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Hello, welcome to another module in this massive
open online course on probability and random
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variables for wireless communications.
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So today, let us start looking at a new topic,
that is conditional probability.
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So today we will start our discussion on conditional
probability.
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So Conditional Probability, Consider 2 events,
A, B. Now, the
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conditional probability of A given B is defined
as the probability of, is denoted by the probability
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of A given B which is the probability of A
intersection B divided by the probability
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of B. So we are now defining a new concept
that is the conditional probability.
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Let us say, A and B are 2 events.
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The conditional probability of A given B,
that is the, what is the, how is the probability
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of A or how is the probability of occurrence
of A affected given that the event B has occurred.
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That is let us say, someone has told you that
the event B has occurred.
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Then how does that affect the probability
of occurrence of A?
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This is termed as the conditional probability.
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That is denoted by the probability of A conditioned
on the event B which is equal to the probability
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of A intersection B divided by the probability
of B.
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And therefore, naturally, this also implies,
that the probability of A given B times the
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probability of B equals the probability of
A intersection B. Or in other words, the probability
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of A intersection B is equal to the probability
of A given B times the probability of B. And
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also, therefore we can write this as…
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So, we said we have probability of A intersection
B equals probability of A given B times the
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probability of B. Similarly, we can write,
we can define the probability of B conditioned
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on A as the probability of, B
intersection A divided by the probability
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of A which basically implies that the probability
of B given A times probability of probability
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of A equals probability of B intersection
A. And now you can see, probability of A intersection
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B is probability of A given B times probability
of B. Probability of B given A, the probability
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of B intersection A which is same as the probability
of A intersection B is equal to the probability
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of B given A times the probability of A.
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Therefore from these 2, we can conclude that
the probability of A given B times the probability
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of B equals the probability of B given A times
the probability of A and both are equal to
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the probability of A intersection B which
basically implies that probability of A given
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B is equal to the probability of B intersection
A times the probability of A divided by the
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probability of B. So the probability of A
given B is equal to the probability of B given
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A times the probability of A divided by the
probability of B.
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All right?
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So we have derived an important property of
conditional probability.
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This is also known as the base Bayes’ result.
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We will explore more about this later.
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So 1st, we have said that the probability
of A conditioned on B is equal to the probability
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of A intersection B divided by the probability
of B. Similarly, the probability of B conditioned
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on A equals the probability of B intersection
A divided by the probability of A. Let us
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now look at an application of this concept
of conditional probability in the context
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of a Mary Pam M-ary PAM modulation.
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Remember, or M-ary PAM stands for pulse amplitude
modulation.
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So let us look at a simple application of
conditional probability.
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So let us look at a simple example of conditional
probability in the context of M-ary PAM.
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Remember, we are considering M-ary PAM for
M equal to 4 symbols.
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And remember, PAM denotes a pulse amplitude,
pulse amplitude, population modulation, which
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is also basically an amplitude shift keying.
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And remember, for M equal to 4 symbols, our
sample space, S consists of the symbols, minus
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3 alpha, minus alpha, alpha, 3 alpha.
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We are considering 4 PAM.
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That is M-ary PAM, pulse amplitude modulation
which is a digital modulation.
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That is this is a digital constellation from
which the digital transmission symbols are
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drawn..
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Further, we are considering, M is equal to
4 symbols.
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That is, we are considering, M equal to 4
symbols and the sample space for this is minus
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3 alpha, minus alpha, alpha, 3 alpha.
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And we also consider an example in which the
various probabilities are given as 1 by 8,
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1 by 4 and half.
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And let us take A as before, let us take A
to denote the set of symbols minus 3 alpha,
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alpha and B to denote the symbol alpha.
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Now what we are asking
is what is the conditional probability of
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B given A. So we are saying, A common descent
, that’s it, A denotes the PAM symbols,
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minus 3 alpha and alpha, B denotes the PAM
symbol alpha.
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Now we are asking, what is the conditional
probability of B given A?
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That is, given that event A has occurred,
that is the observed symbol belongs to the
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set, minus 3 alpha, alpha which means to say
that is the PAM symbol is either minus 3 alpha
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or alpha.
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What is the probability of event B conditioned
on a given A?
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What is the probability of B?
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That is what is the probability that the PAM
symbol is alpha?
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Remember, B is the PAM symbol alpha.
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Right?
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So we are asking, what is the probability
that the observed symbol is alpha given the
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observed symbol is either minus 3 alpha or
alpha.
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And this can be found as follows.
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We have that is what is probability of B given
A?
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Probability of B given A equals probability
this is equal to, the
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probability
symbol equals alpha given A that is symbol
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equals minus 3 alpha or alpha.
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And what is the probability of B given A?
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Remember, from the definition, this is equal
to the probability of B intersection A divided
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by the probability of A. And probability of
A equals remember, probability of minus 3
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alpha, alpha which is equal to, the probability
of minus 3 alpha is 1 by 8 plus the probability
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of alpha equals 1 by 4.
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This is simply equal to 3 by 8.
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Now what is B intersection A?
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B intersection A is basically minus 3 alpha,
alpha intersection alpha which is B and this
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is basically the set alpha.
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Therefore, we have probability B intersection
A equals simply the probability of alpha which
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is equal to 1 by , 4.
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So we have found 2 things.
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We have found probability of A which is 3
by 8 and we have also found the probability
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of B intersection A which we are saying is
basically 1 by 4.
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And therefore, the probability of B conditioned
on A , is simply the probability of B intersection
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A divided by the probability of A. This is
equal to 1 divided by 4 divided by 3 divided
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by 8 which is equal to 1 divided by 4 into
8 divided by 3 which is equal to 2 by 3.
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Therefore we have derived probability of B
conditioned on A equals 2 divided by 2 over
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3.
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Therefore the conditional probability of B
given A or the probability of the event B
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conditioned on the event A is two third that
is 2 over 3.
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Now let us find the other conditional probability.
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What is the conditional probability of A given
B?
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So now let us find, what is, so let us ask
the question, what is probability of A , conditioned
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on B?
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That is, remember, A equals minus 3 alpha,
alpha, B equals symbol alpha.
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That is, we are asking, probability
that the symbol equals minus 3 alpha or alpha
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given B. That is given symbol equals alpha
and therefore what is the probability of A
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conditioned on B?
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We are saying that the A is the set minus
3 alpha, alpha, B is the set alpha.
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Therefore the probability of A conditioned
on B is the probability, we are asking the
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question, what is the probability that the
observed symbol is minus 3 alpha or alpha
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when it is already given that the symbol observed
is B that is alpha.
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And clearly, this probability should be 1.
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Because given the symbol is alpha, this symbol
is alpha, therefore the symbol is definitely
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either minus 3 alpha or alpha.
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Therefore, the probability of A conditioned
on B should be equal to 1.
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But let us check that,
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We have probability of A conditioned on B
equals probability of A intersection B divided
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by probability of B.
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We have already derived probability of A intersection
B equals 1 by 4.
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Observe probability of B equals probability
of basically the set alpha which is equal
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to 1 by 4 and indeed probability of A given
B, equals probability of A intersection B
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divided by the probability of B equals 1 by
4 divided by 1 by 4 which is equal to 1.
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Therefore, probability of A given B is equal
to 1.
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And also observe something very interesting.
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Probability of A given B is not equal to the
probability of B given A.
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Remember, we derived the probability of B
given A. Probability of A given B is 1.
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So probability of A given B is not equal to
the probability of B given A. So probability
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of A given B is 1 which is not equal to the
probability of B given A which we basically
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derived, as which, if you see previously we
basically derived the probability of B given
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A has 2/3rd.
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So this is the concept, what we have seen
in this module is, we have seen the concept
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of conditional probability.
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That is, given 2 events, A, B, what is the
conditional probability of A given B?
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Or in other words, how does observing event
B affect the probability of the occurrence
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of event A. That is the conditional probability
of A given B.
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Similarly we have also seen the conditional
probability of B given A and we have seen
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a simple example to clearly illustrate this
concept of conditional probability.
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And we will look at the other aspects in the
subsequent modules..
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Thank you very much.