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We will now consider an example
that illustrates the

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difference between the notion
of independence of a

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collection of events and
the notion of pairwise

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independence within
that collection.

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The model is simple.

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We have a fair coin which
we flip twice.

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So at each flip, there
is probability 1/2

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of obtaining heads.

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Furthermore, we assume that the
two flips are independent

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of each other.

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Let H1 be the event that the
first coin toss resulted in

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heads, which corresponds to this
event in this diagram.

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Let H2 be the event that the
second toss resulted in heads,

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which is this event
in the diagram--

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the two ways that we can have
the second toss being heads.

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Now, we're assuming that the
tosses are independent.

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So the event heads-heads has a
probability which is equal to

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the probability that the first
toss resulted in heads--

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that's 1/2--

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times the probability that the
second toss resulted in heads,

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which is 1/2.

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So the product is 1/4.

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We have probability 1/4
for this outcome.

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Now, the total probability of
event H1 is 1/2, which means

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that the probability of what
remains should be 1/4, so that

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the sum of these two
numbers is 1/2.

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By the same argument, the
probability of this outcome,

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tails-heads , should be 1/4.

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We have a total of 3/4.

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So what's left is 1/4.

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And that's going to be the
probability of the outcome

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tails-tails .

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Let us now introduce a new
event, namely the event that

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the two tosses had
the same result.

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So this is the event that we
obtain either heads heads or

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tails-tails.

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Schematically, event C
corresponds to this blue

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region in the diagram.

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Is this event C independent
from the events H1 and H2?

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Let us first look for pairwise
independence.

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Let's look at the probability
that H1 occurs

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and C occurs as well.

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So the first toss resulted
in heads.

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And the two tosses had
the same result.

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So this is the same as the
probability of obtaining heads

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followed by heads.

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And this corresponds to just
this outcome that has

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probability 1/4.

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How about the product of the
probabilities of H1 and of C?

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Is it the same?

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Well, the probability
of H1 is 1/2.

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And the probability of C--

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what is it?

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Event C consists of
two outcomes.

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Each one of these outcomes
has probability 1/4.

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So the total is, again, 1/2.

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And therefore, the product of
these probabilities is 1/4.

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So we notice that the
probability of the two events

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happening is the same as the
product of their individual

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probabilities, and therefore,
H1 and C

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are independent events.

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By the same argument, H2 and C
are going to be independent.

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It's a symmetrical situation.

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H1 and H2 are also independent
from each other.

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So we have all of the conditions
for pairwise

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independence.

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Let us now check whether
we have independence.

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To check for independence, we
need to also look into the

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probability of all three events
happening and see

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whether it is equal to the
product of the individual

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probabilities.

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So the probability of all
three events happening--

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this is the probability
that H1 occurs and H2

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occurs and C occurs.

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What is this event?

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Heads in the first toss, heads
in the second toss, and the

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two tosses are the same--

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this happens if and only
if the outcome is

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heads followed by heads.

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And this has probability 1/4.

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On the other hand, if we
calculate the probability of

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H1 times the probability of H2
times the probability of C, we

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get 1/2 times 1/2 times
1/2, which is 1/8.

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These two numbers
are different.

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And therefore, one of the
conditions that we had for

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independence is violated.

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So in this example, H1,
H2, and C are pairwise

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independent, but they're not
independent in the sense of an

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independent collection
of events.

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How are we to understand
this intuitively?

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If I tell you that event H1
occurred and I ask you for the

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conditional probability
of C given that H1

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[occurred], what is this?

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This is the probability that
we obtain, given that the

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first event is heads, the first
result is heads, the

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only way that you can have the
two tosses having the same

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result is going to be in
the second toss also

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resulting in heads.

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And since H2 and H1 are
independent, this is just the

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probability that we have heads
in the second toss.

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And this number is 1/2.

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And 1/2 is also the same as the
probability of C. That's

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another way of understanding the
independence of H1 and C.

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Given that the first toss
resulted in heads, this does

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not help you in any way in
guessing whether the two

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tosses will have the
same result or not.

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The first one was heads, but the
second one could be either

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heads or tails with
equal probability.

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So event H1 does not carry any
useful information about the

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occurrence or non-occurrence of
event C. On the other hand,

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if I were to tell you that
both events, H1 and H2,

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happened, what would
the conditional

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probability of C be?

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If both H1 and H2 occurred, then
the results of the two

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coin tosses were identical, so
you know that C also occurred.

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So this probability
is equal to 1.

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And this number, 1, is
different from the

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unconditional probability
of C, which is 1/2.

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So we have here a situation
where knowledge of H1 having

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occurred does not help you in
making a better guess on

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whether C is going to occur.

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H1 by itself does not carry
any useful information.

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But the two events together,
H1 and H2, do carry useful

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information about C.

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Once you know that H1 and
H2 occurred, then C

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is certain to occur.

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So your original probability for
C, which was 1/2, now gets

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revised to a value of 1.

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So H1 and H2 carry information
relevant to C. And therefore,

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C is not independent from these
two events collectively.

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And we say that events H1.

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H2, and C are not independent.