WEBVTT
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We have already seen
an example in which we
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have two events
that are independent
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but become dependent
in a conditional model.
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So that [independence] and
conditional independence
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is not the same.
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We will now see another example
in which a similar situation is
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obtained.
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The example is as follows.
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We have two possible
coins, coin A and coin B.
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This is the model
of the world given
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that coin A has been chosen.
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So this is a
conditional model given
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that we have in
our hands coin A.
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In this conditional model, the
probability of heads is 0.9.
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And, moreover, the
probability of heads
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is 0.9 in the second
toss no matter
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what happened in the first
toss and so on as we continue.
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So given a particular
coin, we assume
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that we have independent tosses.
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This is another way
of saying that we're
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assuming conditional
independence.
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Within this conditional model,
coin flips are independent.
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And the same
assumption is made in
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the other possible
conditional universe.
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This is a universe in which
we're dealing with coin B.
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Once more, we
have, conditionally
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independent tosses.
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And this time, the probability
of heads at each toss is 0.1.
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Suppose now that we choose
one of the two coins.
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Each coin is chosen with
the same probability, 0.5.
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So we're equally likely
to obtain this coin--
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and then start flipping
it over and over--
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or that coin-- and start
flipping it over and over.
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The question we
will try to answer
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is whether the coin
tosses are independent.
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And by this, we
mean a question that
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refers to the overall model.
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In this general model, are
the different coin tosses
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independent?
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Where you do not know ahead of
time which coin is going to be.
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We can approach this
question by trying
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to compare conditional and
unconditional probabilities.
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That's what
independence is about.
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Independence is about certain
conditional probabilities
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being the same as the
unconditional probabilities.
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So this here, this
comparison here
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is essentially the question
of whether the 11th coin
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toss is dependent or
independent from what
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happened in the
first 10 coin tosses.
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Let us calculate
these probabilities.
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For this one, we use the
total probability theorem.
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There's a certain probability
that we have coin A,
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and then we have the probability
of heads in the 11th toss given
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that it was coin A. There's
also a certain probablility
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that it's coin B and then
a conditional probability
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that we obtain heads
given that it was coin B.
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We use the numbers that
are given in this example.
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We have 0.5 probability of
obtaining a particular coin,
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0.9 probability of heads
for coin A, 0.5 probability
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that it's coin B, and
0.1 probability of heads
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if it is indeed coin B.
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We do the arithmetic,
and we find
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that the answer is 0.5,
which makes perfect sense.
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We have coins with
different biases,
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but the average bias is 0.5.
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If we do not know which
coin it's going to be,
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the average bias
is going to be 0.5.
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So the probability of heads
in any particular toss
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is 0.5 if we do not
know which coin it is.
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Suppose now that
someone told you
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that the first 10
tosses were heads.
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Will this affect your
beliefs about what's
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going to happen
in the 11th toss?
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We can calculate
this quantity using
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the definition of
conditional probabilities,
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or the Bayes' rule, but let
us instead think intuitively.
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If it is coin B, the
events of 10 heads in a row
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is extremely unlikely.
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So if I see 10
heads in a row, then
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I should conclude that there
is almost certainty that I'm
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dealing with coin A.
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So the information
that I'm given
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tells me that I'm extremely
likely to be dealing with coin
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A. So we might as well condition
on this equivalent information
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that it is coin A
that I'm dealing with.
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But if it is coin A, then
the probability of heads
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is going to be equal to 0.9.
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So the conditional
probability is
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quite different from the
unconditional probability.
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And therefore, information
on the first 10 tosses
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affects my beliefs
about what's going
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to happen in the [11th] toss.
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And therefore, we do
not have independence
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between the different tosses.