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I now want to emphasize
an important point.

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Conditional probabilities are
just the same as ordinary

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probabilities applied to
a different situation.

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They do not taste or smell or
behave any differently than

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ordinary probabilities.

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What do I mean by that?

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I mean that they satisfy the
usual probability axioms.

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For example, ordinary
probabilities must also be

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non-negative.

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Is this true for conditional
probabilities?

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Of course it is true, because
conditional probabilities are

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defined as a ratio of
two probabilities.

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Probabilities are
non-negative.

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So the ratio will also be
non-negative, of course as

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long as it is well-defined.

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And here we need to remember
that we only talk about

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conditional probabilities when
we condition on an event that

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itself has positive
probability.

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How about another axiom?

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What is the probability of
the entire sample space,

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given the event B?

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Let's check it out.

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By definition, the conditional
probability is the probability

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of the intersection of the two
events involved divided by the

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probability of the conditioning
event.

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Now, what is the intersection
of omega with B?

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B is a subset of omega.

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So when we intersect
the two sets, we're

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left just with B itself.

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So the numerator becomes the
probability of B. We're

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dividing by the probability
of B, and so the

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answer is equal to 1.

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So indeed, the sample space
has unit probability, even

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under the conditional model.

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Now, remember that when we
condition on an event B, we

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could still work with the
original sample space.

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However, possible outcomes that
do not belong to B are

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considered impossible, so we
might as well think of B

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itself as being our
sample space.

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If we proceed like that and
think now of B as being our

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new sample space, what is the
probability of this new sample

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space in the conditional
model?

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Let's apply the definition
once more.

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It's the probability of the
intersection of the two events

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involved, B intersection B,
divided by the probability of

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the conditioning event.

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What is the numerator?

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The intersection of B with
itself is just B, so the

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numerator is the probability
of B. We're dividing by the

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probability of B. So the
answer is, again, 1.

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Finally, we need to check
the additivity axiom.

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Recall what the additivity
axiom says.

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If we have two events, two
subsets of the sample space

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that are disjoint, then the
probability of their union is

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equal to the sum of their
individual probabilities.

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Is this going to be the case
if we now condition on a

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certain event?

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What we want to prove is the
following statement.

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If we take two events that are
disjoint, they have empty

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intersection, then the
probability of the union is

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the sum of their individual
probabilities, but where now

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the probabilities that we're
employing are the conditional

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probabilities, given the event
B. So let us verify whether

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this relation, this fact
is correct or not.

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Let us take this quantity
and use the

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definition to write it out.

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By definition, this conditional
probability is the

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probability of the intersection
of the first

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event of interest, the one that
appears on this side of

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the conditioning, intersection
with the event on which we are

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conditioning.

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And then we divide by the
probability of the

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conditioning event, B. Now,
let's look at this quantity,

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what is it?

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We're taking the union of A with
C, and then intersect it

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with B. This union consists
of these two pieces.

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When we intersect with
B, what is left is

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these two pieces here.

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So A union C intersected with B
is the union of two pieces.

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One piece is A intersection
B, this piece here.

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And another piece, which is C
intersection B, this is the

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second piece here.

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So here we basically used a
set theoretic identity.

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And now we divide by the
same [denominator]

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as before.

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And now let us continue.

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Here's an interesting
observation.

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The events A and
C are disjoint.

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The piece of A that also belongs
in B, therefore, is

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disjoint from the piece of
C that also belongs to B.

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Therefore, this set here and
that set here are disjoint.

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Since they are disjoint, the
probability of their union has

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to be equal to the sum
of their individual

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probabilities.

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So here we're using the
additivity axiom on the

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original probabilities to break
this probability up into

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two pieces.

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And now we observe that here
we have the ratio of an

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intersection by the probability
of B. This is just

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the conditional probability of A
given B using the definition

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of conditional probabilities.

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And the second part is the
conditional probability of C

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given B, where, again, we're
using the definition of

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conditional probabilities.

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So we have indeed checked that
this additivity property is

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true for the case of conditional
probabilities when

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we consider two disjoint
events.

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Now, we could repeat the same
derivation and verify that it

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is also true for the case of a
disjoint union, of finitely

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many events, or even
for countably

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many disjoint events.

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So we do have finite and
countable additivity.

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We're not proving it, but the
argument is exactly the same

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as for the case of two events.

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So conditional probabilities do
satisfy all of the standard

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axioms of probability theory.

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So conditional probabilities
are just like ordinary

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probabilities.

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This actually has a very
important implication.

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Since conditional probabilities
satisfy all of

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the probability axioms, any
formula or theorem that we

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ever derive for ordinary
probabilities will remain true

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for conditional probabilities
as well.