The law of total probability is another probability law
that gives you a way to reason about cases, which we've
seen is a fundamental technique for dealing
with all sorts of problems.
So the point of cases, of course,
is that you can prove a complicated thing
by breaking it up into, if you're lucky, easy sub cases.
So here's the way to understand the law of total probability
abstractly.
It starts off with set theoretic reasonings.
Suppose that I have a set A embedded in some larger sample
space S. So A is really an event,
but we're just going to think of it as a set.
Now suppose that I have three sets,
B1, B2, and B3 that partition the sample space.
That is B1, B2, and B3 three don't overlap.
They're disjoined, and everything
is in one of those three sets.
So there's a picture of B1, B2, and B3, cutting up
the whole sample space S, represented
by the square or rectangle.
Now of course, these three sets that
cut up the whole space, willy nilly
cut up the set A into three pieces.
The first piece is the points in A that are in B1.
The second piece is the point in A that are in B2.
And the third is the points in A that are in B3.
So that's why we a basic set theoretic
identity that says that as long as B1, B2, and B3
have the property, that they're union is in the universe.
Everything.
And they are pairwise disjoined, then
any set A is equal to the union of the part of A
that's in B1, the part of A that's in B2,
the part of A that's in B3.
And this is a disjoint union, because the B's don't overlap.
That means that if I was talking about cardinality,
I could add them up.
But in terms of probability, I can apply the sum rule
for probabilities and discover that the probability of A
is simply the probability of B1 intersection A, B2 intersection
A, B3 intersection A.
Now the most useful form of the law of total probabilities
is when you replace this intersection, B1 intersection
A, by the conditional probability using the product
rule-- so let's replace it by the probability of A given
B1 times the probability of B1.
That's another formula for B1 intersection A.
And if I do that with the rest of them,
I now have the law of total probability stated
in the usual way in terms of conditional probabilities
where it's most useful.
Now I did it for three sets.
But it obviously works for any finite number of sets.
As a matter of fact, it works fine for any countable union
of sets.
If I have a partition of the sample space S
in to B0, B1, and so on-- a partition
with a countable number of blocks-- then
it's still the case that the probability of A
is equal by the sum rule to the probability
of these disjoint pieces, the parts of A
that are in each of the different blocks
of the partition.
And reformulating that as a conditional probability,
I get the rule that the probability of A
is the sum over all possible i of the probability of A given
Bi times the probability of Bi.
And that basic rule is one and we're
going to get a lot of mileage out of when
we turn in the next segment to analyze and understand
the results of tests that are maybe unreliable.