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The law of total probability
is another probability law

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that gives you a way to reason
about cases, which we've

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seen is a fundamental
technique for dealing

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with all sorts of problems.

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So the point of
cases, of course,

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is that you can prove
a complicated thing

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by breaking it up into, if
you're lucky, easy sub cases.

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So here's the way to understand
the law of total probability

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abstractly.

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It starts off with set
theoretic reasonings.

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Suppose that I have a set A
embedded in some larger sample

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space S. So A is
really an event,

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but we're just going to
think of it as a set.

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Now suppose that
I have three sets,

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B1, B2, and B3 that
partition the sample space.

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That is B1, B2, and B3
three don't overlap.

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They're disjoined,
and everything

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is in one of those three sets.

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So there's a picture of
B1, B2, and B3, cutting up

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the whole sample
space S, represented

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by the square or rectangle.

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Now of course, these
three sets that

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cut up the whole
space, willy nilly

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cut up the set A
into three pieces.

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The first piece is the
points in A that are in B1.

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The second piece is the
point in A that are in B2.

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And the third is the
points in A that are in B3.

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So that's why we a
basic set theoretic

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identity that says that
as long as B1, B2, and B3

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have the property, that they're
union is in the universe.

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Everything.

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And they are pairwise
disjoined, then

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any set A is equal to the
union of the part of A

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that's in B1, the part
of A that's in B2,

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the part of A that's in B3.

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And this is a disjoint union,
because the B's don't overlap.

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That means that if I was
talking about cardinality,

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I could add them up.

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But in terms of probability,
I can apply the sum rule

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for probabilities and discover
that the probability of A

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is simply the probability of B1
intersection A, B2 intersection

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A, B3 intersection A.

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Now the most useful form of
the law of total probabilities

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is when you replace this
intersection, B1 intersection

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A, by the conditional
probability using the product

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rule-- so let's replace it
by the probability of A given

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B1 times the probability of B1.

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That's another formula
for B1 intersection A.

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And if I do that with
the rest of them,

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I now have the law of
total probability stated

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in the usual way in terms
of conditional probabilities

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where it's most useful.

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Now I did it for three sets.

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But it obviously works for
any finite number of sets.

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As a matter of fact, it works
fine for any countable union

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of sets.

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If I have a partition
of the sample space S

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in to B0, B1, and
so on-- a partition

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with a countable
number of blocks-- then

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it's still the case that
the probability of A

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is equal by the sum
rule to the probability

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of these disjoint
pieces, the parts of A

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that are in each of
the different blocks

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of the partition.

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And reformulating that as
a conditional probability,

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I get the rule that
the probability of A

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is the sum over all possible i
of the probability of A given

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Bi times the probability of Bi.

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And that basic rule
is one and we're

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going to get a lot of
mileage out of when

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we turn in the next segment
to analyze and understand

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the results of tests that
are maybe unreliable.