --> 00:38:10,220
an
event probability A is determined without
426
00:38:10,220 --> 00:38:13,990
the actual random experiment as a ratio as
follows –
427
00:38:13,990 --> 00:38:19,710
probability A equals to probability nA by
N, where nA is the favorable
428
00:38:19,710 --> 00:38:29,660
outcome related to the events, sorry; this
there is one t, to the event A and N is the
429
00:38:29,660 --> 00:38:33,690
total
possible outcomes.
430
00:38:33,690 --> 00:38:37,420
That means, we are not exactly following any
particular random
431
00:38:37,420 --> 00:38:42,690
experiment; whatever the observation is available
to us based on this whatever the ratio
432
00:38:42,690 --> 00:38:52,420
of this success out of the total number of
trials, we are assigning this particular number
433
00:38:52,420 --> 00:38:55,050
which is the classical definition of this
probability.
434
00:38:55,050 --> 00:38:59,200
It is implicitly assumed in this event
that all possible outcomes of a particular
435
00:38:59,200 --> 00:39:01,700
experiments are equally likely.
436
00:39:01,700 --> 00:39:09,780
Coming to the throwing of a dice example,
all possible outcome, that is any integer
437
00:39:09,780 --> 00:39:16,920
number from 1 to 6. all this outcomes should
be equally likely, so that probability of
438
00:39:16,920 --> 00:39:23,270
getting 1 should be equals to probability
of getting 2 equals to all this numbers should
439
00:39:23,270 --> 00:39:26,170
be
equals to 1 by 6.
440
00:39:26,170 --> 00:39:30,800
So, from our day to experience, we generally
assign this probability,
441
00:39:30,800 --> 00:39:38,500
.and these when we say that yes, all these
probabilities are equal and equal to 1 by
442
00:39:38,500 --> 00:39:41,450
6, then
all this events are equally likely.
443
00:39:41,450 --> 00:39:46,350
Basically this is a point where this basic
concept of
444
00:39:46,350 --> 00:39:48,640
probability is questioned.
445
00:39:48,640 --> 00:39:49,640
.
446
00:39:49,640 --> 00:39:56,560
And sometimes this kind of critical view is
thrown to this basic concept of probability.
447
00:39:56,560 --> 00:40:03,790
frst, the term equally likely actually means
that equally probable or fair choice, which
448
00:40:03,790 --> 00:40:09,220
is
not always feasible in the practical cases.
449
00:40:09,220 --> 00:40:14,110
It is not possible for that for the practical
cases;
450
00:40:14,110 --> 00:40:23,040
basically having a fair dice or fair, coin
is impossible even though you go for a very
451
00:40:23,040 --> 00:40:27,990
large
number of trials, and how large is large,
452
00:40:27,990 --> 00:40:32,750
to get that particular probability of a head
say
453
00:40:32,750 --> 00:40:41,140
0.5 or for a dice for a particular number,
is 1 by 6 is that, whether they are really
454
00:40:41,140 --> 00:40:45,780
equally
likely that cannot be in ensured always.
455
00:40:45,780 --> 00:40:49,840
So, this is the first thing that generally
face the
456
00:40:49,840 --> 00:40:51,020
problem of this classical definition.
457
00:40:51,020 --> 00:40:56,210
Second thing is that the definition is applicable
to the limited practical problems, since
458
00:40:56,210 --> 00:41:06,970
the equal probability of the choice is hard
to achieve, so even though there are some
459
00:41:06,970 --> 00:41:14,780
experiments that by choice or by the experience,
we say that all this possible outcome are
460
00:41:14,780 --> 00:41:16,119
equally likely.
461
00:41:16,119 --> 00:41:20,070
So for those kind of experiments this definition
can be applied, but what
462
00:41:20,070 --> 00:41:25,970
about the many other examples where this kind
of equally likely events cannot be
463
00:41:25,970 --> 00:41:26,970
assured.
464
00:41:26,970 --> 00:41:31,470
So in such cases the applicability of this
classical definition generally throws the
465
00:41:31,470 --> 00:41:32,820
question here.
466
00:41:32,820 --> 00:41:41,260
.And the third thing is that the number of
possible outcome is infinity, now for a
467
00:41:41,260 --> 00:41:47,290
particular experiment, if I say the number
of possible outcome is infinity, then some
468
00:41:47,290 --> 00:41:49,490
kind
of measure to the infinity should be assigned
469
00:41:49,490 --> 00:41:54,109
to get the ratio of the N.
.
470
00:41:54,109 --> 00:42:03,650
Say for example, if that example is given
like this that in a circle there is another
471
00:42:03,650 --> 00:42:07,830
circle
inside this and one random event is that there
472
00:42:07,830 --> 00:42:14,230
is a chord here, what is the probability?
473
00:42:14,230 --> 00:42:20,710
That some part of this chord will be inside,
this inner circle as well.
474
00:42:20,710 --> 00:42:27,570
So, there are two
chords, I can just draw now one is that this
475
00:42:27,570 --> 00:42:32,670
is passing completely outside, this inner
circle,another is passing inside this one.
476
00:42:32,670 --> 00:42:38,320
Now, if I want to assess the probability of
this
477
00:42:38,320 --> 00:42:43,131
kind of experiment, then some kind of measure
of what is the total possible outcome
478
00:42:43,131 --> 00:42:50,160
should be there, so in this experiment I can
draw any line here and say that these many
479
00:42:50,160 --> 00:42:51,570
chords are possible.
480
00:42:51,570 --> 00:42:57,770
So, I have to have some kind of some kind
of assessment of this area - this area if
481
00:42:57,770 --> 00:43:01,250
it goes
this is outside, and if it goes in this area
482
00:43:01,250 --> 00:43:02,250
this is outside.
483
00:43:02,250 --> 00:43:06,660
So, this can be achieved in
different ways, the first thing is that if
484
00:43:06,660 --> 00:43:15,370
I know the origin of this circle; one way
is that we
485
00:43:15,370 --> 00:43:22,100
can just start from any point, and say that
this is one and there is another chord like
486
00:43:22,100 --> 00:43:23,100
this.
487
00:43:23,100 --> 00:43:28,730
And I should say that this is the angle, which
is my favorable case so all chords in these
488
00:43:28,730 --> 00:43:33,680
area will pass through this circle and all
others are this one, so that this area divided
489
00:43:33,680 --> 00:43:35,490
by
the total area can give the probability.
490
00:43:35,490 --> 00:43:43,890
.Second way of looking at the same problem
is that there is one outer circle, and there
491
00:43:43,890 --> 00:43:46,850
is
another inner circle here.
492
00:43:46,850 --> 00:43:54,190
And I can join two particular chord here like
this, and say that
493
00:43:54,190 --> 00:44:00,190
all possible chords that is in this area are
my favorable case, and all other which are
494
00:44:00,190 --> 00:44:03,930
falling outside that is the non favorable
case.
495
00:44:03,930 --> 00:44:14,119
So, this area divided by total area will give
this will give the probability.
496
00:44:14,119 --> 00:44:16,730
This is a classical example, it is known that
These two
497
00:44:16,730 --> 00:44:23,230
probability some time may not may not match
to each other.So this is also one particular
498
00:44:23,230 --> 00:44:35,240
drawback of this classical theory, and when
we are going to give that favorable ,to non
499
00:44:35,240 --> 00:44:43,670
favorable ratio when we are computing the
ratio of this favorable case and non favorable
500
00:44:43,670 --> 00:44:44,670
case.
501
00:44:44,670 --> 00:44:48,470
So, that definition of how this experiment
is performed whether with respect to the
502
00:44:48,470 --> 00:44:50,830
experiment, this particular thing is more
feasible or this; one is more feasible is
503
00:44:50,830 --> 00:44:53,420
to be
justified first, and then the probability
504
00:44:53,420 --> 00:44:54,420
should be calculated.
505
00:44:54,420 --> 00:44:59,700
And this is particularly in
case when the possible outcome is infinite,
506
00:44:59,700 --> 00:45:02,850
that kind of problem may arise.
507
00:45:02,850 --> 00:45:03,850
.
508
00:45:03,850 --> 00:45:09,960
So, validity of the classification definition,
in application the assumption of having
509
00:45:09,960 --> 00:45:12,630
equally likely outcomes can be established
through a long experiment,; say for example,
510
00:45:12,630 --> 00:45:23,570
that if I take an example of occurrence of
cyclonic storm as random in the time interval
511
00:45:23,570 --> 00:45:34,680
of 0 to t, then the probability that it may
occur in the interval t1 to t2 is equals to
512
00:45:34,680 --> 00:45:41,470
t2 minus
t1 divided by T; so 0 to t, some time interval
513
00:45:41,470 --> 00:45:47,030
I am taking, and I am saying that the
occurrence of this cyclonic storm is random,
514
00:45:47,030 --> 00:45:52,200
and I am implicitly assuming that this
515
00:45:52,200 --> 00:45:56,619
.occurrence is equally likely over any time
in this time interval.
516
00:45:56,619 --> 00:46:00,300
Then, what is the
probability that it will…, obviously this
517
00:46:00,300 --> 00:46:07,290
t1 is greater than 0, and t 2 is less than
T, then in
518
00:46:07,290 --> 00:46:16,350
this area, of course that means that t1 and
t2 are lying in the range of 0 to T, then
519
00:46:16,350 --> 00:46:18,140
the
probability of this cyclonic storm occurring
520
00:46:18,140 --> 00:46:23,030
in this time interval is equals to basically
this
521
00:46:23,030 --> 00:46:27,460
is t2 minus t1 divided by total possible t,
capital T minus 0.
522
00:46:27,460 --> 00:46:32,490
So, in this kind of analysis
when we are assigning this probability we
523
00:46:32,490 --> 00:46:38,000
are implicitly assuming that the outcomes
are
524
00:46:38,000 --> 00:46:39,580
equally likely.
525
00:46:39,580 --> 00:46:40,580
.
526
00:46:40,580 --> 00:46:49,620
If it is impossible to repeat an experiment
sufficiently large number of times, assumption
527
00:46:49,620 --> 00:46:55,750
is made that the available alternatives are
equally likely, this is basically the start
528
00:46:55,750 --> 00:46:58,630
point of
the probability theory.
529
00:46:58,630 --> 00:47:05,150
When this kind of experiments cannot be run
in a large number it
530
00:47:05,150 --> 00:47:12,670
is implicitly, assumed that all possible outcomes
are equally likely.
531
00:47:12,670 --> 00:47:13,670
..
532
00:47:13,670 --> 00:47:21,930
Now, the determination of probability; the
assigning the probabilities P i of a certain
533
00:47:21,930 --> 00:47:29,010
event in an inexact way, second this we have
discussed, just whatever the record that is
534
00:47:29,010 --> 00:47:35,700
available with us, and in that what is the
number of success, and what is the total number
535
00:47:35,700 --> 00:47:40,859
that is available, that ratio gives you that
probability for the particular event.
536
00:47:40,859 --> 00:47:44,930
Second, theory of probability follows certain
axioms.
537
00:47:44,930 --> 00:47:50,400
So, just now we have seen what are
the axioms that probability follows, so probability
538
00:47:50,400 --> 00:47:54,900
of a particular event can be
determined by the deductive approach, and
539
00:47:54,900 --> 00:48:04,530
third one, what we have seen that make a
physical guess of the event based on the probability
540
00:48:04,530 --> 00:48:09,780
numbers calculated P(Ai) and P(Bj).
541
00:48:09,780 --> 00:48:10,780
..
542
00:48:10,780 --> 00:48:15,330
This probability and occurrence, I think this
is needed at the starting of this course to
543
00:48:15,330 --> 00:48:23,390
know that what is this probability, and what
is the actual occurrence in the real field.
544
00:48:23,390 --> 00:48:31,470
Suppose that I say a particular experiment,
say the the rainfall event say that the rainfall
545
00:48:31,470 --> 00:48:36,710
event occurs more than what is the capacity
of my drainage network.
546
00:48:36,710 --> 00:48:42,750
So, this is a
particular event, that rainfall amount greater
547
00:48:42,750 --> 00:48:49,610
than the threshold value which our drainage
system can take care, and I say that this
548
00:48:49,610 --> 00:48:52,830
probability is, say 0.0405; now the thing
is that,
549
00:48:52,830 --> 00:48:55,740
if I say that this probability is 0.0 5,.
550
00:48:55,740 --> 00:49:01,380
now the question is that in the next event
of the
551
00:49:01,380 --> 00:49:08,700
rainfall what is the chance, what is the occurrence
that that particular rainfall will really
552
00:49:08,700 --> 00:49:09,700
exceed.
553
00:49:09,700 --> 00:49:15,990
So, what does this 0.05, what does this number
actually give you to infer?
554
00:49:15,990 --> 00:49:19,480
This is the
topic of this particular slide., Sso the if
555
00:49:19,480 --> 00:49:23,750
P(A) is known the P(A) means that probability
of
556
00:49:23,750 --> 00:49:29,609
particular event is known the answer to the
question of itis occurrence in the next trial
557
00:49:29,609 --> 00:49:32,520
can be based on the size P(A).
558
00:49:32,520 --> 00:49:34,000
So, now this size ofP(A) matters here, if
the P( A) is low
559
00:49:34,000 --> 00:49:38,480
say for example, 0.3 it is given, then it
concludes that onlycertain degree of confidence
560
00:49:38,480 --> 00:49:47,660
that the event A will occur, it does not mean
that exactly you perform ten trials, and
561
00:49:47,660 --> 00:49:56,820
exactly 3 trials will be your success that
is not the case, so in a long terms that is
562
00:49:56,820 --> 00:50:02,859
with
certain degree of confidence, so this is very
563
00:50:02,859 --> 00:50:12,480
important the certain degree of confidence
that is the thing that we can infer from here.
564
00:50:12,480 --> 00:50:16,930
.So, it a measure of belief, this generally
cannot be experimentally tested, so this is
565
00:50:16,930 --> 00:50:19,109
subject thing of the measure of this belief
that with this degree of certain confidence
566
00:50:19,109 --> 00:50:20,380
that
this particular event A will occur.
567
00:50:20,380 --> 00:50:25,560
Second thing is that if the probability is
very close to one; that means say for example,
568
00:50:25,560 --> 00:50:31,260
0.999; it concludes that with the practical
certainty that A will occur in the next trial.
569
00:50:31,260 --> 00:50:35,869
So
if very high probability, then we generally
570
00:50:35,869 --> 00:50:42,660
say that with practical certainty, this will
definitely occur when the probability is very
571
00:50:42,660 --> 00:50:45,930
high, so the probability, when it is very
low,
572
00:50:45,930 --> 00:50:52,941
it is very unlikely to occur, and when it
is very high, generally we say that it is
573
00:50:52,941 --> 00:50:56,580
very
likely to occur, but we can never say that
574
00:50:56,580 --> 00:51:04,359
exactly if you consider, some n numbers of
trial, and it will happen exactly in the probability
575
00:51:04,359 --> 00:51:10,150
number multiplied by this total number
of trials, it may not occur in that exact
576
00:51:10,150 --> 00:51:12,960
number . So, the probability and occurrence
of a
577
00:51:12,960 --> 00:51:17,540
particular event should be understood, it
should be assessed in this particular way.
578
00:51:17,540 --> 00:51:18,540
.
579
00:51:18,540 --> 00:51:24,940
And the last thing, but this is also important
the randomness and causation, so this
580
00:51:24,940 --> 00:51:29,910
randomness links with the probabilistic system,
and the causation links with the
581
00:51:29,910 --> 00:51:30,910
deterministic system.
582
00:51:30,910 --> 00:51:36,010
The randomness is stated with certain errors
and certain range of
583
00:51:36,010 --> 00:51:45,230
this relevant parameters, and causation is
stated with a high degree of certainty if
584
00:51:45,230 --> 00:51:52,100
the
number of outcomes is large enough, but again
585
00:51:52,100 --> 00:52:00,060
basically there is no conflict between the
randomness and causation, because when we
586
00:52:00,060 --> 00:52:06,950
say that there are some laws of nature,
since the theories are not the laws of nature,
587
00:52:06,950 --> 00:52:09,050
both statements are true.
588
00:52:09,050 --> 00:52:12,220
.So, even though I say that a particular system
is deterministic, just for our simplicity
589
00:52:12,220 --> 00:52:14,980
sake
we say that this is a deterministic system,
590
00:52:14,980 --> 00:52:20,630
but if we will really see whether that particular
system is exactly deterministic or not, a
591
00:52:20,630 --> 00:52:23,750
little bit of randomness may be there as well.
592
00:52:23,750 --> 00:52:34,330
So, basically that is why we say that the
basically the conflict between the randomness,
593
00:52:34,330 --> 00:52:42,910
and causation are not there; these are just
basically some concept, and if it is clear
594
00:52:42,910 --> 00:52:45,619
then it
is generally very useful to that.
595
00:52:45,619 --> 00:52:46,619
.
596
00:52:46,619 --> 00:52:56,630
The final concluding remarks is that in this
lecture we have seen that random events are
597
00:52:56,630 --> 00:53:02,730
possible outcomes of the random experiment,
and probability is a measure of the
598
00:53:02,730 --> 00:53:06,450
uncertainty in the occurrence of them.
599
00:53:06,450 --> 00:53:11,359
Random events consists of either single point
or
600
00:53:11,359 --> 00:53:19,300
multiple points on the same sample space;
the relationships among the random events
601
00:53:19,300 --> 00:53:20,990
are
governed by the set theory, and the event
602
00:53:20,990 --> 00:53:22,810
properties; and this set theory and this event
properties will be explained in the next lecture
603
00:53:22,810 --> 00:53:23,810
of this course.
604
00:53:23,810 --> 00:53:24,810
Thank you very much.
605
00:53:24,810 --> 00:53:24,810
.