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Suppose I have a fair coin which
I toss multiple times.

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I want to model a situation
where the results of previous

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flips do not affect my beliefs
about the likelihood of heads

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in the next flip.

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And I would like to describe
this situation by saying that

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the coin tosses are
independent.

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You may say, we already
defined the notion of

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independent events.

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Doesn't this notion apply?

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00:00:29,210 --> 00:00:30,980
Well not quite.

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We defined independence
of two events.

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But here, we want to talk
about independence of a

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collection of events.

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For example, we would like to
say that the events, heads in

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the first toss, heads in the
second toss, heads in the

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third toss, and so on,
are all independent.

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What is the right definition?

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Let us start with intuition.

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We will say that a family of
events are independent if

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knowledge about some of the
events doesn't change my

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beliefs, my probability model,
for the remaining events.

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For example, if I want to say
that events A1, A2 and so on

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are independent, I would like
relations such as the

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following to be true.

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The probability that event A3
happened and A4 does not

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happen remains the same even
if I condition on some

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information about some
other events.

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Let's say if I tell you that
A1 happens or that both A2

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happened and A5 did
not happen.

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The important thing to notice
here is that the indices

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involved in the event of
interest are distinct from the

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indices associated with the
events on which I'm given some

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information.

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I'm given some information about
the events A1, A2, and

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A5, what happened to them.

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And this information does not
affect my beliefs about

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something that has to do
with events A3 and A4.

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I would like all relations
of this kind to be true.

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This could be one possible
definition, just saying that

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the family of events are
independent if and only if any

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relation of this type is true.

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But such a definition would not
be aesthetically pleasing.

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Instead, we introduce the
following definition, which

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mimics or parallels our
earlier definition of

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independence of two events.

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We will say that a collection
of events are independent if

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you can calculate probabilities
of intersections

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of these events by multiplying
individual probabilities.

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And this should be possible
for all choices of indices

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involved and for any number
or events involved.

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Let us translate this into
something concrete.

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Consider the case of three
events, A1, A2, and A3.

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Our definition requires that
we can calculate the

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probability of the intersection
of two events by

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multiplying individual
probabilities.

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And we would like all of these
three relations to be true,

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because this property should
be true for any

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choice of the indices.

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What do we have here?

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This relation tells us that
A1 and A2 are independent.

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This relation tells us that
A1 and A3 are independent.

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This relation tells us that
A2 and A3 are independent.

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We call this situation pairwise
independence.

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But the definition requires
something more.

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It requires that the probability
of three-way

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intersections can also be
calculated the same way by

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multiplying individual
probabilities.

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And this additional condition
does make a difference, as

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we're going to see in
a later example.

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Is this the right definition?

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Yes.

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One can prove formally that
if the conditions in this

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definition are satisfied,
then any formula of

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this kind is true.

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In particular, we also have

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relations such as the following.

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The probability of event A3 is
the same as the probability of

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event A3, given that
A1 and A2 occurred.

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Or the probability of
A3, given that A1

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occurred but A2 didn't.

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Or we can continue this
similarly, the probability of

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A3 given that A1 did
not occur, and A2

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occurred, and so on.

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So any kind of information that
I might give you about

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events A1 and A2--

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which one of them occurred
and which one didn't--

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is not going to affect my
beliefs about the event A3.

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The conditional probabilities
are going to be the same as

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the unconditional
probabilities.

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I told you that this definition
implies that all

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relations of this kind [are]

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true.

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This can be proved.

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The proof is a bit tedious.

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And we will not go through it.