WEBVTT
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Conditional probabilities are
probabilities associated with
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a revised model that takes into
account some additional
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information about the outcome of
a probabilistic experiment.
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The question is how
to carry out this
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revision of our model.
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We will give a mathematical
definition of conditional
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probabilities, but first let us
motivate this definition by
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examining a simple
concrete example.
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Consider a probability model
with 12 equally likely
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possible outcomes, and so each
one of them has probability
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equal to 1/12.
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We will focus on two particular
events, event A and
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B, two subsets of the
sample space.
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Event A has five elements, so
its probability is 5/12, and
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event B has six elements, so
it has probability 6/12.
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Suppose now that someone tells
you that event B has occurred,
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but tells you nothing more
about the outcome.
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How should the model change?
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First, those outcomes that
are outside event B
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are no longer possible.
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So we can either eliminate
them, as was done in this
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picture, or we might keep them
in the picture but assign them
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0 probability, so that
they cannot occur.
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How about the outcomes
inside the event B?
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So we're told that one of
these has occurred.
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Now these 6 outcomes inside
the event B were equally
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likely in the original model,
and there is no reason to
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change their relative
probabilities.
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So they should remain equally
likely in revised model as
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well, so each one of them should
have now probability
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1/6 since there's 6 of them.
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And this is our revised
model, the
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conditional probability law.
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0 probability to outcomes
outside B, and probability 1/6
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to each one of the outcomes that
is inside the event B.
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Let us write now this
down mathematically.
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We will use this notation to
describe the conditional
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probability of an event A given
that some other event B
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is known to have occurred.
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We read this expression as
probability of A given B. So
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what are these conditional
probabilities in our example?
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So in the new model, where these
outcomes are equally
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likely, we know that event
A can occur in
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two different ways.
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Each one of them has
probability 1/6.
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So the probability of
event A is 2/6 which
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is the same as 1/3.
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How about event B. Well, B
consists of 6 possible
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outcomes each with
probability 1/6.
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So event B in this revised model
should have probability
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equal to 1.
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Of course, this is just
saying the obvious.
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Given that we already know
that B has occurred, the
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probability that B occurs
in this new model
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should be equal to 1.
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How about now, if the sample
space does not consist of
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equally likely outcomes, but
instead we're given the
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probabilities of different
pieces of the sample space, as
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in this example.
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Notice here that the
probabilities are consistent
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with what was used in the
original example.
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So this part of A that lies
outside B has probability
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3/12, but in this case I'm
not telling you how that
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probability is made up.
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I'm not telling you that
it consists of 3
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equally likely outcomes.
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So all I'm telling you is that
the collective probability in
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this region is 3/12.
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The total probability of A is,
again, 5/12 as before.
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The total probability of
B is 2 plus 4 equals
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6/12, exactly as before.
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So it's a sort of similar
situation as before.
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How should we revise our
probabilities and create--
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construct--
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conditional probabilities
once we are told
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that event B has occurred?
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First, this relation
should remain true.
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Once we are told that B has
occurred, then B is certain to
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occur, so it should
have conditional
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probability equal to 1.
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How about the conditional
probability of A given that B
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has occurred?
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Well, we can reason
as follows.
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In the original model, and if
we just look inside event B,
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those outcomes that make event
A happen had a collective
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probability which was 1/3 of the
total probability assigned
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to B. So out of the overall
probability assigned to B, 1/3
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of that probability corresponds
to outcomes in
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which event A is happening.
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So therefore, if I tell you that
B has occurred, I should
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assign probability equal
to 1/3 that event A is
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also going to happen.
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So that, given that B happened,
the conditional
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probability of A given B
should be equal to 1/3.
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By now, we should be satisfied
that this approach is a
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reasonable way of constructing
conditional probabilities.
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But now let us translate our
reasoning into a formula.
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So we wish to come up with a
formula that gives us the
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conditional probability of an
event given another event.
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The particular formula that
captures our way of thinking,
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as motivated before,
is the following.
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Out of the total probability
assigned to B--
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which is this--
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we ask the question, which
fraction of that probability
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is assigned to outcomes under
which event A also happens?
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So we are living inside event
B, but within that event, we
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look at those outcomes for which
event A also happens.
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So this is the intersection of
A and B. And we ask, out of
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the total probability of B,
what fraction of that
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probability is allocated to that
intersection of A with B?
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So this formula, this
definition, captures our
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intuition of what we did
before to construct
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conditional probabilities in
our particular example.
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Let us check that the definition
indeed does what
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it's supposed to do.
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In this example, the
probability of the
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intersection was 2/12 and the
total probability of B was
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6/12, which gives us 1/3, which
is the answer that we
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had gotten intuitively
a little earlier.
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At this point, let me also
make a comment that this
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definition of conditional
probabilities makes sense only
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if we do not attempt
to divide by zero.
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That this, only if the event B
on which we're conditioning,
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has positive probability.
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If B, if an event B has 0
probability, then conditional
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probabilities given B will
be left undefined.
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And one final comment.
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This is a definition.
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It's not a theorem.
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What does that mean?
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It means that there is no
question whether this equality
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is correct or not.
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It's just a definition.
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There's no issue
of correctness.
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The earlier argument that we
gave was just a motivation of
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the definition.
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We tried to figure out what the
definition should be if we
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want to have a certain intuitive
and meaningful
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interpretation of the
conditional probabilities.
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Let us now continue with
a simple example.