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PROFESSOR: The law
of total expectation

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will give us another
important tool for reasoning

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about expectations.

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And it's basically a rule like
the law of total probability,

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closely related to it
really, for reasoning

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by cases about expectation.

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So it requires a
definition of what's

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called conditional expectation.

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So the expectation of a random
variable R, given event A,

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is simply what you
get by thinking

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of replacing the
probability that R equals v

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by the probability that
R equals v given A.

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So it's the sum over
all the possible values

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that R might take
of the probability

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that R takes that
value, given A.

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OK, with that definition,
we can state the basic form

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of the law of total
expectation, which

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says if you want to calculate
the expectation of R,

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you can split it into
cases, according to whether

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or not A occurs.

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It's simply the
conditional expectation

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of R given A times
the probability of A,

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plus the conditional expectation
of R, given not A times

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the probability of
not A. So it really

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looks [? as ?] the
same format as the law

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of total probability.

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Now, of course it
generalizes to many cases.

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So the general form
would say that I

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can calculate the
expectation of R

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by breaking it up into
the case that A 1 holds

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times the probability of A
1, the case that A 2 holds

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times the probability
of A 2, through A n.

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And this could very
well, and typically is,

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an infinite sum, where
the [? A i's ?] of course,

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are a partition of
the sample space--

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so they're all the different
cases, either A 1 or A 2

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or A 3, they're disjoint.

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And altogether, they cover the
entire set of possibilities.

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Well, let's use this to get
a nice different and simpler

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way-- more elementary way-- of
calculating the expected number

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of heads and flips.

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So let's let of n be the
expected number of heads

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and flips-- just shorthand,
because the notational

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will be easier to
work with than writing

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capital E brackets of H n.

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So what do we know
about expectation of n?

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Well, I can express it in
terms of the expectation

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of the remaining flips.

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So if I have n flips to
perform, they're independent.

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Then if I perform the first
flip, something happens.

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And after that I'm
going to do n more

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flips, and the expected
number of flips

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is going to be the expected
number on the remaining n

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minus 1 plus what happened now.

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Well, if I flipped a head
first, then I've got a 1

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as adding to my total
number of heads.

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And then I'm going
to do n more flips,

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so the expected
number of flips is

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going to be that 1
plus the expected

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number on the rest of them.

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If the first flip was not
a head, it was a tail,

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then the total expected
number of heads

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is simply the expected number of
heads on the rest of the flips.

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And these are two cases where
I can apply total expectation.

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So by total expectation, the
expected number in n flips

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is 1 plus e n minus 1 times
the probability of a head,

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plus e n minus 1 times
the probability of a tail.

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Well, now we could do a little
algebra multiply through here

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by p-- that becomes a p,
and this becomes a p times

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e n and minus 1.

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So I've got e n minus 1 times
p, and e n minus 1 times

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q-- remembering that p plus q
is 1, this simplifies to being

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simply e n minus 1 plus p.

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Well, this is a very simple kind
of recursive definition of e n,

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because you can see
what's going to happen.

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Subtracting 1 from n adds a p.

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So if I subtract 2 from n,
I add another p-- I get 2 p.

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And continuing all
the way to the end,

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by the time I get to 0,
I've gotten n times p.

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And I've just figured out what
I was familiar with already--

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which we previously
derived by differentiating

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the binomial theorem-- the
expected number of heads

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in n flips is n times p.

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But this time I got it in a
somewhat more elementary way,

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by appealing to
total expectation.